Analysis of Two-Cell Swept Box with Ribs parallel to the Line of Flight under Loading by Constant Couples

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2 8

APR. 1953

THE COLLEGE OF A E R O N A U T I C S

CRANFIELD

ANALYSIS OF TWO-CELL SWEPT BOX WITH RIBS PARALLEL

TO THE LINE OF FLIGHT UNDER LOADING BY CONSTANT

COUPLES

by

D. HOWE, D.C.Ae.

This Report must not be reproduced without the permission of the Principal of the College of Aeronautics.

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• •

"'•

2 8

APR. 1953 R e p o r t F o . 6I4 M a r c h , 1 9 5 3 T K E G ^ O L L B G E O F A E R O N A U T I C S ' ' ' ' ' ' ' ' \ C R A N F I E L D

Analysis of Two-Cell Swept Box with Rihs parallel to the Line of Flight imder

Loading hy Constant Couples ^

hy

-D. Howe, -D.C.Ae.

oOo

SUMMARY

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The method of ohllque co-ordinates^ ^ is used to analyse the prohlem associated with the strength and deformation of a uniform, rectangular, two-cell swept "box beam having ribs parallel to the line of flight. The case of loading "by constant couples is considered, "but no accoiint of root effects is taken»

The rihs are assumed to "be continuously distri"buted, the ri"b boom area, together v/ith the stringer area, "being distributed over the skins. A degree of flexibility is allowed to the rib webs.

Results are presented in the form of cross sectional rotations and stress resultants.

BHF

This investigation was made during the teniire by the author of a Clayton Fellowship avrarded by the Institution of Mechanical Engineers.

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LIST OP CONTENTS Page Notation 1 1.0. Introduction 3 2.0. Theory 3 3.0. Results 9

l+.O. Special Case:- Two Equal Cells 10

5*0 Discussion 1 "1

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NOTATION Oxyz OXYz

^±y

^i

Main system of oblique Cartesian Coordinates

Auxiliary system of oblique coordinates

(1=1,2,3» j=1>2,3) Matrix inversions for rear

and froxib skins respectively.

AJ (1=1,2,3) Areas of rear, main and frontspar booms

respectively.

2 A = A^ + A2 + A^

A-p, A^ Rib boom areas in rear and front cells respectively

t i II II 'R* ^R •^o» ^L s t r i n g e r areas i n » •D-'-K T>-i + r * ' h -iin II tf It ir If t l II II II t l II

a-^, a ' Rib pitch in

a^, a' Stringer pitch " " " " "

To

Half depth of box in direction of z axis.

C. . Coefficients used in expression of rates of section

"^ rotation.

c Half width of box in direction of y axis

c., c» Width of rear and front cells respectively

E Young's Modulus of Elasticity

L - M\.

4 - ^ V 4 - -^ + -^•- ^e' - -^'V

(e -

-^')-\®xx~ dxj' fyy- dyj' l^xy" ax ^ dyj' \®xx- dxJ' \^yy~ ay/*

3V' öU''^

ie* =

~^ + T^ ] >

Strain components referred to axes Oxy; in

V

^^ ^/

rear and front cells respectively.

E

G =

2/A

, g.) Shear Modulus

K.. Coefficients used in expression of stress resultants.

1 J

(L^, M.) Oblique components of couple, axes OXY.

(p, q, r) Oblique components of rotation about axes Oxyz.

(p^ J Q.-I) Rates of oblique components of rotation

S, S' Shear stress resultants in rear and front cells

respectively.

S^ (1=1,2,3) Shear flows in rear, main, and front-spar webs

respectively

T j , T' Direct stress resultants In x direction in the rear

and front cells respectively.

Tp, Tp Direct stress resultants in y direction in rear

and front cells respectively.

