1、;,.NATIONALADVISORYCOMMITTEEFOR AERONAUTICSTECHNICAL NOTE 2392CHARTS GIVING CRITICAL C 0MPRESSIS7E STRESS OFc ONTINUOUSFIJ3T SHEET DMDED INTO PARALLELOGRAM-SHAPED PANELSBy Roger A. AndersonLangley Aeronautical LaboratoryLangley Field,Va.WashingtonJuly1951I. . . . . . . . . . . . . .- . . . . . . . ,
2、. - -.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-1TECH LIBRARY K/U%, NMIullllmllllllllulllllln011L5i24NATIONAL ADVISORY COMMITTEE FOR AERONAUTICSTECHNICAL NOTE 2392CHARTS GIVING CRITICAL COMPRESSIVE STRESS OF CONTINUOUSl?LATSBEETDIVIDIZDINTOP AR
3、AILEL,(XHMM-SHAPEDPANELSBy Roger A. Anderson,Charts giving the compressive-buckling-stresscoefficients forsheet panels of a shape occurring in swept-wingplan forms are presented.The panels are assumed to be a part of a continuous flat sheet dividedby nondeflecting supports into parallelogram-shapeda
4、reas. The stabilityanalysis was perfomed by the energy methcd and the results show that,over a wide range of panel aspect ratio, such panels are decidedly morestable than equivalent rectangular panels of the same area.INTRODUCTIONA highly desimble characteristicfor high-speed flight is to haveaircra
5、ft outer surfaces that r-b free of buckles or waves under allnormal flight conditions. Whether these surfaces remain smooth underflight loads is determined by the skin thickness and the arrangementand rigidity of the internal supporting structure to which the skin isattached. In the past, the suppor
6、ting structure generally divided theouter skin into an array of approximately rectangular panels, butpresent-day swept- and delta-wing plan forms call attention to the factthat skin panels of other shapes such as parallelograms.may occur.The present paper considers the stability under compressive st
7、ressof continuous fkt sheet divided by nondeflecting supports into an arrayof parallelogram-shapedpanels. Wide ranges of panel skewness sml aspectratio Wre investigated,and two orientations of the parallelogramshapedpanels with respect to the direction of the applied stress were con-sidered. The res
8、ults of the analysis are presented in the form of chartsof theoreticalbuckling-stress coefficientsas a function of paiel skewness and aspect ratio.-. .“.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-2 IVACATN 2392xYX1Y .9abb!$twDva%aYk,kx,*vTSYMemS
9、mutually perpendicular directions, rectangularcoordinate systemdirection parallel to x-direction, skew coordinate systemarbitrary direction, skew coordinate systemskew angle between y- and yt-directionsjmeasuredpositive clockwise from y-direction, degreespanel )aytb2%=xinternal bendingexternal work
10、ofenergyapplied stressProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACA TN 2392%mubmn Fourier coefficientsDESCRIPTION OF PROELEMAn idealized arrangement of the supporting structure possible forswept and delta wings is shown in figure 1, wherein a
11、part of a con-tinuous flat sheet infinite in length and width is shown divided by non-deflecting supports into an array of parallelogram-shapedpanels, ofwhich rectangularpanels are a special case. For the case of rectangu-lar panels the assumption is usually made that, if the supporting mem-bers are
12、 torsionallyweak, the plate bending moments arising duringplate buckling under edge compression stress are negligible at the panelboundaries and each panel therefore my be treated as an isolated rec-tangularplate with simply supported edges. %abilityd ata for suchpanels are available in standard tex
13、tbooks, such as reference 1.Wheb a simply suorted rectangularplate in compression is skewedinto the shape of a parallelogram, a certain increase in stability dueto the shape change is achieved,as shown by the numerical examples inreference 2. A further increase in stability is possible when theskewe
14、d panels are part of a continuous sheet because of the restraintthe adjacent sheet panels impose on each other during the formation ofa continuousbuckle pattern. The effect of this type of restraint isillustrated in reference 3 for the case of continuous sheet divided intosquare panels subject to sh
15、ear stress. Restraint due to sheet con-tinuity would be present in the case of parallelogram-shapedpanelsregardless of whether the loading is edge compression or shear.Two orientations of the parallelogram-shapedp“anelswith respectto the principal direction of the compressive stress due to wing bend
