1、,w!.No. 774.-.-TENSION FIELDS IN ORIGIITALLY CUHVED , THIN SHEETSDURING SHNARING STRESSESBy H. Wagner and W. Ballerstedt,rLuftfahrtforschungo. XII, ;O, 2, Mar 16, 1935Verlag voxt R. Olfienbourg, S.hncben uild Berlin.= ,.,=-. . . Provided by IHSNot for ResaleNo reproduction or networking permitted wi
2、thout license from IHS-,-,-. .-: - - .3 117601353FOR AEROI?AUT.ICSNo. 7741661TEiTSIOITI?12?LDSIN ORIGINALLY CURVED, THIN SH3ETS,DURING SHEARIHG ST!RESEUISBy H. Wagner and IT. BallerstedtSUMARYSo-called shell (or stressed skin) bodies or wingscoilsit of outer skin and stiffening sections, the latterr
3、lllllliilgin Tart longitudinally (longitudinal stiffeners)and partly transversely (transverse stiffeners, rings) .The curved slzin is usually so tl.in that it bucklesIoilg before the structure has reached the ultimate load.3uckling is followed fithe appearance of oblique wrinklesii:the sheet.The ana
4、lysis of the stresses iilthe sileet and stif-feners is predicated uporL the dj.rectioll Of tho ri.nkles,particularly the tensile stresses (pri-ncipal stresses) .Tis al?.alysisaxlclt.e calculation of the stresses afterbuckling form the su”bject of the present article. It iil-cl.udes:1. Metal cylinder
5、s with closely spaced longitudi-nal stiffenerso2. Metal cylinders mitl c.osbly spaced .transversoriilgs.As concerns the Ioilgitudiilal stiffeners, the reportforms , aside from minor modificatioils, a repetition ofthe theoretical part of an experimental report by H. Wag-iler at the Rohrbac3. .!etalAi
6、.rplaile Coinpany in .1927, andwliose publication has been permitted hy Dr. Rohrbach.-r-“-;-;-;-*l!U%er Zugfelder iilursyrunglich Eekrummten, dunnen Blechenbei Beanspruchung durch Sckuokr partic-ularly, elongation and axis X*E, modulus of elasticity.cCxprincipal elongation and elongation in the tens
7、ionfield in direction of x r.nd y.Cyy, shear strain.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-. .H.A. C.Ai Technical Memorandum No. 774 3. . . . . ,The premises of the calculation are that the dimen-sioning of sheet and stiffenes” and t-he stre
8、ss -in shearand tension are uniform within the entire range.,Ip a suitably designed flat sheet wall the bucklingload of the sheet versus shear is usually so low comparedwith the permissible stress in the tension field that theinfluence of the compressive tiffness of the sheet maybe disregai-ded, Now
9、 , in curved sheets the %uckling stressis usually substantially higher on the one hand., and onthe other, no sv.ch high stresses are permissible oncebuckling has taken place; otherwise, the wrinkling in thecurved sheet would induce permanent deflections. Thus theanalysis of the diagoilal tension fie
10、lds of curved sheetsmust allow for the buckling stiffness of the sfi.ect.The following considerations arc based upon the argu-ment that tlzc sheet, even after buckling, continuos totransmit the proportion of the total shcarins stress Tcorresponding to thc bv-ckling stress “i-shear o andonl. the prop
11、ortion T - To oxtendiilg hcyond the buckliigload, in tensioil. From this follows: th,e stiffeners which,admittedly, are not stressed prior to the buckling, under-go a stress or,ly to the amount of the excess T - To.Figure 1 illustrates a piece of the shell. From theequilibrium of the forces in secti
12、on AA in the y direction,and, section BB in the x direction, tileproportion T - Togives the forces (compression) in the stiffeners (equation(10), N.A.C. A. Techilical Hem-orandurn Ho. 604, reference 1):x=- (T - To) s tx cot” (la)Thus the teision in longitudinal aitd transverse stiffeilerstecomes:x=-