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2

-t, t' Skin thicknesses of rear and front cells respectively t. (1=1,2,3) Thickness of rear, main and front spar webs

respectively

U, U' Displacements of rear and front cell skins in

X direction

u. (1=1,2,3) Displacements of rear, main and front spar v/ebs in X direction

V, V' Displacements of rear and front cell skins in

y direction

W. (1=1,2,3) Displacem.ents of rear, main and front spar webs in z direction ^RO (X, Y, Z) a

A, A'

/•- A ' ! /^2' '"-'2 i

Displacements of rib web on y axis in z direction Components of force, axes Oxyz

Angle between Ox and Oy axes

Distortion of section in rear and front cells respectively

Constants in equations for distortion of section

e

X

Co , U)

Terms used in expression of coefficients

Poisson's Ratio

Warping of section in rear and front cells respectively

CO J , Ü) ' i

i Constants in equations for warping of section

0) , («2

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INTRODUCTION

The method used in this analysis is essentially that

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developed by Hemp in Part 3 of his work^ ' on the application of oblique coordinate-^ to swept v/ing structures. Pig. 1 shows the construction of the box and the notation used.

The sweep back angle is (711/2 - x ) , and the box is defined by a set of oblique axes Oxyz. An auxiliary set of oblique axes OXYz are also used. The upper and lower surfaces are given by z = ± b respectively, and they are asstimed to be reinforced by closely spaced stringers parallel to the x axis, and closely spaced ribs parallel to the y axis. The skin thickness in the rear cell Is t, the rib boom area A^, rib pitch a_, stringer area A„, and stringer pitch a^. The

K S S

comparable dimensions in the front cell are t'. A', a'. A' and

K K S

a' respectively. The rear spar web is defined by y = c and s

has thickness t^, the mainspar web by y = (c - c.) and has thickness tp, whilst the front spar is given by y = - c, and has thickness t,. The areas of the rear, main, and front spar booms are A., Ap and A-, respectively.

Where the spar and rib webs are capable of carrying end loads, their effective area is considered to be Included in the appropriate boom area, the webs themselves being assxjmed to carry only shear loads. All the materials have a Young's Modulus of E, and Poisson Ratio cr . The rib webs are

considered to be rigidly connected to the spar webs, but are allowed a limited flexibility in themselves.

The effect of root constraint is not investigated, and the box is considered to be loaded by constant couples.

THEORY

with X.

Assume a linear variation of the rotation of the box

Rotation component about x axis P = P^x j

^ i ... (1)

Rotation component about y axis q = q.x j

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i^

-The warping and distortion of a cross section of the box are assiimed to be linear in y

W a r p i n g : - Rear C e l l : - ü>=cö^y + cü2 1

I . . . (2) F r o n t C e l l : - 03' = cojy + coA

S i m i l a r l y d i s t o r t i o n ; - A =w\.y + A O •

^

'^

\

... (3)

Using Eqs (I) to (3)» the displacements

become:-For the Skins;^ U = q.x b slna + «(y) • U'= q.x b slna + o)' (y) j

V = - p.x b sina + A, (y) \ S V'= - p^x b slna + ^s,' (y)

(^R^ y=0 = " ^xx- lb

... (U)

... (5)

For the Spar Webs:- u. = q^x.z slna + (co) . |- /

Ug = q^x.z slna + {<si)^_^^^^ ) * I ^" " * ^^^ Uj = q^x.z sina+ (co')^^,^. | j

The displacement in the z direction, of the rib webs, on the centreline of the box is given by Ref. 1 Eqs. (9U) and (98)

as:-y=

x^ i

Hence:- w. = p,x c slna - e . ^ j 2 • Wg = p^x(c-c^ )sina - e^^. 2=b i ... (7) W3 = - p^x c s l n a - e^^. ^ ,1 / Eqs.