16、ingare considered in the present paper. The two lodtng conditions are:stress acting parallel to a set of panel sides and stress acting perpen- .dicular to a set of panel sides. Both loading conditions are shuwn infigure 1. 1In the analysis the assumption is made that the supporting membersare rigid
17、enough to prevent deflection of the sheet at the panelboundaries but offer no restraint to rotation. A quantitative analysisof the stiffnesses required of supportingmenbers (ribs, stiffeners,shear webs, etc.) to prevent deflection around the edges of parallelogram-shaped sheet panels is beyond the s
18、cope of this paper, but indicationsare that they would be somewhathigher th= the stiffness required tosupport equivalent rectangular sheet panels. - - - - . - - .Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-k NACA TN 2392In order to establish a st
19、ability criterion for the configurationanalyzed, the energy method of analysis is used, and an approximation ismade for the deflection of the plate expressed in skew coordinates (seeappendix A). The deflection function used leads to the exact solutionof the differential equation of equilibrium of th
20、e plate loaded in com-pression for the special case of rectangular panels and also for thespecial case of equal-sidedpanels of arbit=ry skewness subjected tocompressive loading perpendicular to one set of sides. The accuracyof the rest of the data is indicated,where feasible,by comparisonwith the re
21、sults of more accurate ener solutions,which are given inappendix B.RESULTS AND DISCUSSIONThe critical compressive stress for parallelogram-shapedplates IUYbe given by the formula used for rectangularplateswhere the dimension b is the perpendicular distance between the sup-ports alined in the x-direc
22、tion, as in figure 1. me cfitical-stresscoefficient k depends on the panel aspect ratio P (definedas a/b),the direction of the applied compressive stress, and the magnitude ofthe skew angle .For loading in the x-direction (stressacting parallel to a set ofsides), the chart for kx is given in figure
23、2. Note shouldbe takenthat, for a given value of 13,the buckling coefficients indicated foreach curve are associated with panels of equal area. In ofier to facili-tate association of the curves of figure 2 with the geometry of thepanels, figure 3 has been prepared. In this figure, sketches of thepan
24、els of aspect ratios of 1, 2, and 3 are presented along with four ofthe curves of figure 2. In each sketch the male of buckling for thepanel is indicated. Figures 2 and 3 indicate that the stability of theskewed panels is definitely increased relative to equivalent rectangularpanels of the same area
25、 over a wide range of panel aspect ratio. Thisincrease in stability is due partly to the change in panel shape but iscaused mainly by the restraint imposed on the mckleof buckling due tothe presence of adjacent skewed panels. At aspect mtios of 4.5 andgreater, this restraint has largely disappeared
26、and the coefficientsapproach the value 4.The curves presented in figures 2 and 3 were derived by the energymethod of analysis with the use of an assumed deflection function capable “ - Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACA TN 2392 5of
27、giving an idealized representationof the panel Ixcling mcdes. Thebuckling coefficientsthus obtained represent upper limits to the truecoefficients. In order to evaluate the unconservativenessof.the curvespresented in figures 2 and 3, check calculationswere made by energysolutions of greater accuracy
28、. The results of these calculations,whichare believed to be less than 1 percent unconservative,are indicatedbythe circles near the curves in figures 2 and 3. Eecause of the com-plexity of the buckle pattens, check calculations over the entire rangeof aspect ratio were not feasible to make, but it se
29、ems reasonable thatthe remainingparts of the curves would show about the same degree ofunconservativenessas in the regionswhere check calculationswere made.The numerical values of the circles in figures 2 and 3 are listed intable 1.For loading in the y-direction (stressacting perpendicular to aset o
