An approximate solution to the swept wing root constraint problem

8rw69 "' " ^^'^"^ "'''•'-' '-''-'-'

CoA Report No. 98

Kanaalstraat 10 - DELFT

16 MEI 1956

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

C R A N F I E L D

AN APPROXIMATE SOLUTION TO THE SWEPT

W I N G ROOT CONSTRAINT PROBLEM

by

TECHNISCHE HOGESCHOOL

VLItGTUIGBOUWKUNDE Kanaalstraat 10 - DELFT

EEK)RT NO, 98

EEBRa/Jg,1956

T H E C O L L E G E O F A E R O N A U T I C S C R A N F I E L D

lm approximate solution to tlie Svvopt Wing Root Constraint problem

-by-D« Howe»

SUI'lI/Jg

This report presents an approximate solution to the problem associated v/ith the root constraint in a swept v»-ing,

The structure considered is a uniform rectangular single cell box having closely spaced rigid ribso Approxi-mate allovTance is nade for the effect of boon area at the

spar-skin joints,

The stress distribution in the skin is represented by a polynomial in the chordvd.se ordinate, oblique coordinate notation being used. The final eqiaations are derived by use of the theory of minimum strain energy»

Comparison with experimental results has indicated that the theory gives a satisfactory estimation of the con-straint effects, and especially the influence of the spar booms9

Consideration of the validity of the boom approxi-mation shows it to be justified in tv/o particular instances having relatively large bocci area»

-2-1 .

2 ,

3.

i^

5-CÖNTEmS

L i s t of Symbols

I n t r o d u c t i o n

Description of Box

Theory - I n i t i a l Assuciptions

- S t r e s s D i s t r i b u t i o n

- S t r a i n Energy

Results

Cconents - Validity of appro;

Fap;e

3

4

4

4

4

7

ation 11 - Unswept Case 12 References 12 Appendix 1 - Terms in Strain Energy Equation 15

Appendix 2 - Special cases - l) zero boco area 14 2) zero sv/eep 14 Figure Details of Box»

•3'

SïilBOlB

A Spar boon area

A. . l i a t r i x inversions i n d e f i n i t i o n of s t r a i n s

B T + T'

v/ w

C.(i=:0'3) Coefficients i n expressions for s t r e s s r e s u l t a n t s

D T - T'

w w

E Youngs Modulus

F. . Coefficients in strain eneigy equation

G Shear Modulus

K. . Coefficients of equations for Cp and C ,

1 / Cfblique couple components

M

03!y System of oblique axes

S Shear s t r e s s r e s u l t a n t i n skin

S , S' Shear flovra i n r e a r and front spar webs

w

_respectively

T., Tp Direct stress resultants in skin

T^, T' Loads in rear and front spar boons respectively

U Strain energy

Z Normal shear force

ap Rib pitch parallel to x axis

a Stringer pitch parallel to y axis

b Half depth of box

c Half width of box

e e Direct and shear strains in skin

XX xy

I

Span of box parallel to x axis

t , t* Thickness of rear and front spar webs respectively

W ' W jr JT - « /

a Included angle between axes Ox and Oy

EA A,,

11

c

E/i A

11

X 1 + /5

w 1 + 3 ^

-V-1» Introduction

The root constraint problem of a swept box is complex, Based on the oblique coordinate theory of Hemp (ref, l ) , the

solution presented is an approximate one for the case of s

uniform rectar^gular swept box having closely spaced rigid ribs» The type of solution, which is a strain energy

analysis, has been used by Hemp (ref, 2) for a box having zero boom area. In this report an approximate extension to cover

the case of discrete booms is given, 2» Description of Box

The box is shovm in Fig, 1, The notation used is that of Ref, 1, Skin reinforcement is by stringers and ribs v/hich are both assumed to be distributed and contribute to the

slcin thickness. The spar v/ebs are capable of carrying only shear loads, and are of unequal thickness. The spar booms are of equal area. The root end is built in,

3, Theory - Initial Assijmptions

It is assumed that the skin spanwise direct stress resultant, T., can be expressed as a polynomial in the

chordivise ordinate, y» Terms containing povrers of y greater than two are considered to be negligible. The skin chordvvise direct stress resultant, Tp, is assumed to be zero, shear equilibrium being maintained by the ribs,

Stress Distribution

T^ =C^(x) + 2C2(x) J

^y^3(^(^f 1

j (1)

T„ = 0 !

₂

J

where C^(x), C2(x) and C-(x) are arbitrary functions of x, Since for equilibrium ;- -r— + -r— = 0

^ 3 X oy

(c , x ? 1 . 4 ^ . ^ ^ 1

S = - 0 C + i

_{i o c d X „2 d X ^3 d x y}

T^*^ r ^ + ' H T^ (2)

where C i s a f u n c t i o n of x , o

-5-Equilibrium at the spar boom joints requires

J-dT S = - ( s ) + - J ^ w ^ 'y=c Ö X d T ' S» = (S) + - r - ^ w ^ ' y = - c Ö X dT / dC, dC, o W / _ 1 £ S, = -r + C C + -z + -3— -r , w Ö X l o d x d x d x

