r

TECIC.L E ORLDU S

rA:IoiTAL ADVISORY CO::iTTEE FOR AEROIAUTICS

:co 1362 LL) L iRESSE.

AT:O;:.L i,ut JMMIflEE FOR AEROPIALJTK

fLT, rLi., WA, .

ACCURATE CALCULATIOT OP MULTISPAR CAITTILEVER A1TD

S:::ICA:;TIL:VER ?TiTGS 7ITH PARALLEL TEBS U1ER DIRECT

AITD INDIRECT LOADIG

By Egen Sn&er

Zeitchrfft f!r F1utechik un Lctorluftschiffahrt To. 20, Octoor 23 1931

Verlag von R. Cldenbourg, :.;nchen und. Berlin

17ashi nt all arc:, l932

https://ntrs.nasa.gov/search.jsp?R=19930094754 2020-08-01T16:45:34+00:00Z

NATI O1TAL ADVISORY COLIMI TTEE POR AERO1T..tiJTI Cs

..:' .T.EcKN:rCAL uE1R1Du:t •' .......

.ACCURATE :'CMCULAT.iO1 . OP :iiULT;I SPAR 'C.TILEVDR 'AITD

SEItiTTILVER YIJGS 7ITH PARALLEL 7EJ3S UiTDR DIPCT

AND. .::IND'IREC:.'L.O;ADI'i'TG . ..':H. ':'

. .3y: 'Euge' :;Ser:'

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iLinim':st?ru::t.'ura.l 'we.igh : :a wing i. :ensr.edwien wi'thin .bouifd''.o'f 'the selete.d suppo:t sy.sten',' the';:truc' tur'al..'.m'aterial 'háving.'the' '.hihe'st eff .iciency. •(o h' for teiaslo'i, E/Y for buckling) is so disposed that its cross-sect'ional''areas are everywhere' utilized. u to th str'eiith limit. In particular, uneconomic bending stresses (bec"se of tne only locally restricted, maximum stress of t..ie mate-rial) should'be avdid:ed where'ver'o ,ss.bl'e'. his 'c't"ipula tion of resolving all stresses into pare tension end con-piesiOn intiniaosthe''uti1ity'o'f truss beams, "and thü epleins their ever-increasiig use in the design of the modern large airplanes. The outor snpo of a cantilever 'w'.g'nd. with it he ' height of the 'bearn is determined by. the aorodnathical1y defined. airfoil. . The problem then is to decide on the typo of costruction, a matter wxiich do-pddá"oii tho"naturó of tio extornally ' imp osod loads. Th'bii 'roultant can bc readily 'resolved 'into componont p 'árallol' and' normal 'to the wing'chord. ' The parallel cd'm-ponont' is'invcriabiy, as a glan'&o at'Piure 1 shows, a srfill fraction of 'the normal component and, in addition' oncotutors tho wing chord 'as boa height; hence, *111 n6t 'ocomo docisivo for beam shao ox' calculationo In diroct contrast, t10 normal'copnoit 'i prodominnt and, boauso of the' mall beam height, decisive; aside from that Hthe' points of application of th,eir resultants are extréely variable. (See fig. 1.) In 'view of this ar'iabilit,' the lightest weight is oontingent uoii the' whole struc ture helping to c'arry the lo.d, i.e., a very higi combined effect between all support'i-ñgpart'. By'c'om'bined effet we imply, for example, the subsequently examined nultispar

* tf Z . genauén Berechnung viel eliiiig-parallol 'stdgiger.,. ganz-und halbfreitragend.er, mittelba undunmitto lbar 1las-teter Pl&gelgeri'ope. Zeitschrift fur Plugtechnik und mc-torj,uftschiffahrt, October 28, 1931, pQ• 597-603.

2 ,. N .A. C,A. Tecbnica1 0., ..6S2

wing, in which a spar not oay, orries "tIiiload. but at the same time is helped. by adjacent spars through the agency of the ribs 0 Combined effect is impossible in a two-spar wing unless the':'bon'd b.etwe'&n .'.ib nd. spaJaffor&s'-'tiie' transmission o ' a torsion moment to the spar andthe latter itse.L± is resistant to torsi.on. A two-spar lattice wing has nô''cothbined effect In three-spar or multispar wings torsional stiffness of tie iiidi'vi'd.ual. sa-r is. no longer necessary; resistance to bending suffices0

In our case, i 0 e 0 , ultispar cantilever wing, the oar-allel webs are interconnected. by transverse ribs.

