1、I1.JdNATIONAL.ADVISORY COMMITTEEFOR AERONAUTICSTECHNICAL NOTENo. 1361.DEFORMATION ANALYSIS OF WING STRUCTURESKuhnMemorial Aeronautical LaboratoryLangley Field, ?W3. *,LIBRARY COPYtn APR2$IQ=UUggwwmm=ixJtJ more refined tress theories have thereforebeen developedover a period of years. Theories of thi
2、s natitiaare applied to theproblem of calculating the i!eflections,particularly of wings.Bending as well as torsionel deflections are discussed for wingswithout or with cut-outs; T%very. simple approximation formulas are therefore-purposes when conventional structures underloading are being section
3、 aiongside cut-out(coamingtTin,half-length of cut-out; half-length of carry-throughbayfractions defnedby equation (15).,. depth of box tieam,.depth of front sp.,-. . mEPARmCIRY DISCUSSION“,., .,.312emen.-. i? dx “ “:”-“dy=.,two sections ,ibtance dx(2)-.The tilde (-) l.sused %?oughout tie present pae
4、 to indtc.ate.etresseso a verticalbending moment M appled to as a result,the tendency ta warp differs from section to section, and secondaryetreeses are set up by the resulting interference effects. Simikrly,the elementarybending theory is =trictlyvalid only if the appliedload is a pure bending mome
5、xit. In actual wing structures, thebending momente are produced by transverse loadB, and the shearstrains in the covers proihacedby these loads tiolate the assumptionthat plane cross f3ectionsremain plane. As in the torsion case,interferenceeffects between adJacent sections produce secondarystresses
6、,Stress theories that take these interference effects intoaccount are unavoidably more complex and less general than theelementary stress theories, They necesmrilymake use of simplifyingand restrictive asamnptionsparticularly regazWng the cross sections,in order to keep the mathematical”-cdmplexi.ty
7、within bounds= The.effect of these amnunptions on the accuracy of the calculations canbe minimized (except in the regions around large cut-outs) by thefollowing-procedurb:(1) The elementary stresses are calculated for the actual.oross sectionsQ(2) The secondary stresses produces.by the interferencee
8、ffectsare calculated using cross sections simplified as ?muchas necessaryor deetiableIn conventionalwing structures with reasonably unifoma loading(constant sign of bending or toraionelnmmnt alo span), adequateaccuracy can often be obtained even when highly simplified crosssections are used. This re
9、mark applies to stress calculationsand.even more forcefully to tleflectioncalculations,becauseanystipulated accuracy of the deflections can bo achieved with a lowerorder of accuraoy in the streses. Although this factis quite wellom it will be demonstrated later by means of an example for thetorsion
10、case as well as for the bending case.a71a11.“Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACA NO. 2351 7Advanced stress theories of torsion end bending in shells havebeen developedby a number of authors, striking d3ffere.ntcompromisesbetween accu
11、racy, complexity, end generality. The stress tieoriesselected in the present paper as basis for calculating the deflectionsWe *ose of references 1 and 2, b references 3 ma 4, these theortesHave been shown to be reaeonabl.yadequate for stress analysis, andconsequently they are amply accurate for the
12、deflection analysis ofconventional structures,TORSIONAIU!LYSISDiscussion of fundamental case,- The structure that will bediscussed as j?undementalexsmple is a box of doubly symmetricalrectangular crom3 section as shown in figure 2(a), with infinitelyolosely spaced rigid bulkheads, built-ti rldly at
13、one end andsubJected to a torque T atthe free end. (E43efig,2(b) Llhe ,cross section is an idealized one, that is, the walls are as t. .%)The terms ?bn (or ?h represent.)added to the streeses.?%(or ?hin order to obtain the true sbeeeb. In wfng boxes, h/ iscorrection term that must becomputed”by the
14、elmaentary theoryusually much smaller than b/, and is consequently only littleless than unity. The correction terms are thereforenearly as lareas the etmwses calculatedby the elementarytheory and me thuso%viously important.The fundamental.relations given in reference 1 peimi.tthederivation of a dift
15、erenti.al.equation for the anglm.of twist, whichappears as a function of the torsion-bendingparameter(i2)Boxes approximating the proportions found in wings have a length Lsuch that it is permissible to settanh lg.% 1.Provided by IHSNot for ResaleNo reproduction or networking permitted without licens
16、e from IHS-,-,-u “ For such proportions, the solution oftakes the fozm (z .1q.l-=9tie differential equtonThe angle of twist is plotte in figure 3, with q taken as unityfor simplicity., Xtis apparent that the!correction to the elementary“theory,in regions not close to the root, is approximately a con
17、stant.At the tiy, with e-fi= O,1(14)For conventional vings, . is of the order of 10, =a the correctionterm that must be added to the tip twist calculated byathe elementarytheory woula therefore amount to about 10 percent if tho wing wereof constant section and if the torque iereapplied at the tip. “
18、Ac.tualwinge are tapeedand csxry a distributed torque, but thesetwo deviations from the simplecase tend to offset eaoh o%her.intheir inxfluenceon the twist curve; the calculation ustmade.maytherefore serve as a rough indication of the order of magnitude ofthe twist cwrection. A stipv.latadmaximum er
19、ror of 2 percentinthe tip twist - which is about the best forconvenience, the fommila for k gtven in reference 6 is reproducedin appendix B. The followtng formulas given heretn can be deduced .readily from the Yesults iven in”ihe”referencc. .The =itude of the X-Woup acting oneach adjacmt full bayat
20、the Junction with the cut-outbay is given by. .X + -1)t (20)and consequently,by formula (16), each of these bays has a twistcorrection(21),Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-. NACA TN No. 2.361 23* where the subscript fb denotes full bay
21、. It should be noted thatthe correction is positive, that is, the twl.st of a full bay isincreased by an adjacent cut-out bay,The relative twist between the end hlltieads of the cut-out haycan be written In the same form as that for full bayswhere the subscripts coco ?co+.co (22)denote cut-out bay.
