the stress distribution in a swept-back box-beam with_2837

7/29/2019 The Stress Distribution in a Swept-Back Box-Beam With_2837

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R. & M. No . 2837(13,903)A.R.C. Technical Report

M I N I S T R Y O F S U P P L Y

A E R O N A U T I C A L R E S E A R C H C O U N CI LR E P O R T S A N D M E M O R A N D A

T h e S tr es s D i s t r i b u t i o n in a S w e p t -b a c kB o x - b e a m w i t h P e r p e n d i c u l a r R i b s

ByP . B . H O V E L L , t3.Sc.

Crown Copy r i gh t Rese rved

L O N D O N " H E R M A J E ST Y 'S S T A T I O N E R Y O F F I C E1954

F I V E S H I L L I N G S N E T

~ , , ~ k : ~ ; . ~ . : '- ~ -a ~ : , ~ ~ ~,~.1~ ,~4.~ ~ , ~ ~,I . A T ! o . A L A E ~ o r u d f i ~ : , ; ~ : -E, .qT A ) I . . I ~ H M E N T

" ~ , , ,, k t ,, 4 , B ~ . u ~ . i

j / - . d O T i. % , - ',, ::


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h e Stress D i s t r ib u t i o n in a S w e p t - b a c kw i t h P e r p e n d i cu l a r R i b s

B o x - b e a mBy

P . B . H O V E L L , B . S c .C O M M U N I C A TE D B Y T H E P R I N C I P A L D I R E C T O R O F S C I E N T I F I C R E S E A R C H ( A I R ) ,

M I N I S T R Y O F S U P P L Y ~ - " ~ ~ ~ f ' ~ , ~ - ~ ~ : . . . . . . . ~,~4,'~ 4 -~ ( IOT I % 4R e po rts a n d M e m o r a n d a N o . 2 8 3 7

D e c e m b e r , z 9 5 oJ

s o l u ti o n i s o b t a i n e d f o r t h e d i s t r ib u t i o n o f i n t e r n a l l o a d i n a s w e p t - b a c k b o x - b e a m o f r e c t a n g u l a r- s e c ti o n u n d e r a n y s y s t e m o f e x t e r n a l lo a d i n g . T h e t h e o r y , b a s e d u p o n s t r a i n e n e r g y , i s c o n s i d e r ed e x a c t w h e ns o l u t io n f o r e n c a s t r6 r o o t c o n d i t io n s i s g i v en . E x t e n s i o n s o f t h e m e t h o d t o c o v e r r o o t f l e x i b il i t y a n d

e r i c a l e x a m p l e o f a s i m p l e t w o - s p a r b o x i s i n v e s t i g a t e d f o r a r a n g e o f s w e e p b a c k a n g l e o f 0 d e g . t o 4 0 d eg .a n e n c a s tr 6 r o o t c o n d i ti o n . T h e a n a l y s i s s h o w s t h a t a p p r e c i a b l e r e d i s t r i b u t io n s o f l o a d i n th e b o x , c o m p a r e de p t b o x , a r e o b t a i n e d w h e n e i t h e r t o r q u e o r b e n d i n g m o m e n t i s a p p l i e d t o t h e b e a m .

PhbAn0P

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k l , k ~ , k 3 , k 4k l ' , k ( , ha ' , k4 '

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t , , t s '

L I S T O F S Y M B O L SV e r t i c a l l o a dS p a r d e p t hD i s t a n c e b e t w e e n t h e s p a r sT h e t r a n s f e r c o e f f i c i e n t a t r t h r i bA n g l e o f s w e e pr i b p i t c hB o o m a r e a ( f r o n t S p a r )B o o m a r e a ( r e a r s p a r )R a t i o o f b o o m a r e a s i n b a y s 1 , 2 , 3 , 4 ( f r o n t s p a r )R a t i o o f b o o m a r e a s i n b a y s 1 , 2 , 3 , 4 ( r e a r sp a r )b t a n O /pY o u n g ' s m o d u l u s - b o o m m a t e r i a lM o d u l u s o f r i g i d i ty - s k i n s a n d w e b sT o p a n d b o t t o m s k i n t h ic k n e s se s

. E . R e p o r t S t r u c t u r e s 9 8 , r e c e i v e d 5 t h A p r i l , 19 51 .1


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t~

CC 'DD 'FF 'H tLIST OF S Y M B O L S - - c o n t i n u e d

Web thicknessesRib thickness, equal except for rib 1Rib thickness of rib 1

Defined in Tables 1 and 2

1 . I n t r o d u c t i o n . - - T h e problem of the stress distribution in swept-back wings has beeninves tigat ed in a numb er of reports. Wittr ick 1 (1948) dealt w ith regions of the wing surfaceremote from the root, on the assumption of rigid ribs lying in the direction of flight. The morepractical case of flexible ribs was investigated by Mansfield2 (1949) and He mp 3 (1950). The ydet erm ined the influence of rib-flange stiffness on the stringer sheet when t he ribs are at anglesother than 90 deg to the stringers. Their solutions involv ed the det ermi nati on of stressfunctions which satisfied certain boundary conditions, and are applicable only at distancesfrom the root of the wing. An approxi mate the ory for the effects of root restrai nt was alsogiven by He mp s assuming that rib webs are rigid in shear and tha t the mode of deformatio nin the restrained condition is similar to the mode with no restraint.The present report treats the problem from a different aspect and uses a minimum-energycriterion to de term ine the stress distribut ion in a t wo-spar box- beam for any system of loading

and tak es the root res traint into account. The solution is exact for a parallel rectangul arbox-beam in which flexible ribs are perpendicular to spars that carry the whole of the bendingloads in concentrated booms.2 . S t a t e m e n t o f t he Pr o b l e m a n d M e t h o d o f S o l u t i o n . - - F i g . 1 shows a rectangular two-sparbox-beam with an applied load at the intersection of the rth rib and the rear spar

D

/ /

Z

Front

R e a r Spa r, p

F IG. 1 . Two-s pa r box-beam unde r ve r t i ca l load P .