13、 T cot CL (2a)Y = - q T tan a (2%)where, for the sake of brevity:Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-IToA:c.oAe Technical MemorandumT-T s ty=TOy- - _.23y the same ar?;ument the stress ofNO* 774(3a)(31)the she-et is theresultant stress of
14、luckling stress in shear To and, aT- -rOtensile stress -.- . However, the tensile stresssin a cos 04decisive for the dimensioning o.* the “sheet is simply putatTCf= -sin a cos aand the principal elongation of theTc -.,sheet at(4)r. E sina,COS a(5)The minor inaccuracy introduced with this simplificat
15、ionis well within the degree of accuracy of the assumptionthat t?ae stress after buc”ling i shear, is To.DIRI!CYION OF WRINKLES IN SEX3TS WITHC?IJOS3LYSPACED L01TGIT?J31NALS,Z+ is presumed. in the following derivation that thespacing of tl.e transverse rings is greater than that ofthe longitudinal s
16、tiffeners - by at least twice as much.The spaciilg of the longitudinal stiffeners is presumed tobe small relative to the radius. 3or example,In order to compute the direction of the wrinkles inthe tension field, we first consider the elongations cxand Y of the sheet In x and y directions. The elon-g
17、ation cx is given through the force in the IongitudihalProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-IT.A.C.A. Technical Memorandum No. 774 5stiffener. ._ . . . . . . (7X . .x=- ;g cota (6)As concerns the determination of the elongation Cy of thesi
18、leet panel iil y direction, we refer to figure 2. Undershear the originally unstressed curved sheet is stretchedstraight while forming tension wrinkles. The small dis-tance -tx Cy, for which the straight conitection ofpoints Ot and Ui is shorter after loading than thecurved connection of points O an
19、d U before loading, isnow calculated. The shortening due to the straight stretch-ing alone is ( = tx/r),Besides, the two lon;itudinal stiffeliers O and U ap- .preach each other to the measure ofas a result of the compressive stress in ty.e cross% lilen-ers . The desired shortening of CU to ():uf bec
20、omes,1 tx3 ,y- tx Cy= + m P“- “ F- xCOilsGqUeiltly, tl.c elongation in transverseis (see (2b):1 txa T .- cY=-24r2 tan andirection tanawhich is approximately evenly distributed. over le,ngth t.r“of tk.e stiffener, P.ildsli;”ntlybend ir.nard for a rea-ilrcalculable deree f. This deflection f modifies
21、Cyfor an amount cf; that is (see (7),To illustrate: If several adjacent lonitudinalsbeild for the sane anount of f (fig. 3) , then cf has thefvalue ; but as the deflectioiz Of these longitudinal iszero at- the points Colinecting the transverse stiff eners, .wemay write approximately:Cf=h+ (11)Thus w
22、ith Ey written iil(8) conformable to (10) ratherthan to (7) , it givesProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-1“.M.A.C.A. Technical Memorandum No, 774 7where%y a =Ccotct-ffl (13a)1.*”, . . . . . . or .,-.,ox , , .,.a = _ cot3 a - q“r (131J).I
23、TOW follows the discussion of equations (9) to (13).Iilaily case, only one of the two equations (13) is to beused,. In the aisence of external longitudinal forces othe-r than the shearing forces (torque) , (13a,) is suitable,whereas (131) is preferable when the permissible compres-sion Ox of the Ion
24、gitudinalsis given from the start;that is, chiefly, when tlie.she-n isto be dimeilsioned. forconcurrent action of transverse and longitudinal forces,If accompa,ilied by simultaneo-is bending of the longitudinalstiffeners, th.er. x denotes the tension of the fiber ofthe longitudinal lylng on the skin