Consistently witii our preced.ng stipulation, 7e now vièuaIiz-e spals and.'rib, s as tiiss beam orgrilláge. 'The norie1 leeds are taken up and. transmitted by the normal lft-triiss of the spar0 The equelly nor r ial rib-truss to-gether witn the spar lift-truss form a eam grillage end. effect tne requisite oond.. Tne individual lift-trusses of tins grillage are not rigid in torsion nor is t.'e gril-lage stable a&ainst loads acting in its plaer To tae up the small perallel loods tne snear strengtn of a corru-gated. metri s:in is ased exclus.ve]..y, w.cri also takes up tne eir _oads and. their transiission to the lift-truss flages of the ' sar' in foth ' oan equiva1ent'lod The unloaded. t n'sverse ,struc'ti,re (r'o.$) mer8ly sørves."to 'pro-ducé the combined effect and. , revents overstes.s: in,. the. metal skiñ These ribs .ma3. the complete s'ructure stat-• ically indeterminate. Such designs should. ord.iaarily e avoidéd.fór wing s't'ructux'é 'because the stat . ically i'nd.'. terminate structure i usuaiy heavier thax tiie.'equivIent statically d.étermi:ate stzucture, due to t1e i,nomplete.. usacility of tne sections of t.ne mer.bors. But tnis par-ticular case is an 'oxcoptiqn. . The great •aria'b:il'±ty of th& location of the applied.: load. produces, in, view of . the combined. effect through the statically indeterminate lat-tices, such a f'ar.xoaching ,unl.oading f . to rc:.sectivo member undr load, as to maio the comp.otp statically' in-dotorminate structure lighter. in.si.to'.of" it.,''. This coth-binod' óffect, moroo'or, ' bocomos vitally iiporttnt. when, du to fniluro of one port, tho roravining obcrs mast taie over tne load. o±' the otner0

For our purposes we select tue spar spacing at 1/10 wing c.nord., a'a secifica1ly, e arrenge seven spar lift-trusses.

N.A.C.A. Technical iiemorandum To. 662 3

Rib spacing and. thickness are defined in keeping with the fact that the combined effect is considerable even with .,.fevr small ribs and. shows but little increase then in-creased in :n'iimber or size. " -Liaximum combined effect even with comparatively small ribs is then obtained ihen the latter are in the vicinity of maximum spar doulection, i,o,, at the wing tips.

In accordance viitia calculation, we prosuiuo equal stiffness, in bonding in all six ribs. (See' fig, 2.)

- After t.hus' explaining the , most essential points , for .o,btaini-ng the requi'si 'te "lightest" wing, there remains the s.tat:'ical anlys .i: s . 'of the'co'mplete struôture,' so as to be able to decide upon the dimensions of the individual parts.-. This is' done 'by an unüial niethod,. the r1ncipal á.vantges of which are: ' ',, ' " . " . ' -

l,Preedom.of static examination from special.load. case by -introducti.ón'of influence lines,' consequently.one investigation 'covering -'a]l 'load cases (pull_up.: glide, steep, dive, inverted. flight, 'pull-up in inverted flight, etc.). . : -. - -' '--' '. . ' '

2'.:' Simple inclusi'on of' direct loads in the static ana].ysi s, . '.'. '

3. Provision for actual' support conditions on fuse-lago ' and. eventual bracing. ..

.4• Regard as to limit: of rib stiffness.

• 5, Calculability of covering effect.,

• 6. Examination o±' each spar inertia moment occurronco.

7.- Utmost possibility of approach to actual load. at-titude. -

- 8. Special qualification for calculating extremely large and. "all-wing" --airplanes.

; In conclusion, it thj be- observed that the study is equally applicable.,- with suitable modii.cations, of course, to wings wi-th varying 'chord, and.' that tho consid-ération of a -load variable along -the spars (tapering at the tip, for ex.mple), ' present' s ó obstacles.