Spar


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e .g . , WXYZ,the front spar booms will be influenced by the flexibility of the spar CDXYof the rear spar ABWZ ; and conse quently in the distribution of the direct

ative ly simple solution can be found when the box is a two-spar beam. The reariffness of the box-b eam and the bend ing stiffness of the front spar. Thus a certaintion of the load P will be transferred to th e front spar, an d fur therm ore this transferonly be effected at t he ribs. Each rib can be assumed to transfer a proporti on ~ of the

s of P and Z~ (r = 1 to N ; N bei ng the numbe r of ribs). A solution is obtained byting the strain energy of this system and minimising the energy with respect to each

The Ap pl icat ion o f the Method to a Tw o-spar Rec tangular Box-beam (wi th ~o taper) wi thThe beam is of rectangular

The skins and webs carry shear load only and do not co ntrib ute to the bendin g str engt hThe root of the bea m is encastr6.The ribs are perpen dicula r to the spars, a nd are free to warp out of their planes. The yhment to the skins and webs, but cannot transmit a bending mom ent in theirThe transfer of load from one spar to the oth er can take place only at th e ribs.The distr ibutio n of loads will be such th at the strain en ergy of the sys tem is a minimum.Buck ling of the skin and web plates does not occur.

Discuss ion o f t he assumpt ions . - -The assumption (b) does not preclude a corrections. The choice of the equiv alent area to be added to the boom area will vary betwe en

ss. In th e case of a predo min ant ly torsional loading the wails will be bent and

should be appre ciated t hat a correction of one-half the skin area is not exact, for an error

the beam. The stress distribution obtained by the meth od will be most accurate when the

; if the ribs lie in a direction other th an at right-angles to the spars, the rib mome nt,

in comparatively simple terms would be most difficult. The assumption that thection to the spars is significant at rib 1 ,where additional terms wou ld

3


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4 . L o a d T ra n s fe r at a R i b . - - W h e n t h e l o a d , ~P i s t r a n s f e r r e d f r o m o n e s p a r t o t h e o t h e r ,t h e b e n d i n g m o m e n t w i ll b e t a k e n b y t h e s h e a r fl ow s i n t h e t o p a n d b o t t o m s k i n s o f t h e b o x -b e a m a s s h o w n i n F i g . 2 ; a n d t h e b e a m w i l l b e s u b j e c t t o t h e s e l f - e q u i l ib r a t i n g s y s t e m o f l o a d sa s s h o w n i n F i g . 3 .

h

F I G . 2 .

~ - ( s h e o r f l o w - l b . / i n . )

AP--fi-

S e l f - e q u i l i b r a t i n g s y s t e m o f l o a d s a t r ib .T h e h a n d l i n g o f s i m u l t a n e o u s e q u a t i o n s , e q u a l in n u m b e r a n d v a r i a b l e s to t h e n u m b e r o fr i b s w o u l d o b v i o u s l y b e u n m a n a g e a b l e , a n d a c o m p r o m i s e w o u l d b e r e q u i r ed . I t is t h o u g h tt h a t 5 r i b s ( i . e . , 6 w h e n t h e r o o t r i b i s i n c l u de d ) s h o u l d g i v e t h e l o a d d i s t r i b u t i o n w i t h r e a s o n a b l ea c c u r a c y a n d a ls o w i t h r e a s o n a b l e c o m p u t a t i o n . I t w o u l d b e a d v i s a b l e t o l e a v e r i b 1 (seeF i g . 3 ) a s d e s i g n e d a n d r e p l a c e t h e a c t u a l r i b s b y 4 r i b s o f e q u i v a l e n t t o t a l s h e a r s t i f f n e s s .T o r e t a i n a c e r t a i n d e g r e e o f g e n e r a l i t y t h e s p a r b o o m s w i l l n o t b e c o n s i d e r e d u n i f o r m i n a r e aa l o n g t h e w h o l e l e n g t h o f t h e b e a m , b u t v a r y i n g i n s t e p s a t e a c h r i b a n d b e i n g o f u n i f o r m a r e ai n e a c h b a y .

R i b 0

R i b IR i b 2

I ~/P/h Rib 3

.

R ib 4

' l/ / ~ / ~ " - - . ./

R i b 5

Front Sparh

11

~ - z .. .. . R e a r S p a rF I G . 3 . L o a d d i s t r i b u t i o n i n b o x - b e a m .

I t w i l l b e n o t e d t h a t F i g . 3 d o e s n o t i n c l u d e a n a p p l i e d l oa d , b u t i t is c o n v e n i e n t t o o b t a i na n e x p r e s s i o n f o r t h e s t r a i n e n e r g y o f th e b e a m u n d e r t h e s y s t e m o f l o ad s a s s h o w n a n d i n t r o d u c et h e e f f e c t s o f a n a p p l i e d l o a d a t a l a t e r s t a g e i n t h e a n a l y s i s .4


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4.1. Stra in Energy o f Direct Load in the Front Spar Bo om s . - -The r a t e o f c h a n g e o f d i r e c ti.e.,2(P/h)25x.T h e r e f o re s t r a i n e n e r g y o f b o o m b e t w e e n r ib s 4 a n d 5

_ _ 1 ( _ p / ]g ) 2 ( 2 ,~ 5 ) 2 k C X 2 d x- - ~ AE0

k 4-- 2AE (P/h)~P~{ 2~)2/3}"D i r e c t l o a d i n b o o m a t r i b 4 = 2(P/h),t~]).T h e r e f o re s t r a i n e n e r g y o f b o o m b e t w e e n r ib s 3 a n d 4

I ~ { 2 ~ ] ) + ( 2 ~~= ~ ( P / h ) 2 A - g + 225)x}2dx0

_ _ l ( p / h ) 2 ka ] ) a< ( 2 2 5 )2 @ ( 2 2 ~ @ 2 2 5 ) (2 2 5 ) @ ( 2 2 4 @ 2 ; ts ) 2/ 3 }~ J 2 ~s i m i l a r l y fo r t h e o t h e r b a y s , e x c e p t b e t w e e n r i b 0 a n d r i b 1 w h e r e t h e s t r a i n e n e r g y o f t h e

f b a n 0= (p /h ) 2 1.__AL , ( ( 2 ~ + 4 2~ + ~ 2~ + 8 X ~ ) ]) + ( 2 2 ~ + 2 ~ + 2~3 + 2 ~

d 0+ 2 x ~ ) ~ } 2 d x

]) 3---- ( P /h ) 2 ~ - ~ ( 2 2 2 + 4 2 3 + ~ + S ~ ) 2 z+ (225 + 42~ + 02, + 82~ )(2~ + 22 2 + 2;t3 + 2 ~ + 22~)z2+ (22~ + 222 + 22~ + 2,l~ + 22~)2z~/3}

e r e z = b t a n 0/]).4.1 .1 . Stra in energy o f d i rec t load in the rear spa r boom s . - -In a s i m i l a r m a n n e r t h e s t r a i ny o f t h e r e a r s p a r b o o m s c a n b e e v a l u a t e d ; a t y p i c a l e x p r e s si o n , f o r t h e b o o m b e t w e e n