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-1

-Eqs.

ik)

to (7)

the skins. ^xx - öx = 'll^ ^^""^

e = ^ = A

yy dy i

are used to obtain the strains

®xx = ^A ^ ^^^"^

'yy = ^ '

V = ax + ay " "^1^ "^^^ ^ "^1 ^iy = "^1^ ^^^" +

The box is loaded by cons

.*. X = Y =

The Stress Resultants are T., and Tj are Tp and Tp are S and S' are •'• ^xx = ^1^1 + ^ 3 ^

V = ^21^1 + A23S

^xy = ^31^1 + ^ 3 ^

Compatibility of warping ! ü)^(c-c^) +co2=(ö-i( or rewriting:-CO. =0)7 + ("^2 -"^2^ ^ ^ ( c - c ^ )

Using Eqs. (8) and

(9):-q^b slna = A^^T^ + A^ 3

•^21^1 "^ ^^23 -p^b slna = A3^T^ + A33

tant couples Z = 0 restricted. functions of y only zero constant

^ix = ^ïl'ï +

^ 3 ^ '

1 ^;y =

H ^ n -•

^23^' V

''iy = ^31''Ï

*

'SsS' 1

at the mainspar requires;-2 - C^ ) + 0) ^

3 = A^'^T^' + A]3S'

in

"i

3 - A^ = A^^T] + A^3S' - h\ = 0

3 - co^ = A^^T^' + A^3S' -coJj

/ Using . e 0...« J ... (8) ! ' ... (9) .. (10) .> (11)

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e

-Using the s t r e s s - s t r a i n r e l a t i o n for the spar

webs:-I

^ 1 - ^ * i \ ^ a z + a x (w^c + co^) [ S^ = Gt^ •! p ^ c s l n a + ( !«-, ( c - c . ) + "Jpi'i Sg = Gt2 ' ; p ^ ( c - c ^ ) s i n a + -^—! ^ ^ j-f (-co'c + Wp) I S3 = G t 3 -J - p ^ c s l n a + ^^| { . . , ( 1 2 ) E q u i l i b r i u m o f t h e s p a r f l a n g e j o i n t s r e q u i r e s t h a t : -S^ + S S3 - s ' S - So -

s'

= 0 = 0 = 0

V

. . . ( 1 3 ) E q . ( 1 3 ) i m p l i e s t h a t : - S. + So + S^ = ÖT- = 0 _2b

For overall

equilibrlum:-L^ = 2bc S^ - 2bc S3 + 2b(0-0^)82 2b- Sc^ + S'Cg.!

and using Eq. (13):-L^ '1 _{-c^S - C2S'}

or:- s' =

- s

-r

°2 = - S ^1

ÏÏbcJ

^1 ÏÏbc^ ... (1U) M^ = 2bE(SA)(q^b slna) + 2b(T^c^ + T^'c2) ... (15) where T^k = A. + A2 + A, / Substituting

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S u b s t i t u t i n g from E q . ( 1 3 ) i n t o E q . ( 1 2 ) : -S ' G t ^ c = - P ^ s i " < ^ + (-co^' c + (Op b e (ci _ o M i""'-! ( c - C , ) + 0)p ] G i ^ P ^ s l n a + ^111- + — b T ^ c T T (CO^C + COg) b o L . . . (.16) E q s . ( 1 6 ) a n d ( 1 0 ) g i v e M a n d co' 0)

^°^°-°i^ r (s-s')

= TT-Z ! 4.^ /r.U \ + G C t g l c - c ^ j

*i°;

Ü)

, _ ^^('^-^j) I' (s-s') _s:l

2 " G Cg I t g l c - c ^ } "*• t3C I . . . ( 1 7 ) U s i n g E q s . ( 1 1 ) a n d ( 1 6 ) : -Sb a n d G t ^ c -^ -^31 ^1 "^ ^ 3 3 ^ S h _ A t m l , At q » -G t ^ c - ''^31 1 "*• ^ 3 3 Cop 2(0. 1 c t 0 ) 0 2«1 + - T . . . ( 1 8 )

E l i m i n a t i o n o f co. a n d col from E q . ( 1 8 ) by u s i n g E q . ( 1 0 ) y i e l d s : '