4 ... N.LC.-. Technical .emorandum..lT&

Here also it is essntia1 :0.. rentiate between extena1 loads paaIIel and.nomal .the.pin of the grille The p arlll loads are bai.aiced. y the shear strength of the shin; the normal • .i. oads are treated as lows (see fig. 2) lith tiae stipulated. four-point sun-port tne complete spars and. two ri-bs, let us say, x 6, and.. x 7, suffice'to maintain stabi :lity with the - suit that 12 ribs may be altogether removed. Since each rib as beam on seven supports. is five .ties statical-ly in-determiat, the total nuthb.er.. of the statically indeter-minates . is . 12 X 5 60. Substituting the word. "rib" for "spar," and viceversa, y.eld.s the saminal reult-.

Gëiéral rülé: igrillage, extranoously.siipoorted by four conditions (limitations of displacement and distor-tioii)-withmemberhintable...in torsion--is externally-stat-ically detérthinate añdinternally as -rnaiy ...times statical-ly indeterminate as it has internal joints (from which four members emanate). Every xternal additional cndi-tion 'ti-ulates ä exterriaf static- indetermiratness Al-beit tuis total nanber of statically inde-terurete cuan-titie is noci reduced to 30, Decease of the symmetry of the :ad:S -au exaôt t±oatment b conventional method o± statics is still , out of the questio.n, so- recoti.r.se is had. to the rnetho of differ&ntial equations as broached by Professor E:.nst Melan, Vienna. Instead. of the truss beams we substitu solid bear viithinrtia thoment :j = F-if /2.. In-addition, we disrd the hitherto cistoar r , butvëry arbitrary assumption of cantilever wing rigidly restrained. at the fuselage. It.is felt. that this free hodr" d.agram, as illustrated in Figure 2, is more nearly correct. It is the sane as considQring the -two. semi.-wing, :conñecte di-rectly or by means of a central fx'eme, as static. unIt on which the load, now .symmetridal to the airplaie axis, is imDo:sed and vrhérin' the load. encounters its opposing forces in the attachment. forces of fu,selage or in d.rect wi-n'g loads -as encountered. when approaching the Uallw.ngu type of airplane, where engines, tanks, cabins, controls, etc., are mounte. on thewing. .Since these are conditions vcry : Iargel r deendentupon the repecte type of con-struction, we confine ourselves to representing the meth-od with a statica1ly.determinatoarrangemnt of the oppos-ing forces, as I ibatëdin Figuro2, and.e1aining any occurring changs tnd.Or other support •conditionè during the-s. i.ig calciilation That this method is.aticu1r-

ly suited. for tin e c2lcalatJ.on of giant airpla-e tpes, such as the Junkes, Ruthpler, and. Donier is obvious. An-

N.A.c.A. Technical liemorandum No. 3S2 ' 5

other feature from tne structural point of view is tnt the perfectly free 'truss is un.stablo.:against asymmetrical normal loads, sucn as are imposed 3J eiloro-i effect, in spite of the covering, to the extent that, duo to the lack ofany. :t or,si,onal. stiffness of the .grillge. membe'r.s, the ént:tr. gria.can, without resistance, ' .efor in't6 a hy-p-r.b.oloi'd.al area,. because all..gr.liage thenbers.,reain straight. This i'istability can be remedied bj restrain-ing.a.t least four joints of the grill,age., ;say',,.fou.r.fuso-. lcgo attachment points relative to the grillage or else by resorting to spatial stiffening. In the numerical ec-ecution, we confine the accuracy to tiat ootained oy slide rule. " . " ,' . S