])3= } (P / h )2 ~ k ' 2 ( ( 2 ~ , + 42 ~)2 + ( 2 2 , + 4 2 ~ ) ( 2 ~ + 2 ~ , + 2 4 )

+ ( 2 2 3 + 2 ~ , + 2 ~ ) ~ / 3 } .4.2 . Stra in Energy o f the Top and Bottom Sk ins in S hea r . - -The s h e a r s t r e s s i n t h e t o p a n d(1/ts)(2~)(P/h), (1/ts)(2 + 25)(P/h) , e t c. , a n d t h e s t r a i n e n e r g yp a n e l s w i l l b e (P/h)2b])/Gts (2~)2, e t c ., a n d f o r t h e t r i a n g u l a r p o r t i o n wi l l b e (P/h)2b])/Gt,+ 2 2 + 2 ~ + 2 , + ~ ) 2 2 / 2 ] .4.3 . Stra in Energy o f the Spar Webs in Shear . - -S imi lar ly t h e s t r a i n e n e r g y o f t h e s p a r w e b se v a l u a t e d , a t y p i c a l t e r m w h i c h re f er s to t h e f r o n t s p a r w e b i n t h e b a y b e t w e e nn d 3 w i l l b e

_ (p/h ) ~ hi ) (23 + 2~ + 2~) 35


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a n d f o r t h e w e b b e t w e e n r i b s 0 a n d 1 w i l l b e= (p / h ) ~ h p (~1 -- ~ + h3 + 4 , + Z~)Yz.

4.4 . Strain Energy of the Ribs . - -Since t h e b e n d i n g m o m e n t d u e t o 2 P i s p r o g r e ss i v e ly r e a c te db y c h o r d w i s e s h e a rs i n t h e t o p a n d b o t t o m s k i n s t h e r e w i l l b e n o b e n d i n g m o m e n t i n t h e r i ba n d t h u s t h e c o n t r i b u t i o n b y t h e r i b t o t h e s t r a i n e n e r g y o f t h e s y s t e m w i l l b e t h a t d u e t o au n i f o r m s h e a r o f P / h . p e r u n i t l e n g t h , i.e.,bh ~.S t r a i n e n e r g y o f t h e r i b = ( P / h ) 2 ~ -

5. Total St rain Energy of the Bo x-B ea m .- -Th e t o t a l s t r a i n e n e r g y U o f t h e s y s t e m c a n b ew r i t t e n f r o m t h e a b o v e e x p r e s s io n s a s f o ll ow s : - -

U = (P/h) ~ [k~(2~)~/3 + k~{(2Zs)~ + (2;.~ )(2~ + 2;~5) + (2~,~ + 2;,5)~}+ k~{(22~ + 4; .~)~ + (2~t~ + 4;~)(2~.~ + 2;~ + 2, ~ ) + (2;,~ + 2,~,, + 2;~)~ /3}+ k~{(2;~ + 4;~ + 6;~) ~ + (2; ,~ + 4~.~ + 6; ,~)(2;.~ + 2;~ + 2 ~ + 2~.~)+ (2~,~ + 2~,~ + 2;t~ + 2~,~)~/3}+ { (2 ~ + 4;~ + 6;t~ + 8~ )2z + (2~2 + 4~.~ + 6;~ + S~.~)(2;~ + 2;~ + 2~.~ + 2~.~ + 2~,,)z~+ (2~ + 2~2 + 2;% + 2 ~ + 2~)~z /3}]+ (P/h) ~ [ k ~ ' ( 2 , ~ ) ~ / 3+ ks ' ((22~ ) ~ + (22~)(2Z~ + 2).~) -- (2X~ + 2;.~)~/3}+ k2 ' ((2 ,t~ + 4~ ) ~ + (2 ~ - ? 42~)(22s + 2 ,~ + 2~ ) -- (22~ + 2 ;t~ + 22~)~/3}+ k~'((22~ + 42~ + 6~ ) ~ + (2,~ + 42~ + 62~)(2;~ + 2 ~ + 22~ + 22~)

~(P/h)~-~Yt, ~ ~ + (~ + ~)~ + (~ + ~, + ~)~ + (~ + ~ + ~, + ~)~

~ ( P / h ) ~ , [ ~ ~ + (~+ (~ + ~

+ (P/~)~-t~[~~ + (~+ (~ + ~

~ ( P / h ) ~ , [ ~ + (~1 p h y h b

+ ~ + ~ + ~)~z/21+ ~o)~ + (~ + ~ + ~)~ + (~ + ;~ + ~o + ~)~+ ~ + ~ + ~)~z/21+ ~)~ + (~ + ~ + :~)~ + (~ + ~ + ~ + ~)~

+ ~ )~ + (~ + ~ + ~)~ + ( ,h + :~ + ~ + ~)~1

6


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T h e a b o v e e x p r e s si o n c a n b e r e g a r d e d a s t h e ' g e n e r a l ' e x p r e ss i o n fo r t h e s t r a i n e n e r g y o f. T h e f ir st a n d s e c o n d s q u a r e b r a c k e t s r e f e r t o t h e f r o n t a n d r e a r s p a r b o o m sy , th e t h i r d a n d f o u r t h g i v e t h e t o p a n d b o t t o m - s k i n c o n t r i b u t i o n s , t h e f if th a n dg i v e t il e c o n tr ib u ~ d o n s of t h e f r o n t a n d r e a r s p a r w e b s a n d t h e s e v e n t h b r a c k e tt h e c o n t r i b u t i o n s o f t h e r ib s .

T h e p r o c e d u r e a d o p t e d t o o b t a i n t h e s t r a i n e n e r g y f o r a l o a d P a p p l i e d o n t h e f r o n t s p a r a ti s to r e p l a c e 2 n 2~ b y (2 nZ ~ - - n ) a n d 2 , b y (2 ,. - - ! ) i n t h e f r o n t s p a r b o o m a n d f r o n t s p a r w e bt i o n s r e s p e c t i v e ly , a ll o t h e r t e r m s r e m a i n i n g a s w r i t t e n i n th e g e n e r a l ex p r e s s io n .w h e n a l o a d P i s a p p l ie d t o t h e r e a r s p a r t h e g e n e r a l e x p r e ss i o n f o r t h e s t r a i n e n e r g ya i n s u n a l t e r e d e x c e p t (2 nX , - - n ) r e p l a c e s 2 n ;~ , a n d (2 , - - 1) r e p l a c e s 2~ i n t h e r e a r s p a rm a n d w e b c o n t r i b u t io n s r e s p e c t i v e ly .