22、The lelementary;ltwist $Co is the twist that results from tho deformations of the “members of the cut-out bay when the end bulkheads are prevented from.warping out of tl.1.eirples; the walls then act as beams restrainedbJ end moments in cuch a manner that the ten.gsntsto the elastic curveat the two
23、ends of each beam ?.emainparallel. The twist correction A%.is the tvist Vnat would result if e members of the cut-out bay wererigid and the end bulkheads were warpo,out of their planes, theamount of warping being determined by the tirque T and the X-groupacting between the cut-out bay and the ed,cen
24、t full bay., Application of the methoflof internal work tithe stressesgiven in reference 6 yields for the elementary twist .,+(l-. k)2 ; dS k-”,C(l - kJ2+- 4(21i- 3j2 d33EA#lc2 hh L ,3EA3bhJ.(23,). ., ,For a full-width cut-out, k = 1, and all the terms containing(1 -.k) disappear. ., From the geomet
25、ry of the”structure,the twist correction forone-half oftilecut-ut bay (from the midpoiat to a bulkhead) is,.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-14 NACATIV NO 1361whichmaybe writtenBy Ma.xwelllsreciprocal theorem, or by direct comparison o
26、f theformulas, end by v.seof formula (A3) orayendix, A, it can bseen thatBy definition.TF=px -where p is the coefficient g.venby formula (Al) of append.ix,4,and X is Siven by formula (20). The twist correction for theentire length of the cut-out bay can therefore be written in theformWco =2Aq7fi+x (
27、24)Ier to Ye consistent with all assumptions made, the torqueused in evaluating formulas (22)+a (24) should be the torque actingin the cut-out ay, The vgJ.ueaof T for the two adjacent fullbays, however, should be calculated.for tie torques actulY actingin these bays. .When the cut-out 1s small.,no c
28、losing bul.kheadeare providedin general. In this case, the chmges in stress the so-called shear-lag theories are refinedtheories of lending flmwhich the eftectmof ese trains are takeninto account. The engneering theory of shear lag developed inreference 4 is basedon the ue of simified cross sections
29、 suchas that shown in figure 79 A.%eam with such a cross sectionmaybe used, therefore as example to illustrate the relative i.mpqrtmnce01 shear-lag effects on stresses and on def.lectiona-m order tokeep the formulas as simple as yosible, the discussion will beconfined to a cantileverbeam cd?comtant
30、section, fixed to a ri.dabutment and subjected to a vertical ldad. “P.“atthe tip of eachshear web.Reference 7 shows that the analytical.solutlon of the stressyroblem for sucha beam is chexacterizedby the shear-lag parameter(26)which plays a Eimilar role intorsion-bendingparameter Kadvanced torei.ont
31、heory. Thein tho flange, in the centralrespectively,the advancedbending meory as theglvenby expression (X2) in theanalytical formulas for the stressesatriner, and in the cover sheet are,(Cp%l+T,w=T)sinhKx cosh KL(27)(28).cosh Kxcosh KL) (29) Provided by IHSNot for ResaleNo reproduction or networking
32、 permitted without license from IHS-,-,-. N4CA TTNO. 1361_* withMc 4;= = rP;I (30)(31where t is the thiclmessFor-conventional wingsimplificatd.onthe stresses at tho root of “thebom.ncan he writtenin the form. .OF-( )%=L13. +).$.(32).(33)-,., T = o,The “deflectionat the tip can be calculated from the
33、 work . ,eqvation -”. . ,fi *1 I I3L!? however, i?.-the spar caps are light ( 6, a conditionProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-,NACA TN No. 1361 21which will be fulfflled in practically all conventional wings, andif the loading is reason
34、ably uniform, the stress correction can bewritten in the fozmwhere d is the half-length of(40)the carry-through bay and alluantities appearing in the fraction (exce t x) em measured at the “,8representative*, within the carry-through bay, the correctionmay be assumed to have aconstant value equal to
35、 the root value. Application of formula (39and approxtite integration consistent with tho ordelof accuracy offozznula(40) gives the deflection correction for the tip of the beam(41), Because the correction small, it willbe sufficientlyaccurate to assme that it decreases linearly to zero at the stati