To comprehend the s.t.atiç behavior of 'the cothplete :ing structure and in particular-of the combined effect. betwen the individual spar walls, as well as for: provid-ing the requisite: bases of the proportions., w rely 'on the influence lines. •. Our influence ']ines, somewhat at variance with their orthodox definition, have as ordinates theinfluence of an, equivlont:load-o' 1 per unit--length.. of spar, 'extendiflg over oe whole spar (from x =' 0 . to x. =13 ii our example), hon this load constant by ' it-self,, travels from spar to spar in rib d.ir.e,.ctin Z1Q: di rect rib lading being presumed herein - to.the requi:site magnitu.e :(.s.par moment, rib moment, isplacement 'Of .join) at 'the respective point. . ,Altogther there are seven dif-fer,ent load., positions.'.' When. one spar,., y = constant sets up a uniform: load along its whole longth, all other spars are unloaded. ' Eence tho..co putati'o,',of all, requisite, quantities for "each of those, seven 'load po'si.tion.:s makes it po.ssibl.e''to..p.lot each influence 'lne.o,f .these,quan.'ti-ties. By virtue of the symmetry of the wing structure to the center spar, it is necessary to investigate only four instead of seven 'at,titid'e&.,,To' assure"inclusio.n. of di-rect wing loads. t'he' procedure 'i's as '±ollows: arbitrary, direct wing :lo'ad'.s, are. always known. ,and constant in mag-nitude. and.' .pJace 'hence'.ae counted .amo'ng the',,other known e'x;t'er'nal loads 'w.hich mere1r 'have to comply' with' the general ',cod'i't,io'hs. bf: ext,ernal loads' oi the,, fre grillag.. In order to be able to retain our method' of defining the influence lines we select one of the ttunit loads' of the whole win'g :(all spars loaded w'th. , 1. per: 'rne t ei lengh), bearing in mind the relation between actual weight of the direct load (engines, cabin, fuel tanks, etc.) and in di-, rect actual wing loading,, a. corresponding . part of the dl'L

rect lç ,ading ply, during loading of one of the m spars, the m part of the direct loading at the requisite

6 H.A.C.A. Technical liemorandum To,'62

points, c1 efine tue reactions recessary nereto Coccurring eitier'a's body attachment forces'o a'srdi.rect ontrol, forces on the "al1wing' 1 airplane);: 'after whih we can proceed as in tuie ord.inar case.

To illustrate Let an average, equeble spar loading of 150 kg/rn oe synonymous to tne mean actual wing load-. ing; an enginb' of 800 kg actual weight iz assumedly mo.un' ed. on the top" of the wing in form of a. single..load.. Our: influence 1mb calculation was .carried . oilt with a stipu-lated, equivalent spar 'load Of 1000 'k/rn t . Consequently,: we must, in oider to maintain similar loadiñ corj.dition's,'.." determine a' corresponding part of the direct loading

(800 X = 5330 kg) in view •o.f. the. , el'ationship be-

tween actual weight of direct loading (8OQ . 'kg) and mean surface loading (respectively,. of' the qi4vient spar load 150 kg/rn' dependent thereon) . So when 'loading the wing: with the originally assumed equivaln load, (.1000 kg/rn'.)' over all spars and 'using up the computed irect' loading'.... proportioi (5330 kg), this system of forces must be brought' into equilibrium by normal fo'rces'.ep1ied. at th,óse poits "viilere, by nature, fuselage , attachment forces or' control. forces must occur.0 Our rnetho of defining the. influence line.s stipulates loading of. only one of the rn spars at one, time', sci that obviously only the mth art of the direct load. jortion must .be applied, beOause'al'l' m spars together are to'correspond to. the complete portion again. After defining the reactions for this loading attitude, the calculation can be carried through for"thi's case. The method is applicable to any airlane attitude, even for case .0 where, as is known, no 'comDonont of the'd.irect loading must, be incurred in the normal direction of. the wing. .

Before beginning the actual cal:culation',' the inertia moment of spars and ribs and some iotation must' be' de-fined. The sars are assumed s t.russ-cantileer''o equal resistance in bending, tapering lineai'Iy"in hOi'ght from: the root toward the wing tip. Countiug ± 'beginiiThg .t the ring tip and stipulating an equivalent load q per meter of length, we have ' ;:: :..........:... :

Mx = a x2/2; ' Mx/Vx; 7x. =: MxtCY' q' X 2 /2 G

and therefrom,h/2

N.A.C.A. Technical Memorandum No. 662 7

which, with h = a x reveals moment of inertia

= q a/4Gz 3 k x1,

wherein k is the spar inertia moment at the first joint x = 1. Besides, we stipulate equal stiffness in bending in all spars, and equal and. constant inertia moment of

magnitude k in all ribs, so that J k.