T h e s t r a i n e n e r g y , U , i s m i n i m i s e d f o r ~ 1 22 . . 25 a n d t h e e q u a t i o n s OU / ~ 2 , = 0 [ r = 1 ,. . 5 ] c a n b e s o l v e d f o r 21 . 25. I t w i l l b e r e a l i s e d t h a t t h e r e p l a c i n g o f 2 2 , b y (2 2, - - 1 )n o t a f f e c t t h e c o e f f i c ie n t s o f 2, i n t h e e q u a t i o n s ~ U/~A~ = 0 a n d h e n c e t h e c o e f f ic i e n ts o f( s : 1 . . . 5 ) i n t h e e q u a t i o n s ~ U l n a , = 0 (r = 1 . . . 5 ) w i l l b e t h e s a m e f o r a l l p o s i t i o n st h e a p p l i e d lo a d . T h u s t h e p o s i t i o n o f t h e a p p l i e d l o a d w i l l o n l y a f fe c t t h e c o n s t a n t t e r m o fq u a t i o n s ; t h is f e a t u re , w i l l c o n s i d e r a b l y r ed u c e t h e c o m p u t a t i o n i n v o l v e dt h e s o l u t io n o f a b o x - b e a m u n d e r a s y s t e m o f d i s t r i b u t e d l o a d s. T a b l e s 1 a n d 2 g iv e t h ef ic i en t s a n d c o n s t a n t t e r m s o f t h e s i m u l t a n e o u s e q u a t i o n s .

6 . T h e E f fe c t o f a C u t - o ut i n th e B e a m . - - A s i m i l a r m e t h o d o f a t t a c k c a n b e a p p l i e d t o d e t e r m i n et o n ti le lo a d d i s t r i b u t i o n o f a c u t - o u t b e t w e e n t w o o r m o r e r i b s i n t h e t o p o r b o t t o mn . I n s u c h a c a s e n o t o r q u e c a n b e t r a n s f e r r e d t o t h e r o o t a c ro s s t h e c u t - o u t b y s h e a r s ino p a n d b o t t o m s k i n s. T h i s i n fe r s t h a t t h e t r a n s f e r a t t h e o u t b o a r d r i b o f t h e c u t - o u t m u s th e t r a n s f e r t h a t h a s b e e n e ff e c te d o u t b o a r d o f t h i s r ib . I t w il l b e n o t e d t h a t t h es f y i n g o f t h e f o l l o w i n g r e q u i r e m e n t s w i l l b e a u t o m a t i c , i . e . ,(a) T h e t o t a l v e r t i c a l s h e a r w i ll e q u a l t h e v e r t i c a l l o a d s a p p l i e d o u t b o a r d o f t h e r ib .(b) T h e m o m e n t o f t h e f r o n t s p a r w e b s h e a r a b o u t t h e r e a r s p a r w i ll e q u a l t h e m o m e n t o f

t h e lo a d s a p p l i e d o u t b o a r d o f t h e r ib a b o u t t h e r e a r s p ar .I n g e n e r a l t e r m s , i f t h e o u t b o a r d r i b o f t h e c u t - o u t i s t h e r t h o f t h e N r i b s th e c o n d i t i o n t o

N2,, - - ~ 2,.S= r+ t

S u b s t i t u t i o n i n t o t h e g e n e r a l s t r a in - e n e r g y e x p r e s s i o n s i m i l a r t o s e c t io n 5 w i ll r e d u c e t h em b e r o f v a r ia b l e s t o N - - 1, a n d t h e m i n i m i s i n g o f t h e f u n c t i o n U w i t h r e s p e c t to ~, ( S = 1,r - - 1 , r + 1 , N ) w i l l g i v e th e n e c e s s a r y e q u a t i o n s f o r t h e s o l u t i o n o f t h e p r o b l e m .7 . T h e E ff e c t o f F l e x i b i l i ty a t t he R o o t . - - T h e s t r a i n - e n e r g y f u n c t i o n h a s b e e n f o r m u l a t e d

e n c a s t r 6 c o n d i t i o n s a t t h e r o o t. S i n c e t h e d i s t r i b u t i o n i n t e r m s o f l o a d a t t h e r o o t is k n o w ns n o t d i f fi cu l t t o a d d e x t r a t e r m s t o t h e g e n e r a l s t r a in e n e r g y e x p r e s s i o n t o i n c l u d e t h e e f fe c te i t h e r a x i a l l o a d f l e x ib i l it y o r s h e a r l o a d f le x i b il it y . I t w o u l d b e n e c e s s a r y t o e s t i m a t e t h ei o n d u e t o u n i t l o a d a n d o b t a i n t h e s t r a i n e n e r g y d u e t o th i s fl e x ib i li ty . T h e a d d i t i o n a ls w o u l d b e m i n i m i s e d w i t h r e s p e c t t o 2 r (r = 1 t o N ) a n d i n c o r p o r a t e d i n t h e c o e f fi ci e nt sc o n s t a n t s o f t h e ~ U/~2r ---- 0 e q u a t i o n s g i v e n i n T a b l e s 1 a n d 2 .8 . T h e A p p l i c a t i o n o f t he M e t h o d to a N u m e r i c a l E x a m l b l e . - - T o i l l u s t r a t e t h e e f f e ct s o f s w e e p -o n t h e l o a d d i s t r i b u t i o n , a b o x - b e a m is c o n s i d e r e d f o r a n g le s o f s w e e p f r o m 0 d e g to 4 0 d e g .o m e t r y o f t h e b e a m i s s h o w n in F ig . 4 a. T i le s o l u t i o n of t h e s i m u l t a n e o u s e q u a t i o n se n a lo a d P is a p p l ie d a t e a c h r ib a t t h e f r o n t a n d r e a r s p a r s a l lo w s t h e t r a n s fe r - c o e f f ic i e n t