25、 of the sheet; that is,the ieaitension of th,is fiber in longitudinal direction.tIn a design # will probably be chosen beforehand,and the loading T will also be given, The deflecti.on fOf tlie longitudinal stiffeners - usually of little influ-ence - may %0 estimated first, at (say f = 0.3 cm) or put
26、at zero. This affords the left-hand side of (12). Thisis followed by assuming arbitrary valves for a (say 10,20, 30), ,and rea. ?.iilg on fiure 4 the corrcspoildin.g valuesof cot a conformsolc to the k-nomn coordinate (left sideof (12) and the axis of tile allscissa. computeaccording to the chosenco
27、t (xa values from (13a) or (133);that is, from eitherFF= K%G=; a + ;:cot a = (14a)The plotting of the two Uill”kf2 cot a against a,once frOi fiure 4, then from one of (14) gives an inter-section point which denotes the correct value of.,cot u.,Lastly, the-atiie of thtcnsioll stress is: III 111I III
28、II I II I I I Illnlmmlll I I IlmProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-r =s =tx =Y =Given in kg, cm400 E = 700,000.,06 r = 5018 rx = 140 Y = 1.2f = 0.4 (estimated)The lmckling stress in shear is:2()kg=0.1-33 .:+ 5.3 :? E = 125 gzrTo zfor vhi
29、ch the left side of (12) gives 19.9.How choose three values for a: say, 10, 15, 20,and read, once from figure 4, with an ordinate of “19.9 on -the abscissa, the t.ree vaues: cot CL= 2.3, 2.1, lm80No external axial forces being present, compute from (14a)corresponding to the three chosen a values: co
30、t a =1.98, 2.177 29309 The: plot both cot a curves againsta (fig. 5) and find that both values agree a cot a =2.14; consequeiltlY, cot a = 2.14, a= 25.1 .DIRECTION 03WRINKLES IN SHEZTS WITHCLOSELY SPACXD TRANSVERSE STIFFEH3R5 (RI1fGs)l?or the following, it is presumed that the spacingof the transver
31、se stiffeiers (ri:lgs) is, at the mostabout half. as great as that” of the longitudinal stiffeners,in addition to being small compared witi. the radius of thecylinder, say ty. r. Such an experimeltally obtainedtension field is shon in figure 6.Tile analysis Of the elongation attitude preferablyproce
32、eds from constant principal stress 0, particularlyProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-IN.A.,C.A. Technical Memorandum No. 774 9principal elongation 6 within the whole tension fieldrange and equal direction angle a, The elongation Cy, _,%P
33、aYi.Tec.t.i.O.Jl(perlplleral.direction) at each -point ofthe tension field, is”readily ascertainableIigure 7 shows two views”of a metal cylinder betweentwo transverse stiffeners (rings). Because of the flatstretching of the sbeetfilers, each fiber appea?s aschord of the base circle. lUe mid-ordinate
34、 P of the seg-ment ist # .“ ,P = tan2a., (16).,l!his mid-orclinate represents the contraction o.fthe cylin-der in the center between the rings. The periphery of thecircle beiig proportional to the radius, the elongation ofthe field ieripheral.ly in the center between the rings dueto the contraction,
35、 is(17)As a result of the flat stretching of the fibers thetension field lies ons hyperboloid of rotatio; hencecoiltraction and elongation in y direction are unlike atunlike points x of the height. Assume all variablequantities over x to he denoted by such as p:, kt.The contour of the hyperboloid of
36、 rotation is a paralolawithin the degree of accuracy of our analysis; that is,P 1, +I”d is definitely estab-lished “at each point. All other quantities of the elonga-tion field can be expressed through(20)(21)Cy ! being variable along the height of the cylinder,these two elongation quantities are al