1'Totation: spar spacing h = 1; rib spacing 1 = 4,

- 1 - 4 . - _____ - 1 Px - - 'r1y - - -,

EJx Ekx3 EJ Ek

- 4.000. (p 1 = cp13 -

- . - 0.148. (p C,) 3 11 Ek

= =0.032. Ek

P 7 =

- - 0.500. l2

Ek

= =0.063. s)4 ?1

= 0.019. '9

Ek

0.019 Ek

As cantilever beam of equal resistance in bending, the spar demands, consistently with the foregoing, an inertia moment conformably to a cubic parabola. A truss beam with constant flange profile and linearly changing height would yield a square parabola. For that reason the flange pro-files must be graded. These conditions, which are alto-gether subject to the particular case in p oint, are ac-counted for here by using in each spar panel the maxiritum instead of the mean ±nertia moment for defining cp. For designating (p the number of tho joint following the panel was used.

cp is 1ndependent of y, r is independent of x and y. The displacoment a joints x, y norraa1 to plane of grillage is Zxy.

The moments at the joints in direction x: L_1 ,;

L+1,y

The moments at the joints in direction y: M , _ 1 ;

M,y.

8 F.A.E.A, Techiiicál :eahn•o.62

The su p ort p essure of the freely borne beam át:oit x,y set up by external loads, is T1 = 4. The support pressure of the moment loading b± the fieely supported beam at joint x,y induced by this member is P = 8/3. (3 l y tne spars y = constant are loaded, that is, 'ni-forLly w.th ut loading 1 pr nut length, hence

f1 =lXX2X4=, because •l=4,f•=.

= 2) Force and isplacement areountébsitive in

the same direction. The bending moments are positive when the center of curvtureof the defoithed eiibei lies on the side of negative Zy.

The deelopment. of the diffOreial equations begins with ClaDeyron' s three-moment equatiin

L 4 2(P ± (2x+ L+ +

+ 6 ( X±1X - z XX-1 = (PXX + ____

+i: lx I lx x+i

modified for our purposes to:

CPX Ly_+ 2(TPx + cpx+i)Lx + P+i L+, ±

+ 6/i (z±1 2 Z x ± z 1 ) = - 6 1i (x.x+1). For abbrevtior, re write

6 P./i (cp + Uxy

+ 2(+ + X+1 L+i &( L)

x M_1 + 2(41x ± ± ' x+i : •: x+1 = '9(V :)

i/i (z-+ 1 - 2 z + z_ 1 ). = D.-(z)

1/h (Zy.-j - 2 Zy + y-i) = D(z)

which afford the partial differential .equatiois..

+ 6 D1 x(.z .) = -: ,1 \

, y ( M) + 6 D y (z) =

(2)

N.A.C,A. Techn..ical biemr.andum 1Ov 662

Eacjoint.having three cGordi.nated unknown..-ouantities (L,Mz) a tiird equation is needed and. obtained from tne e quilibrium of tne joints perpeidicular to tne plaae of the grillage. The proportio'n'of ornents " L the jcints in direction of the external load. is:

1/i (Lx i, y - 2Lxy + .Lx^ 1 )'= D x(L)

Moments,;M at thojoiiit on-tribte.: -• •., 1.:

1/h (M,+ 1 - 2 M1 + Mx,_ 1 ) D(M).

Cont.ri'hiition :f 'externa]. Ioad:.TJ :. i .•' •

;;Thus fbllow:s : .th .e third. equation. :.f t,he....siiu1taneaus sy.st sin: . .. .. , .. : ..

D x(L) + ' y() = Txr. ..

Since our lift structire has free boi-idaries oily, the conditions stipulate that the moment in direction nor-mál: :t the boundary mus..dappear; : .Mxy' .= :9 0. The. cross stress in the. f.ir.at. panel outside of, the .aiea must therefore become zero. This is proportional to tne diffërenc ioents at' the ....

y = 0; Mx,_i - Mxy0 consequently; = 0

= - = 0;':,' •:

=

x- T . ...T -' X-1,y '. 1y C .. it . ........: '':'. . :

- X-i,/

x = n; L +i,1 - = 0 0.

If the wIng structure has more .. .than four fuselage at-tachrrients, or if outlying joints are prevented from shift-ing br strutting, as in the semicantilever wing, for in-stance, the attachment 3reures mpo sod. Othese 'redin-dant attachment points are statically indeterminate in their normal component.. . .:,:

These:pre.ssres :are •temporarily..•introuced..zs unknown ext er.a1 .l.oad.s an. the:.procedur.e is. as fl1ów.;..