g . 4 b t o b e c o m p l e t e d .7


9/20

The substitution of the appropriate values of 2 in the expressions of 4.1, 4.1.1, etc. determinesthe spar boom loads and the web shear loads for the various loading conditions.Figs. 5 and 6 show the distribution of direct load in the booms for bending and torque loadsrespective ly applied at rib 5. Figs. 7 and 8 show the spar web shear loads under similarconditions.A check of the present method is obtained by comparing the distribution at zero sweep withthat derived by Williams' method of R. & M. 1761.8 .1 . R e m a r k s o n th e R e s u l t s . - - R e f e r r i n g to Figs. 6, 9 and 10 it will be seen that the intro-ductio n of sweep into the box- beam affects the root const raint for torsional loading. For atorque applied at rib 5 the axial constraint boom loads at the rear-spar root are reduced, at40 deg sweep, to 60 per cent of their value at zero sweep and furthe rmore the reductio n is approx-ima tel y linear with the angle of sweep. It is of interest to note, however, tha t the spar webshear loads, Fig. 8, are hardly affected at the root by the sweep, and at 40 deg sweep are stillmore than 90 per cent of their value at zero sweep.Figs. 5 and 7 show that sweep also materially affects the distribution due to bending loads ;when these loads are applied at rib 5, the rear-spar boom direct load is increased by more than33 per cent for 40 deg of sweep, a nd th e rear-spar web sh ear load is increased by app roxi mate ly50 per cen t for 40 deg sweep. The front spar has corresponding decreases.Fig. 11 is of interest in tha t it shows very good agreement, at zero sweep, with the dist ributio nof direct load in the spar boom s as calcu lated by Wil liams ' met hod of R. & M. 1761. Willi ams'theory was based upon an assumption of ' rigid ribs of infinite number ' and curve II is obtainedby the present method with an assumption of infinitely stiff ribs.The effect of the flexibility of the end rib ( i . e . , rib 5) in the distribution of the spar boomloads is shown by the curves II I and IV of Fig. 11. The torque is applied as differentia l shearsto th e webs at rib 5 ; and t he flexibility of tha t rib will ten d to produce larger differential shearsin the webs inboard of rib 5 than those induced, by the root constraint effect, when the torqueis applied as Batho shear.9 . C o n c l u s i o m . - - T h e effect of sweepback in a rectangular box-beam (no taper), with ribsperpendic ular to the spars and in which the bend ing loads are taken b y spar booms is considered.On the assumption that transfer of load between the spars can only be effected at the ribs, itis shown that the strain energy of the box-beam can be expressed in terms of the applied load,the box geome try and N variables (N being the numb er of ribs). The minimising of the strain-energy function with respect to each rib variable produces N linear simultaneous equations ofN unknowns, t he solution of which will give the load distri bution in the beam.It is considered that sufficient accuracy can be obtained by replacing the actual ribs with5 ribs of equivalen t shear stiffness, and tables are i ncluded in the report to enable the coefficientsand constants of the 5 simultaneous equations to be quickly evaluated for encastrfi root con-ditions.It is also indicated h o w the m e th od can be ad apt ed to include the effects of axial and/or shear

flexibilities at the root. T h e pr ob le m of a cut-out in the top or b ot to m skin is also discusseda nd the load distribution in te rm s of the rib transfer coefficients for a typical case is included.

A num eri cal ex am pl e is investigated an d the transfer coefficients are included in the report.W h e n a t o rs i on a l l o a d! n g is a p pl i e d to t he b o x - b e a m t h e d i re c t l o ad s in t h e s p ar b o o m s a t t h er o ot { a s s u m e d e n c a s t r e i n t h e e x a m p l e ) a r e s h o w n t o b e c o n s id e r ab l y r e d u c e d f or l a r g e a n gl e so f s w e e p w h e r e a s t h e s p ar w e b s h e a r s a t t h e r o o t a r e n o t m a t er i a ll y a ff e ct e d. F o r t h e s a m eb e a m i n b e n d i n g t h e re a re s ig ni fi ca nt c h a n g e s o f b o t h s p ar b o o m a n d s p a r w e b l o a ds at t h e ro ot ,

8


10/20

a s a c o n s e q u e n c e o f t h e s w e e p - b a c k t h e r e a r s p a r i s m o r e h e a v i l y l o a d e d t h a n t h e f r o n t s p a r .s p e c ia l ca s e o f z e r o s w e e p b a c k a c o m p a r i s o n i s m a d e w i t h t h e d i s t r i b u t i o n o f l o a d in t h em s f o r a t o r s i o n a l l o a d i n g o b t a i n e d b y t h e c l a s s i c a l m e t h o d o f R . & M . 1 7 61 a n d

e n t i s o b t a i n e d .

W . H . ;v V ittr ick . . . . . . . .E . H . M ansfield . . . . . . . .W . S . H e m p . . . . . . . .D . W illiam s . . . . . . . .

R E F E R E N C E STi t l e , e t c .

Pre l imina ry ana lys is of a h ighly swept cy l indr ica l tube under to r s ionand bendin g. A.C.A. 39. Ma y, 1948.Elas t ic i ty of a shee t r e inforced by s t r inge rs and skew r ibs , wi thappl ica t ions to swep t wings . R . & M. 2758. Decem ber , 1949.On th e appl ica t ion of ob l ique co-ord ina tes to problems of p lanee las t ic i ty and swep t-back wings . R . & M. 2754. Jan uary , t950 .The s t r e sses in ce r ta in tubes of r ec tangula r c ross- sec t ion undertorque . R . & M. 1761 . May, 1936.

9


11/20

T A B L E 1

Coeff icients of X De term ina nt C = ( P / h ) ~ , = ( P/ h) 2 D = ( P / h ) ~, F ' = b p b h )F = (P/h) (P/h) H = (P/h)

D' = (P/h) ~ It])' t ' . G

21 22 23 24 25C z 8 + 2D z 4Cz ~ -[- -~Cz~ -+- 2D z 8Cz" + ]C z 3 + 2D z 12Cz ~ + -]Cz3 + 2D z 16Cz ~ + SCz ~ + 2 D z = R+ ( F + F ' ) z + 2 H o ~ + ( F + F ' ) z + ( F + F ' ) z + ( F + F ' ) z + ( F + F ' )z4Cz ~ + Cz 3 + 2D z+ F z + F ' z

8Cz ~ + ~Cz ~ + 2D z+ F z + F ' z

-]Ck~ + 8Cz + 8C z ~+ ~ C z ~ + 2 D + 2 D z+ 2 F + F z + C ' k 1'+ 2 D ' + 2 H + 2 F ;+ F ' z~-Ckl -t- 16Cz+ 12Cz ~ + 8Cz~+ 2D + 2D z -+- 2F+ F z + ~ C ' k /+ 2 D ' + 2 F ' + F ' z