36、onlying at a distance IN frcm tho root. X% should be noted that. formula (41) gives the correction caused by shear-lag action onlyfor one cover of the box. Also, , K, and so foriih,characterizethe half-section; therefore, .% mustbe takenas the shear forcein web.If the stringers do not carry through
37、at the root, the rootsection must be considered as a full-width cut-out, and the methoddeecribein the next section is.applicable.Calculation of deflections for winRs with cut-outs.- The stressin a strir?gerinterruptedby a cut-out Mops to nearly zero at theedge of the cut-out (fig. 9) unless the cut-
38、out is very small “andextremely heavily reinforced. It is common practice to cute thestringer stresses near a cut-out by applyi the ordfnary bending.theory to the cross section of the hox after multiplying the cross-sectional areas of the stringersty en effectiveness factor. Theprocedure is simple a
39、nd is well adapted to computing effective.moments of inertia that are adequate in most cases for a deflectionanalysis by the standard procedure of integrating the M/EX ourve.,.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-The most severe te of cut-
40、out is the full-width cut-out atthe root of the wing, which is frequently encountered in yracticein the form of a zero-lengticut-out (stlingersbroken Qt the rootjoint) or in the form of aftilte-length out-out (wheelwell or gastank bay). Fie 10(a) shows a free-body diagram of the sectionof the cover”
41、between the outboard edge of the cut-out and asection A-A some distance farther outi Shear-lag calculationson typical wings show that the stringer wtreeses at the section A-A are reasonably cloee to those given by the elementary theory whenthe distancebetween the section A-A and the root is a rather
42、 smallfraction of the semispan. T3ndorthese circumstances, the problemcan be simplifiedby removing the edgeshears and.increasingthetotal.force MA at the section A-A to equal it should, thereforo,beadequate for deflection analysis in au-cases, because the effectof a very small cut-out,on the dcflocti
43、cms is negligible., -“,.A theoretical di.fficulbyarises when the-out-out is so closeto the rootthat there is appreciable ihbrfer6nce between the sti,essdisturbance producedby thecut-out and the ,disturbancecaused by theroot. Tliisconditionmay be.said to exist when the distance xbetieentheroot and th
44、e inboqrtidge of the cut-out is suchthat KxKO.4, where K Zs the shear-lagparameter definedbyformula (24)for a eection halfway between the root and the edge ofme cut-out. .FOP such cases, the following approximate procedureis SuggQatea: _ . . .(1)Make allowancefor the effect of $ho cut-outby “determi
45、ningthe effective b th.(Al)(A2)(A3)IrIthese expression, a is the length of bay n; T is tie(accumulated)torque in the bay, and the remaining ikbnensionalterms are-defined by figure 2SThe flange forceu X (fig. k(b) produced by the interactionbetween adjacenbays are calculated fron we set ofiequatlons.
46、The positive directions of T and X are defined by the arrowsin figure h(b).Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACA TN NO. 1361:,AFPZNtCCXB27DetemainationThe constantof Constent k for Torsion Box with Cut-Outk appearing in equations (20 t
47、o (22),whichdetermines the division of the torque between t?evertical .7lsand the horizontal walls, is detened by the formulawhere10.zCn 1.k“ 11; CnL-.,1khcl=hl+=%2c3=c= 4bdC4 “%24hdc5=.C6 = a% 0C7 = G b2d23E 22()LC2+:c = 3E A2la a2C9 = 3E A3.!10 = 32G ad3E A4(A5)cf2 = C2c = C3(J11+SC4C4 =C5=$P+:F5C
48、6f = $6C7= CTb“8b+2cC81 Cp = C9Clo “ &o.,The dimensions appearing im these expressions aredefined in figure 5. men the net section is very narrow, theProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-!33 “-MCA No. 1361te1211SC7 EUldC8, CTt and C ayereplacedfor greatercOn-venience”of computation3Y the terms%2d2c7a = 3EI c -!Ta,. .where I isthe”moment of,inertia of the netthe spar