10 .N.A.C.A. Technical Memrand.urn No.662

1. Elastically restrained joint z k Ty wnen rèpresents the uikxown attachment p.ress.ure: àt joint

x,y As shown elsewnere, we nave

ii and

1-0

[1/i (- -:2.Xx..+X±)+t'()1 (_/w),

wherewith : :

k j 4/() 2 •+ x^l)+tj(x)]

becomes t.h •eneral incëment equation for defining: the statically indeterminable attachment pressure at each re-dundant elastically restrained joint-.

For redundant fuselage attachment the premise of rig-idly -restr.aed joints1 ordinarily should -hold.

- .2. Rigidly restrained joifls: Z1y 0, consequently,

txy Q.f andt.Ie simple equation for defin.ng the exter-

nal statically indeterm±.naes with r-igidly restrained joint iS: -

1/i 2 .+ x-1,) + ti ()_Q

wherein the unknown is contained within . t1 () aS

external load.

Now we -proceed to Professor LIelan's method of resolv-ing the differential equations: .

1. TIie general resolutions Lxy M y Z xy of the un-

homogeneousness, simultaneous system of partial dif.feri-t..al equations, ...... . .. .

,x.(cpL) +6 Dx(z) = Uxy y('1iM). .+ .Dy.(z) :

Dx(L) + Dy(IA) = -

can be developed from tne proper resolutions Y 1 , Ti1 of the homogeneous simultaneous system of the same partial differential equations,

N.A.C.A. Ta nical Lmo.randm..To....S52 .11

&x(qL) ±:6.D (z)= . .o.,: .•, y(i.jiL)+ .6 Dy(z . ) •:O,....

Dx(L ± Dy(U) = Xz,

and. from the. resolutions X and. j of n nhOogene:o\s system in the following form: ... :

Lxyo

X1 Tb., IIxy =

Y1 ZXy TI1

2. For defining the proper esolutions of the honioe-neu.s system of . par.tia. d.ifferei-iti.al. equations: the.. simul-taneous system resolves itself with . :'

. 1 y' = ?m my' zy.: x

into two groups of .ord.inry ftif'erential equations each:

,x(cpX). + 6 Dx() = 0, Dx(X) - 0 for X and. "

and.

y(Y) + 6 Dy(TI) 0, Dy(Y) .= for Y a Ta.

They reveal ? and. Xu . as root& of: the frequ.ncy equation. produced. by making the coefficient determinants of that system of equation = 0 •which taks the place of the d4fferential equation.

3. For defining resolutions X1 and. of the unhomo-geneous ystem the load. terms and. Txj ar.e.d.volo;;eci

according to the proper resolixtion.s T determined., above.

=11()j T: 0.t.(x),TIi .

whereby, if the r j are regulated, ..

ui.Cx ) 1'.Uxy.:y:.and. i(x) TxyTIiy.

The reCuit ithe'simuitan•aou. system o,.ordi,nary dif -ferential equatioiis

&x( X) ± 6 Dx()= ''i +

for X and. j.

12 N.A.C.A. Technical Memorandum No. 662

4. Now the general resolutions

are computed conformably'to the equations - m - m -

Ly = Z X M1 = Y1 Zy 1=0 1-0 1=0

by matrix multiplications.

This is a brief outline of the general.metbod of..'es-olution and we pioceed to the concrete calculation of the first loading attitude (spar 0 loaded.). . .

In this case the unhomogenéous, .simultaneous.partial differential, equations read

&x(cpL) + 6 Dx(z) = - u, y(4tM) + 6 Dy(z) = 0

Dx(L) + Dy(M) = -

wherein . .

Uxy = 0 for all y 1, 2, 3, 4, 5, 6

Uxy = 6/i x 8/3 (cpx + = 4 (cp + P+i) for .y = 0

andTx = 0 for all° y 1, 2, 3, 4, 5, 6

T 4 for y0 ...- .0

The proper resolutions ofthe homogeneous equations

&x(L) + 6 Dx(z) ,: y(M) + 6Dr() 0,

Dx(L) +. Dy(M) =. z

are developed in direction y (of the unloaded members).

By the agency of

= lx Y 1y Uxy = Xmx my' xy x TIy

the homogeneous, simultaneous system of..p artiaIdifferen-ti.l equations can. be resolved into two groups of two or-dinary differential equations with constant coefficients each •0

0

:jA 1 C... . T chnia]. : flemorandu o. 632 :1113

x(cpX) + 6 Dx() =.Q, Dx(X)1 for X and.

and. . . 1

r(4iY) .± 6.Dy(Tl) .: 0, . Dy(Y).=(..T for Y and.T,

wnerein the second group for Y and. 'fl is of prriiary i-t erest.