30、f sides), the equations (B2c)etist for n = 2, 4, 6, . . .; and equations (B2d) fid (B2e) do not efist.The buckling coefficient kx for loading in the x-direction was calcu-lated for the panel configuration = 60, p = 3 for which -thisbucklepattern applies, and a value kx = 6.95 =S obtained. S restit W
31、aSobtainedwhen four afs and four b*s were used in the deflectionseries (Blc).Antisymmetric buckling, periodic over 2a, b.- For antisymmetricbuckling, pericdic over 2a and bt, the infinite set of stabilityequations derived from the function (Bid)may be written directly fromequations (B2a) to (B=). Fo
32、r this buckle pattern, equations (B2a)exist for m = 1, 3, 5, . . . and n= 2, 4, 6, . . .; equations (B2b)exist for m = 2, 4, 6, . . . and n = 3, 5, 7, . . .; equations (B2c)do not etist; equations (B2d) etist for m = 2, 4, 6, . . .; and equa=tion (B2e)does not etist.Calculationswere carried out for
33、=.60, = 1.2, a panelconfigurationfor which this buckle pattern applies. An adequaterepresentationof the deflection was obtainedwhen four ats andfour bts were used in the deflection series (Bid)which gave a valueof kx = 16.22.As was pointed out in the beginning of this appendix, these moreaccurate so
34、lutionswere carried out only for those panels for which themale of buckling was relatively simple and clearly indicated by theresults of appendixA. In the absence of such information it would benecessary to investigateall conceivablemales of buckling for a givenpanel configuration. _.Provided by IHS
35、Not for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACA TN 2392 231. Timoshenko, S.: Theory of Elastic Stability. McGraw-Hill Book Co.,Inc., 1936, p. 382.2. Salvadori, lrio G.: Numerical Computation of Buckling Loads byFinite Diffennces. Proc. A.S.C.E., vol. , no. 10,
36、 Dec. 1949,p. 1471.3. BudiansQ, Bernmdy Con.nor,Robert W., and Stein, Manuel: “Bucklingin Shear of Continuous Flat Plates. NACA TN 1565, 1948. . . Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-24 NACA TN 2392TA13LE l.- CRITICALCOMBINATIONSOF ax AND
37、 Uy l!YP13SOF STR12SSACTINGON CONTIGUOUS FLAT SHEET DIVIDEDINTOPARMiIZCXXAM-SHAPEO”PANELS(DATAFOR CIRCLESIN FTGS.2 TO 5)kxfi% 1k#DUXt._; uyt=bp b2Stressconibination BucklepatternP k= ky symmetry(1) PeriWcig = 3000.5 0 19.33 s 2a !.577 9.60 0 s “2a t1.0 6.74 0 s 2a t1.155 6.62 0 s 2a t1.155 0 4 s 2a
38、at3.0 0 1.41 s a at* $ = 4500.5 24.12 0 s 2a =!.6 0 14.32 s 2a at1.0 0 6.499 s % =?U. 46 s b?:;14 o : s : 2bt1.414 5.99 2 s % at1.414 7.46 1 s bt2.0 6.36 0 A-S : bt3.0 5.11 0 s bt3.0 0 1.74 s : t$.6000.6 0 19.29 s 2a at1.0 0 9.25 s a t1.2 16.z o A-S a btL 6 12.32 0 s bt2.0 0 4 s ; at2.0 4.0 3.43 s 2
39、a at2.0 7.14 2 s 2a at2.0 7.97 1 s a bt2.0 8.75 0 s a bt3.0 6.95 0 A-S at3.0 0 2.56 s 2 at. l.s- symmetricalaboutpanelmidpoint A-S - antisymmetricalaboutpanelmidpoint.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-(I14rGeneral2.- DATA DETR%UJGIIudal
40、 function wg!=o” I $ = 1.501.01.62.553.0:12111111 To 00 00 00 00 00 0123111 = 45 I0.6 1 1 o 1.9 1 1 273 1 11.2 1 1 1 11.4 1 1 322 11.8 1 1 4j3 12.1 1 1 1 12.5 1 1 572 1 13.2 1 1 3 1 14.2 1 1 4 1 1111111o00111111111222-001741/31/41:21/23/54/74/7Provided by IHSNot for ResaleNo reproduction or networki
41、ng permitted without license from IHS-,-,-TABLE3.- DATA DEFINING THE BUCKLING MODES IN FIGURE 5kx p/N (l J/N d/M = o“, 30k 1 1 0 05 1 1 0 06 1 1 0 0$ = 454 1 1 0 05 1 1 0 06 1 1 0 01 1 1:.5 1 1 32 1gL6Q”4 1 1 0 01 1.2 1 1 32 3j21 12 2i!. 5 “1 1 3 3a71a15 .-Provided by IHSNot for ResaleNo reproductio
42、n or networking permitted without license from IHS-,-,-.,II1IIImm.9.illl11 l-lProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-24 20 16 k, 12 -8 -4 o Results of energy solutions of higher accuracy m03J t I I I I 1 I I 1 I I t I I I I I -o .5 Lo L5 2.0
43、 2.5 3.0 3.5BFigure 2.- Theoretical buckling stresse6 for parallelmgrem-shapedfor stress acting parallel to a set of sides.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-29NACA TN 23920 Results of energy.solutions of higherkxaccuracy,W,m8 .36 0. I I
44、 I Ikx 8 450 I I I Ikx16 8 )600 I I I Io .5 Lo L5 2.0 25 3.0 35BFigure 3.- Illustration of the variation in panel stabilitywith changesin panel geometry. (Idealizationof buckling mode shown in each panel.).- -. . .Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-