%-Ckl + 16Cz+ 12Cz 2 + SCz a + 2D+ 2 D z + 2 F + F z+ ~- C'k / -- 2D"+ 2 F ' + F ' z] Ck~ + %~Ckl + 3 2 C 2+ 16Cz 2 + ] Cz 3 + 4D+ 2 D z + 4 F + F z+ - ] c % ' + c ' k /+ 2 H + 4 D ' + 4 F '+ F ' z

~2-Ckl -]- 24C z + 16C z 2+ SC z ~ + 2D + 2D z+ 2 F + F z + ~2-C 'k1 '+ 2 D ' + 2 F ' + F ' z

~-Ck2 + ~2-C 1 + 48C,~+ 20Cz ~ + SCz ~ + 4D+ 2D z -- 4F + F z+ ~ C ' k ( + % ~ C 'k /+ 4 D ' - / 4 F ' + F ' z

~ ! C k l + 3 2 C z - [- 2 0 C z ~+ C z ~ + 2 D + 2 D z+ 2 F + F z + ~ i C ' k l '+ 2 D ' + 2 F ' + F ' z

~zCk~. + !2_s_ rb3 " J ' ~ l+ 6 4 C z + 2 4 C z ~+ C z ~ -+ 4D -- 2Dz

4 F + F z + ~ - C 'k ~ '+ ! ~ a C ' k l ' + 4 D '+ 4 F ' + F ' z

= S

- - - -T


12/20

T A B L E 1 - -c o n t i n u e d

2~ 2~ 2~ 2~ 2~12Cz ~ + ] C z ~ + 2 D z+ F z + F ' z

16Cz ~ + ] C z ~ + 2 D z+ F z + F ' z

~ a C k l + 2 4 C z+ 1 6 C z ~ + ]C z ~+ 2D + 2 D z + 2 F+ F z + ~ - C ' k /+ 2 D ' + 2 F ' + F ' z

~4-Ck~ + 32C z

~ C k~ + ~- C k~+ 4 8 C z + 2 0 C z ~+ C z ~ + 4 D + 2 Dz+ 4 F + F z+ ~ o - c ' k / + ~ c ' k /+ 4 D ' + 4 F ' + F ' z

]Ck ~ + ~-Ck~+ ! ~ - C k ~ + 7 2 C z+ 2 4 C z 2 + C z 3+ 6 D + 2 D z + 6 F+ Fz + -]C'k~' .- / 2 H + 6 D ' + 6 F '+ F ' z

~ C k~ + ~2-C ~

~-Ck. + 92-Ck~+ 2-2-Ck~ + 96 Cz+ 2 8 C z ~ + ] C z ~+ 6 D + 2 D z + 6 F+ Fz + ~--c 'k/+ .-3-,-, ,~ + ~-2-C'k/+ 6 D ' + 6 F ' + F ' z

+ 2 0 C z ~ + ]C z ~+ 2 D + 2 D z + 2 F+ Fz + ~-C'k~'+ 2 D ' + 2 F ' - ? F ' z

+ 6 4 C z -? 2 4 C z ~+ ] C z 3 + 4 D + 2 D z+ 4 F + F z+ ~ - C ' k g + z ~ - C ' k /+ 4 D ' - ? 4 F ' + F ' z

+ 2-~-Ck~ + 96 Cz+ 2 8 C z ~ + ] C z ~+ 6D + 2 D z + 6 F+ Fz + ~-.C'k.'+ ~-C'k( + ~-~-C'k/+ 6 D ' + 6 F ' + F ' z

- - 1 2 8 C z - I - 3 2 C z 2+ C z ~ + 8 D + 2 Dz+ 8 F + F z + 2 H+ ~ c ' k / + ~ c 'k ~ '+ i~2-C'k~' + ~-~-C'k/+ 8 D ' + 8 F ' + F ' z

= U

= V


13/20

T A B L E 2Constant of ~ Determinant

bO

L o a d a p p l i e da t

Rib 1 F /SRib 1 R/SR i b 2 F / S

Rib 2 R/SRib 3 F /S

Rib 3 R/S

R

~Cz 3 -+- 2Dz0

2Cz ~ + ~Cz ~ + 2Dz

04Cz ~ + -~Cz3 + 2Dz

0

S

2Cz 2 + -~Cz3 + 2Dz0

-~Ckl @ 4Cz + 4Cz ~+ -~Cz3 + 2D'+ 2Dz~C'&' + 2D'-Ck~ + 8Cz+ 6C z ~ + - ~Cz+ 2D + 2Dz

~C'k~' + 2D'

T

4Cz 2 + ~Cz 3 + 2Dz0

-Ck~ + 8Cz+ 6Cz ~ + ~Cz ~+ 2D + 2Dz9C'k1' + 2D'~CG + %s-Ck~+ 16Cz + 8Cz ~+ ~Cz + 4D-- 2Dz~c '&' + ~-c '& '+ 4 D '

U

6Cz ~ + ~Cz 3 + 2D20

-6Ck~ + 12Cz+ 8Cz ~ + ]Cz ~+ 2D + 2Dz-6C'k~' + 2D '-CG + ~6-Ckl+ 24Cz + IOCz ~-- ]Cz ~ + 4D-- 2Dz~o c'& ' + ~ - c ' & '+ 4 D '

V8Cz ~ + ~Cz s -+- 2D2

0232Ckl + 16CZ+ ]OCz~ + ~Cz ~+ 2D-~ 2Dz~-2C'k1' + 2D'

+ 32Cz + 12Cz 2+ ~Cz 3 -+- 4D+ 2Dz~-c'&' + ~c%'+ 4D'


14/20

TABLE 2 - - c o n t i n u e d

Load appliedatRib 4 F/S

Rib 4 R/S

Rib 5 F/S

Rib 5 R/S

R6 C z ~ + ~ C z ~ + 2 D 2

0

8Cz ~ + -~Cz~ + 2 D z

S

1-~--Ckl + 12 C z+ 8Cz ~ + -~Cz~+ 2 D + 2 D z

~ -C ' k~ ' + 2 D '

~ - C k l + 1 6 C z+ l O C z ~ + { C z ~+ 2 D + 2 D z

~2.C'k1' + 2D '