Jb_ Gc of q | , r ( c - C . ) + 2 c j

f; - h- ^31^1 ^33^'- ^31^1- ^33^^-21 c(c\)*

+ 00, ( c - c ^ ) - 2 c i 2^ c ( c - c . ; I I 1 ..' ( 1 9 )

Substitution from the first pair of Eqs. (11 ) into Eq. (15) and elimination between the resulting two enuations

gives:-M '^1 =

T

-52(A^3S'-A^3S)+A]^(^ - EA^33A.S)j/A^^fEA^^.rA+c^i+C2A^^/

f- 'i / •

c^(A^3S-A^3S')+A^^(2| - E A ^ 3 2::A.S')J /A^^^EA^^ijA+c^l+CgA^^j >(20)

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8

-Using E q s . (17) and (19) and r e a r r a n g i n g : '

A^^T^' - A3^T^ - S 2b c b 33 "^ Gt2.c^C2 "^ Gt^c^ - S' A' + 2b c • t 3 GtoC.c _{2"1"2 " " 3 " 2 |}G t , c , . . (21) Eqs. (20) and (21) g i v e : -^31^51^1 "^ 4 i ^ ° 2 •" A : | ^ E r A ) [ _{2b c} 1 A\^IEA^^ZIA\ C^] + A^^Cg j " 1^33 •*• Gt2C^C2 + G t ^ c ^ j l ^ S^31^?1°2 ^ A^^^(c^^ A ^ ^ E 1 : A ) ) r 2bc

/ A : J ^ | E A ^ ^ S A + c^j + A^^C2 J p 3 3 "*" Gt2C^

C2 ^ G^^o^lf

:b_|i

1^31^11 " ^31^11

jAq^ (EA^^ ITA + c^j + A^^C2

M

_{_1}

. . , (22)

2b

E q s . (22) and (ill-) e n a b l e S, S' t o be found and t h e n c e T^, T^' from Eq. (20)

co! and a. follow from Eqs. (1?) and

(18):-c o ' = Ü) , = A' T ' ^31^ ^ bS 2 ^ 2GtoC S' 2-2 2 ^ 2GtoC bS' Si •*• 2 2^1 A33 ^ 3 3 -b r 1 1 ( 0^2'•^3 *2J b r 1 1 r Gc^-t^ t 2 ] . . . (23)

Prom Eqs. (23) and

(II):-P = ^ ^1 djc cosecaj . _, „ j b I 1 1 l( bS' • ^31^1 + S JA33 -^ GH:^ j tg -^ t^ h| - Gt2C^ 2b ^1 ~ dx dq _ coseca b

^ 1

'^^

+ ^ 3 ^1

^ ^ (21+)

The displacements can be found using Eqs. (2U)j (23) and (17). The strains follow from Eq. (9)

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- 9 3 . RESULTS ^1 S ^1 ^ 2 ^ 3 Pi V>/here:-= = = = = = ^11 ^21 ^31 \ ^ % i d x • L. L, L^ ^1 L. = C + + + + + 11^1 ^ 1 2 ^ 1 ^ 2 2 ^ 1 ^ 3 2 ^ 1

\2 ^S

^ 5 2 ^ 1 + C^2^1 ^ = ^11 ^1 -^ ^2 ^1 S ' = K^., L^ + K^2 ^1 V(25) - ^ - ^21^1 + C22M^ "^1 ~ dx K, ^^ = sc2(A^3^x - A ] 3 ; 0 + A^3A^^ n E . D A ^ / ' i + b e X jj.. ,. A ' ^ 1 1 ^ 1 2 = - ^ 1' ^ 3 ( ^ 2 + ^ 1 1 - ^ ) - ^ I 3 ° l ] -^ 2ÏÏ7-J]^ = ) C - | ( A ^ ' 3 > - - ^ H 3 ^ ^ "^ ^'13 ^11'*" ^'^-'-^ ; / U b e K

^ JAJ3(c^ 4- EA^^SA) + A^3 C2] + 2 ^

K, ^2 = K, ₂₁ _i+be K, C2 >-T C ' - - ^

^ 1 " riSë

22 ~ 2b:Ke c ^22 1 K.