37、so variable. Nowthe whole peripheral displacement of the upper ring srelative to the lower, is denoted by Y tYi and thechang,e in distance of the two rings, by C* tye Thesequantities are obtained by integrating Vi and c overthe lengtL ty of the cylinder (over x). The resultingshear strain is(22)Iilt
38、he calculation of the chailges in Cx ty it must,in addition to Cxl , be observed that the surface ele-ments of the tension field lying on the hyperboloid of ro-dp tation, slope relative to the cylinder axis;that is, that the ring spacing is less than the length ofthe (approximately parahoi.c) meridi
39、an of the surface ofthe tension field:With due regard to (21) and (16), (17), (18), and (19), thecalculation finally gives the specific approach Cxboth riilgs atc = c (1 - tan2 a) + tan2 CL q + ( )or, in different form:c - “xtan2 a = -. -.(ayc-)._+ Sk0,f(23;(24)Provided by IHSNot for ResaleNo reprod
40、uction or networking permitted without license from IHS-,-,-N.A.C.A. Technical Menorand.urn No. 774 11For computing the direction of the wrinkles withgiven size of sheet and. stiffeners, we write the valuesof (5), (23), (2b) , and (17) in equation (24) for C, Cx, ., .,.Y and “Ck. It gives(25)whereby
41、 a has the significance denoted “in (13a) or (13-0),Figure 8 was plotted conformably to equation (25) .The manipulation of these equations is precisely asfor closely spaced, longitudinal stiffeners.Translation by J. Vanier,Nat ional Advisory Coi,!mitteefor Aeroilautics.REIEREITCE1. TTagner, Herbert
42、: Ylat Sheet h!etal Girders with VeryThin Uetal Web.Part I - Geiler.alTheories and Assumptions. T.11.2:0. 604, N.A.C.A., 1931.Part II - Sheet lIetal Girders with Spars Resistantto Bending - (lbli.ueUprights - Stiff-ness. T,lfi,Oc 505, U.A,CoA., 1931,Part 111 - Sheet Hetal Girders with Spars Resistan
43、tto Bildiilg- The Stress in Uprfgkts -Diagonal Tension Fields. ToM, To. 606,it.A.C.Ae, 1931,Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-1!.A=-N.A.C.A. TechnicalMmoraadwn No. 774 Figs. 1,2,3,5,5,?o A i3CtiOXWz71+- . + f-m”t ionattitudeof theteasio
44、nfield.u+=l”vk-:WSection,4+3 70 75 a 20z .:2”%$Pigure 5.- Ihmple fordeterminingthedirectionof thewrizikleso! W+ 7 I-7 -tt/ v Tu.“Wwe 20- Thetensionfieldina Ourvedpieceof theshellis ooqx+Irableto thatin a flattensionfield$gjgaccompaniedby an additionalapprowhOf both, flaaiwh -. -fi i, .-1 I.T-”t ay-”
45、-i-”-l- /oyiladeruader 81MMW experhmt.3Ygure 7.- Teasioafieldolosely ._ stiffenediap=-directioa;effectof 8traightstretchingof fibersoa theelongattiaattitu& of the teamioa-f%eldoomponenta.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,- -N.& C*L IwhtX
46、ical., .,= -. . ,.,ManorandmIo ?OONo. ?74 ?igse4,8=1-k 40 . . I II Iclosely Spaced =“ ! IkmPT 0?545a- -6 + 2 -7 0I I I I I ! 1longitudinalI20 iv” 40 I I lillmJJdd& l!llllllllllll 11 I ! I 1 I t I 1 I3 45 z $5 cof a i 49 -L1fJ7 46 45-O 4s- linder with C10fM31y r&ced 10Itudinal stiffmers:relationbetwe
47、endistanceand sizeof stiffeners,shearloadr anddirectionofwrinlile%.30,?0fo.- . . . . . .0 70”llJj3i1 . I aPI30” 40” 50” 60” aIolllllttt 1! I I t I I , I I I I I I45 2 #5 co+ a f 4948474645Figure8.-Metalqylinderwithcloselyspacedcircumfermtialmtiffmers:relationBetweendistance-d mise ofstiffeningsections,shear anddirectionof wrinUosOProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-,IDO NOT REMOVE SLIP FROM MATERIALDelete your name from this slip when returnin