T

C G + ~ a C k l+ 2 4 C z + IO C z2+ -~Cz + 4D+ 2 D z-C'k; + ~-C'k~'+ 4 D '~6-Ck~ + ~- C k~+ 3 2 C z + 1 2 C z ~+ -~Cz~ + 4 D+ 2 D z

5 ~ - C ' k ; + ~ - C% '+ 4 D '

U

-~Ck~ + ~ C k2.+ ~6-Ckl + 36Cz+ 12Cz ~ + -~Cz+ 6 D + 2 D z~C'k ; + ~-C'k~ '+ ~6-C'k~ ' + 6D '

+ z 6 -C k l + 4 8 C zq- 14Cz 2 + ~Cz 3+ 6 D + 2 D z~-c 'k~' + ~- C'k ;+ 6-C'k '~ + 6D '

V

C k~ + ~6-C ~+ z 6 -C k l + 4 8 C z+ 14Cz ~ + -~Cz+ 6 D + 2 D z C ' kd + ~ - C ' k (+ l__O_~F',b"~ ~1 + 6 D '{Ck~ + ~s-Ck~+ ~6_Ck~ + ~_~_8_rb+ 6 4 C z + 1 6 C z 2+ - ~ C z + 8 D + 2D 2~C'k~' + ~-C'k~'+ ~ 6 - C ' k ( + ! ~ - C k l .+ 8 D '


15/20

I

l~IB 31 ~ 1 8 4 -

I~IB 5 FI~ONTS P A R

B O O M A I~ E A = 4 I N S ~-R I B T H I C K N E S S - 0 . O 4 . I N SS K I N T H I C K N E S S - O, C) ~ I N SW E ~ T H I C K N E S S = ( 3 .1 0 I N S

SPAI~

F IG . 4 a . G e o m e t r y o f b o x - b e a m i n n u m e r i c a l e x a m p l e .

%

l ] . . . . . . . . IIo o o o o - -' T-o o o o coo o o

:o ~ - i o,o - ~ - ~oo o o o o o o o o o o

,~ oo o~, 2o o .o o o o o o " 6 c~ 6c~ ; o o o. . . . . . 77

o o o o o o o o o o o o i o o o o o

- ~ ~ 1 . . . . . ~o " I ' - :o~

o o o o o o o o o o o o o o o o o o o o o o o o

o o ~ o o o o o o o o o o o o o o o oo o o o o o o o o o o o o o o o o o o o o o

. . . . . . 1 ~ ~ . . . . I . . . . . . .o o o o o o o o o o oo o o o o o o o o o T ? T . . . . . . . . . . . . . . . . . .., , , ,

I I , o ~ ~o o . . . . ~ o o ~ o ~ ~c ~ o o, oo i~? ? ? ? ? o o o ? ? ? ? ? . . . . o ? ? 7 7

1 1 ] 1 1 I I I l l ~ ' - ~ . . . . . - ~ - ~ - -~ ' ? ~ ' ? 7 ? ? ?

-i ca ~ ~ aa ~ ~a a~ 0a ~a a~ ca ~ ~ ~ aa-

o o o o o o o o oo

o o ? o ? o ? o ? o 7 ? ? ~

~a ca ~ an aa a~ a~ aa a~

cd~J

J

o0' 0000

M

1 4


16/20

BOOMLOAD

JOP

I ' - - . . .

F"-'-.~. II - - - " - ' 4 " - '- - - - ' ~ '" - - - - - ~ pI

. . . . I

4. P

P.P

P OS I T ION OF R OOT( F~ON T S P A R ) R IBFIG. 5.

- ~ _ ~R E A R

RIB

R I B 5 v

S P A R 0 SWEEPBACK. . . . . . . . . I O ~ S W E E P B A C K- - ~ O S W E E P B A C K3 0 S WE E P B A C ~~.OSWEEPBACK

f R O N T SPAR

I / i -,.-,iRIB 2 RIB 3 RIB 4- RIB 5Boom-loads. Bend ing loads applied at rib ,5 .

3 P

~ PR IB 3 ""-,..JRIB R@5

R E A R S P A ~ ~ p

P OS IT I ON OF" R OO T ( ; R ON T S P A R ) te ' 3 0 e ' ~ o e f l o R p B J .

t-

I " 1 .

-3,{:FIG. 6 .

R i B ~ R I B 3 RIB4-

F R O N T S P A Q

O ~ S W E E P B A C K. . . . . . . I O S WE E P B A C K

- - 2 0 = S W E E P B A C K- - 3 0 S W E E P B A C K

. . . . . 4 0 ~S WE E P B A C K

Boom loads. Torsional loads applied at rib 5.

15


17/20

2.5P

1"SP

WEB5NEA~ LOAOAT ~OOT I'OP

'SP

O o

LOADS APPLIED AT RIBS..+

LOADS APPLIED AT RIB 3.2;....~,.~.-----'S/ REAR SPAR WEB.

~ F R O N T SPARW E B .LOADS APPLIED AT ~IB 4.~ /LOA DS AP#LIED AT RIB 5.

I 1 itOo 9.0 30 400AN~LE OF' SWEEPBACK,

Fig. 7. Spar web loads at root due to bending loads.

'!'-I.OP

W EBSHEAR LOADAT ROOT~0,5~

~ L O A O S APPUED AT RIB 3.~"-LOAD5 APPLIEDAT RJB4,

~ L O A D S APPLIEDAT RIB 5.SHEAR F ALL TONGUEWERE TAKEN BY RATHOOE. NOROOT RESTRAINT)

FIG. 8.Ji 1

i 0 o - ~_Oo 3 0 ' ,4.0oSpar web loads at root due to torsional loads.

16


18/20

3 p

Fro. 9 .

BOOMLOADS

" ' . ) ~ . . . .

/- 3 g

Boom loads for equal torques a pplied at ribs 2, 3, 4 and 5. Zero sweep.

F~G. 10.

BOOMLOADS g

-IP

-2P

/ //

Boom loads for equal torques applied at ribs 2 , 3, 4 and 5. 40 deg sweep.

17


19/20

P_.OPA L L R IB S IN FIN IT EL Y S TIF f. . . . .

,I.OP

"FR R IB 5 ~'LE~JBLE/ O T H E ~ I ~I BS F L E X I B L E .

W'~LLJAMS (R & b4 1761)

__ "Nr . R IB 5 IN~ ' IN ITELY ST I~ '~"O T N E R ~ 1 8 S g ' L S X I S L E .

RIB IF I G .