31 " nfe"

K 32 2b>'€ C 2 > 2b)<ê K,

W

K, •51 Ube X .be K, _•i|2 K, ₅₂ ;^(c^+ Cp)

2bVr~

~ ïbPe" \ ( 2 6 ) '11 c o s e c a I ' • ^ 8b e

l ^ i u ; ° 2 ^ 3 ^ ^ " ^ 3 ^ ^ +^13^11 ^^E.SAf

- l^A^ _ J l _ VJL . H - A l 3 Gc^ ft^ ^ t g J '12 '22 '21 c o s e c a I 2b 2v. c o s e c g i|b2i^ : ^ 1 ^ ' 1 ^ 1 3 ^ ' l

-A>°2l

(27) / r \ " 0 0 0 0

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1 0

-)< = k\^

{EA^^

ük + ^^\ + A^^Cg

'k' - A A ' - A ' A

[i =1 A3^A^^C2 + A^2(c_^ +

A ^ ^ E . S A ) A ' A33

X =1 A3^A^^c^ +

A 2 ^ ( C 2

+

A ^ ' ^ E . C A )

A33

-b ( c ^ + C 2 ) ^ ( C ^ + C g ) ^ * 2 ° 1 ° 2

b

I I

Gt3C2 I i

e = ACg + M'C^

Gt^c^

/ (28)

SPECIAL CASE

Two Equal Cells

t^ _ t2 - t3 - t^

_{°1 = °2 = °}

t = t'

(Equal cells with constant web and skin thicknesses)

( A . , E . S A L. \ / \

^1 = ^1

'rh^

"^ ^ 1 / / ^"^^2° "^ EA^^.è7A)

s =

s'

= -

_^bc

'1

^1 - " ^3 - 8bc

8 2 = 0

ƒ

V

/

.. (29)

K. A ^ . ZJA

11 "" 8bc • (2c+A^^E

"Ek)

L^

^ 1 =

"1

Bbc

K. _{31 " 8bo}

1 ^ 2 ~ 2b(2c+A^^E5:?A}

K 2 2 = 0

\^

= 0

K,

V(30)

•51

1 !

8b c I

K32 =

\2

= K52 - °

/ / 0

11

..«..«

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'11 '12 '22 coseca \ (l + cr) ^ 3 3 ^ 1 3 ^ ^ ^ 8bc I Et^c "^ 2b "" 2b(2c+A^Ê. S A ) coseca A. = G 21 ₎₂ M l Ub^(2c+A.Ê..EA) coseca A^ 1-2b^(2c+A^Ê.SA) ... (31) "X = O = M e = 2 X c = 2|iC K = 2A^^c + A^Ê.^CA ... (32) DISCUSSION

It can be seen from Eqs (26), that there is a contribution to the oblique shear stress from the "bending couple" M.. This contribution is dependent upon the value of y, and it can be shown that >- itself is dependent upon the relative values of A. _{R» ^R*}and t in the two cells.

For the case of constant rib pitch, with the ratio of the rib boom area to skin thickness the same in both cells,

I.e. 'R the value of >: is zero. Under these

^R* ~ ^R*' '

conditions, there is no oblique shear stress due to M,•

The results for the special case of two equal cells v/ith constant web and skin thicknesses given in Eqs (29) to

(32), are directly comparable to the single cell results, Eqs (78), (83) and (lOO) of Ref. 1.

Complete analysis of the box beam subjected to constant couples is achieved by using the above results in conjiinction with the relevant parts of Ref. 1 '^ 3.2.

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12

-REFERENCES

1) Hemp, W.S. On the Application of Oblique Co-ordinates to Problems of Plane Elasticity and Swept Back Wing Structures.

College of Aeronautics Report No. 31 Jan. 1950.

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o

>\ It » -T\ o ui O C: t : ^