Bearing in mind the interconnections

.. . ; = (

A 2 /h.• ,. . . 5

(A2 ... difference of second order, e.g.:

•Y. .. Yx+i. 2 Yx . Yx_i),

wherein, - . .consta.nt -.- -

It

the equations

. ( y) +. 5 Dy() = 0', D(Y)-...•

can be . vrrit ten in the form.: -5

• • 5- S ••

5

ni1 (A2 +6) y1+6\21=0

= n X.J " Th

With S

= h and i = h2 X1.'1 '4'

we obtain the normal form - - - . - - -

(A2+6)Hi+6A2i=0

- :- ..: •- A2 — (k). '•• .. . . S S - -

- L:-. - -

with the bound.ary c-ond.tions for : .: •. .• 5- s -

14 N.A.C.A. Technical Memorand.um iTs. 662

y = 0 .. HT_1 y 9 ' : • . = =

The normalized solations. for y and' ofhs system f eqaations with sucn Doundary condito'is iay oc found in tne previously mentioned voluie of Bleich- elai The &esj d.v'a1ue :are .:..

Yj = H1 ......

To determine th so1utions for 'a T. of the Un-homogeneous equations, we first develop the load terms

and Txy according to the . pioper'resolutions 'ra:

Uxy = u)'; Ui()

=t'(); (x)

=Txy 'y.

In the construction of the ' uj(x) it is to be ob-

served that Ux1 = 0 for all y other than zero', hence

the sum itself reduces.to one single term; likewise in the construction of t'(x)...Txy = 0 .f.o all y other than zero. In addition, the reactions at the joints x 6,7 are here also to be treated as external' 'loads.

Now follows the calculation of and ' from

& X ( CPx 5E) + 6 Dx(1) = - Uj()

D(X) + X1t1 t = - ti(x)..

It being impoásible to give a general solution of this unhornogeneous simultaneous systemof ordinary differ-ential equations of the second order with variable coeffi.-cient in closed form, numerical resolution is resorted to, Resolved from the second equation and. written into the first j is eliminated, and X yields

& (ç' - 6 D D(X1 ) = - u(x) + 6

(a differential equation of the fourth order for Xi).

I'.A.b.A. TecbiicaX ;rnimio. 662

Specifically atpoiit

X X-2 ^ X Xl ^ 'X xX F. + + 'x+ -

= w1/ h 2 U() ± 6/i (tx+1 +

- - .;: '. •- r• wnerein

= - 6/12 c = 24/12 + h 9

= - 2 [18/12 - (cpx + +) w1 /4 h2]

This equation is to be posed. for x 1 in the n-pan-el; sytem.thefirst.tithe 'for x 11 - 1' the 1at time, and. for x for_all. i beteen._ Agreeably: to the..re&an d.ant unknowns X_ 1 , X 0 , X, X-- 1, there are four bound-ary conditions for the free boindariec: '

x = O,..X_ 1 = X 0 0 13...X13 = X 14 = 0.

Because of the symmetry of the supporting structurëti.' question, from the remaining twelve unknowns, it is para-mount that , ' . : -:

x l21 X 2 = X i1 X3,.'Xoetc.,

so that only tie equations from x .1 to x 6 have o be worked out. ' - -

This group of six equations, with six unknowns, is now developed for i 0, 1 1, 1 = 2 to_i -= 6 and. re-solved.,yield.ing the numericalvalues for •X1.

As to ' j, iecourseis had to the second equè.tonof

the unhothogeneous simultaneous system and. results in0

ii _ .:-Wtw1 fl/i (X+1 - 2 +) + '()];:-. •:: wherefrom can 'be readily computed. For I = 0 .... I = 1;' this fomula, however, leads to the unknown-a1i.te. per dent :.and. in:.thisrane i is defined'by'resoirig.:

the first equatiôn.of-the unhomogëneou simu1tàneüs-tern. In this manner the numerical values for Ek ' aré-'cdrnuted and all momentat the spar a±i&-the jointscanbe-e±presed according tothe f6rmuIas

16 N.ASC.A. Technical Memorandum No, 662

L 7 = • E X11 t'T.