I~t s z mg3 ~b~, ~J8 51 1. A c o m p a r i s o n w i t h R . & 3 { .1 7 61 a t z e r o s w e e p b a c k o f d i s t r i b u t i o n i nb o x - b e a m f o r t o r s i o n a l l o a d i n g .

J430 2 Wt.17/6b~0 N,8 1/54 D&C o. 34/2631S

PRINTED IN GREAT BRITAIN


20/20

R o & M o N o o 2 8 3 7

Pu b l i c a t i on s o f th eA e r on au t i c a l R e s e ar c h C ou n c i l

A N N U A L T E C H N I C A L R E P O R T S e l f T ~ A E R O N A UT I[ CA L R E S E A R C H C O U N C K ~(~ousre vo zu ~ s)1 9 3 6 V ol . I . A e r o d y n a m i c s G e n e r a l , P e r f o r m a n c e , A i r s c re w s , F l u t t e r a n d S p i n n i n g . 4 o s . ( 4 1 s . l d ).

V o l . I f. S t a b i l i t y n d C o n t r o l , S t r u c I n r e s , S e a p l a n e s , E n g in e s , e t c . 5 o s . ( 5 1 s . l d .)1 9 3 7 V ol . I . A e r o d y n a m i c s G e n e r a l , P e r f o r m a n c e , A i rs e r e w s , F l u t t e r a n d S p i n n i n g . 4 o s . ( 4 1 s . l d .)

V o l . I I. S t a b i l i t y n d C o n t r o l , S t r u c t u r e s , S e a p l a n e s , E n g i n e s , e t c . 6 o s . ( 6 1 s . l d .)1 9 3 8 V o l . I . A e r o d y n a m i c s G e n e r a l , P e r f o r m a n c e , A i r s c r e w s . S O S . ( 5 1 s . l d .)

V o l . I I. S t a b i l i t y n d C o n t r o l , F l u t t e r , t r u c t u r e s , S e a p l a n e s , " W i n d T u n n e l s , M a t e r i a l s. 3 o s . ( 3 I S . d . )1 9 3 9 V o l . I . A e r o d y n a m i c s G e n e r a l , P e r f o r m a n c e , A i r s c r e w s , E n g i n e s . 5 o s . ( 5 s . l d .)

V o l . I i. S t a b i l i ty a n d C o n t r o l , F l u t t e r a n d V i b r a t i o n , I ns t r u m e n t s , S t r u c t u r e s , S e a p l a n e s , e t c .63 s. (64s. 2d.)1 94 o A e r o a n d H y d r o d y n a m i c s , A e r o fo i ls , A i rs c re w s , E n g i n e s , F l u t t e r , I c i n g , S t a b i l i t y a n d C o n t ro l ,S t ruc tu re s , a nd a mi sce l l aneous sec t ion . SOS. (5IS. i d . )

1941 Aero and Hydrodynamics , Aero fo i l s , Ai r screws , Eng ines , F lu t t e r , S t ab i l i t y and Con t ro l , S t ruc tu res .63 s. (64 s. 2d.)1942 Vo l . I . Aero and H ydr ody nam ics , Aero fo il s , Ai r screws , Eng ine s . 75 s . (76s . 3d .)

Vo l . 1 I . N o i se , Para chu t e s , S t ab i l i t y and Con t ro l, S t ruc tu res , Vib ra t i on , W ind Tunne l s . 47 s . 6d .(48s. 7d.)I943 Vo l . I . Aero dynam ics , Aero M ls , Ai rscrews , 8os. (81s . 4d .)

V o l . 1 I . E n g i n e s , F l u t t e r , M a t e r i al s , P a r a c h u t e s , P e r f o r m a n c e , S t a b i l i t y a n d C o n t r o l, S t r u c t u r e s .9os. (9IS. 6d.)1944 Vo l. I . Aero a nd Hy drod yna mic s , Aero fo il s , Ai rc raf t , A i r screws , Con t ro l s . 84 s . (85 s . 8d .)

V o l. I I . F l u t t e r a n d V i b r a t i o n , M a t e ri a ls , M i s c e ll a n eo u s , N a v i g a t i o n , P a r a c h u t e s , P e r f o r m a n c e ,P l a t es , an d Panel s , S t ab i l i t y , S t ruc tu res , Tes t Equ ip m en t , W ind Tunnel s . 84 s . (85 s . 8d .)A N N U A L ] R E P O R T S O F T H E A E R O N A U T I r C A L N E S E A R C H C O U N C ] I ] ~ - - -1933-34 is. 6d. (is. 8d.) 1937 2s. (2s. 2d.)1934-35 xs. 6d. (IS. 8d.) 1938 IS. 6d. (IS. 8d.)Ap ri l I , 1935 to Dec. 31, 1936. 4 s . (4s. 4d.) 1939-48 3s. (3s. 2d.)

~ N D E X T O A L L R E P O R T S A N D M E M O R A N D A P U B L ] [ S H E D ~ N T H E A N N U A L T E C H N ] I C A LR E P G R T S ~ A N D S E P A R A T E L Y - -

A p r i l , 1 9 5 o . . . . R . & M . N o . 2 6 o o . 2s . 6 d . ( 2 s . 7 d . )A U T H O R l IN DE N T O A L L R E P O R T S A N D M E I ~ O R A N D A O F T I - E A E R O N A U T I C A L R E S E A R C HC O U N C I f L - - 19o9-1949 . . . . . R . & M . N o . 257o. 15s. (15s. 3d.)][N DE XE S T O T H E T E CI-E N-/C AL R E P O R T S O F T H E A E R O N A U T I C A L R E S E A R C H C O U N C ] K , - -

December 1 , 1936 - - June 30, 1939.Ju ly 1 , 1939 - - June 30 , 1945 .Ju ly 1 , 1945 - - Ju ne 30 , 1946.J u l y i , 1 9 4 6 - - D e c e m b e r 3 1, 1 9 46 .J a n u a r y I , 1 94 7 - - J u n e 3 0 , 19 47 .l u ly , 1 9 5 1 . . . .

R. & M. No. 185o.R. & M. No. 195o.R. & M. No. 2050.R. & M. No. 215o.R. & M. No. 2250.R. & M. No. 2350.Prices in brackets include postage.

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i s . 3d. (~ s . 4d.)is . ( is . id.)is . ( is . id.)is . 3d. ( is . 4d.)IS. 3 d. (IS . 4d.)IS. 9 d. (IS. Iod.)

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