The d.isplacement at the joints, developed from -

Z1 xy = . ti•i.

contain, in ieioI:assuming z 60 , z70 , z,' •.-z. = 0

a displacement (or torsion) which the grillage executes as rigid body and which is readily eliminated by 1ine in-terpolation. (This displacement (or torsion)sho-uld nt be confused with the previously cited hyperboloidal curva-ture of the complete s.truture.)

For the. second load, attitude (spar I loaded) the un-ho-meneous simultaneous eQuations read

o x(ipL) .+ 6 D.x(z) = -u, y . (M) :- SDy(z) =0

DX(L) + Dy(M)' =

whei n

tJy = 0 for a1. y = 0, 2, 3, 4, 5, 6:

= 6/i X 8/3 (p + Px+i) 4(Px + CPx+ i )for y = 1

and. . 0

9 or all y = 0, 2, 3, 4, 5, 6 T4 for y1.

The solutions of t.hoe hooeneous system of equa.tion are identtcal *ith thoe for 1od.in.g cnditionI. ,. The so-lutins X , and. Ti of the unhomo.geneous equations are a-fined as before and so also the d.ésired unknowns, The other load. attitudes (third., fourth .attitudQ) now can be e-mpted as the others, so that, because of the symmetry

the structure to the median spar, the moments at the joints, at the ris . and joint displacements are knwn for any possible load. attitude and. the influence lines can be ptted. The eva1uatio of thee lines with the p erti-ent load. attitude then yieldsthe thoments at the joints &. therefrom all stresses in the members,

The numerical data of the discussed.'four load. atti-

t'uL are tabulated. : and. apDeidedin-Tab1es..I to. IV.

ll.A.c.A. Techz.ical Iteinorand.um 1b.. 662 17

no combined effect were introduced b,r suitable ribs between the ina.ivithial spars, eacn spar would have to car-ry i'ts full rpoi'tioxi fhel°oad, and under these ondi-tions the moments at the spar joints are readily, computed

from Lx at

L 0 = 0, L 1 = - 8, L3 - 72, L 4 = - 128,

L 5 = - 200, L 6 = -. 288.

In order to represent clearly the results of the stat-icca1culation and for ready reference in a particular case the influence lines for all moments at spar joints, rib joints and displacements of joints must be drawn. Because of the symmetry of the beam to spar III, plotting of the influence lines is confined to those for spars 0 to III. In Figure 3 is given a sample of the actually drawn influ-once lines, those remaining being readily apparent from the data in the tables.

...A.side from the exact determination of the internal loads of the grillage, the exact amount of its deflec-tions is not only of interest from the general point of view, but it is rather ,precisoly these whica onable us to draw cincluions with an accuracy unatainah1e heretofore, regarding thL statical combined üffect. between grillage and a metal skin, be it of smooth or corrugated strip.

In a great majority of ruoden large cantilever or sem-icantilever airplanes the wing forms a lattice beam whose members consist of spars (from 2 to 10, according to spe-cia]. design) and ribs. Spars and ribs may be solid or lattice, the latter being more economical in most cases.

Loads in form of attached bodies, engines, propellers, controls, useful loads, wing braLng, etc., are impressed at arbitrary joints of the lattice beam (comprising both wings) in accordance with the particular case of a canti-lever, semicantilever, indirectly, or more or less direct-ly loaded wing. Such highly statically indeterminate lat-tice beams yald readily to accurate determination by means of differential e quations, regardless of type of

LA. C..A 'Tohii cal Memo i'thin •No.. .,6.2

• 'l.øa&,. c'haig'e 'b'' -inom.Onts o' 1nèrt'ia'. s's -'and 'ibs, spax pacing, bód' hmnt coridi ti on's , Becauso -the• load. vary, 1i use of influence hues is recommended,

In the present report the computatin isactua1ly carried through for the case of parallel' "sari of 'qual resistancein bending without direct loading, including plott'in:g f tthë nfl'cionco l.'ne for other cases the meth-od of calculation is explained. The develo pment of large 'size airplanes can be speeded : up br accurte iothods of calculation, such as this. '

Translation 'b J •Van'ior, ' ', :, '. , •" ' National Advisory Committee H ' ",'.' •

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Fit:s. 1,2