. ^ )

dT* / dC, dC„ dC,

s . = - J 2 : _ c ( c - — L + - ^ - - ^

w o x t o d x d x d x

.(3)

O v e r a l l e q u i l i b r i u m w i t h t h e a p p l i e d l o a d i n g a t any g i v e n s e c t i o n s r e q u i r e s | -lo L, = 2bc (S - S ' ) - 2b 1 ^ w w' S dy -c and >c I.L = 2b (T + T ' ) + 2b T, dy 1 ^ w w' I 1 ^ 0 - c Yilriting : - B = T + T' ^•^ w . , . • ( 4 ) D = T - T' w w

and substituting from Equations (l) - (3)

S-ii = a°'(°o*f r l ) - ^ ° f (5)

H, = 2b jE + 2o (0, + Cj)' (6)

Equations (5) and (6)

yieldj-J^

2 ^ 2 1 dP

^o = o. 2 " 3* d X ~ 2K5 • dx ,

^ I (7)

^1 - 4bc °3 2c

J

6

-T. = y r ^ - f- + 2C„, ^ + C^

1 ijiic 2c 2 c 3

8bc ^ 4 dx o [lib 2 öxj* °\^3 ^2j dx^ y y 2 J d x 1^

•^1 1 dP Z c ' ^ 2

w ' ' 8 b c ' ^ 4 d x ' ^ 4 b " ^ 3 d x

s i = . i L - . i ^ + Z - . . £ f i

w " 8bc ' ' 4 d x 4 b ' ' 3 d x

dM^

since -^^ = 2 ,

.(8)

The spar boom l o a d : - T = EA(e )

^ w ^ xx'y=c

T' = EA(e )

w ^ xx''y=-c

and since e = A. .T. + A.S (see Ref. l ) ;

-XX 11 1 13 • / •

= ^ 1 [ ( ^ l ) y = c - ( ^ i W c j ^ -S3 ( ( ^ V = c - ( ^ W c j

(s) + (s)

^ 'y^c ^ ' y = - c .

D_

EA

T7riting /S =

Y =

EA A,

c

EA A.

11

-^ J

.(9)

and s u b s t i t u t i n g from Equation ( 8 ) s

-2 -2 ,-2^

r e d B

d j c ' ( 1 + ^ ) B = ^ -Y L , /9IL 2 2 ' 1 ^ 1 Y C s dC,

D = 4 /9o C^ + Y»c

4 b 2 b

dB Yc.Z

4b

i - 2 v = > i j ^ - W 3

dx

2b

.(10)

(

J

Equation (10) reveals the coupling between B and D which is a function of sweep (A.,) and boom area ( A ) . Since the presence of this coupling term has the effect of doubling the order of the final equations, and as Y will he small for

-7-small boom area, the following assumption will be made in calculating the strain

energyi-2 d B „ Y . — Ö - = 0

dx

.(11) The validity of Equation (ll) when Y ^ S not small is discussed ija §5. Thus

_{I- B = ^ - r r : ^ + % - (3/5-1) T 4 + ~ T - ^}

I.

I/O 2 d C , , ^ X d X • 2bX i+bX 3\ '•''^ ^ d X • X

=^ = ^^=°a*if(f-^)

where X = (1 + /?) ^J Substitution from Equation (11) into Equation (8)

(12) \

givesI-M

T. =

'1

YL.

1 ~ /*cX " 8boX ^ c • ^2 ^° yk ' d X ^ / , 2 \

n2 M

C°

L.

„

/„ 2v dC^ r 2) dC,

Ö 1 _ Zy • f 2 y \ _ 2 .jw_iL-( V — ^

^ = ~ 8b^ 4boX •*• U + '^ " 2 /•°« d X •" X 2 .) •2'* d X

8bc 2tboX I 3

L. „ dC„

„ 1 Z c 2

^w= Sb^"^4b*3''^*dT

„, 1 , 2 c 2

^7 = ~8bc^4b"3*'*^'<ix

^ 2 / - d x - X - ^ 2 ^

(13) where ca = (3^ + 1) Strain Energy

The strain energy in the skins is given

byS-Us = 2 I j (i T^.e^ + iSe^)dx.dy =

0

o J -C V o

•ïC

(A^^T^+2A^^T^S+A2^s2)dx<$r

o '-i' - C

since e = A.,T. + A^_ S (see Ref, l ) , xy 13 1 33

-8-The strain energy in the spar

websi-f)l

_S'

U,= 2b 1 1 j ^ . e_ + :;^ .e^J dx = ^ |

\-f ^ ^,

\ t xz t' xz; ij \ w w /

b

G

{^/^ s'2

w w

U o

w w y

dx

The s t r a i n energy i n the spar

boons!-• j l

^B = 2

^fT , ( e ) + T'(e ) j dx = ~ I

(TJ

+ T'^)

21 w ^ xx'y=c w'' xx'^y=-cy EA j w w '

dx

The strajxi energy i n the r i b s i s zero as the r i b s are considered

to be r i g i d »

Since, by the P r i n c i p l e of llinimum S t r a i n Energy, a

small a r b i t r a r y increase i n the s t r e s s r e s u l t a n t s and i n t e r n a l

loads must r e s u l t i n zero change of s t r a i n energy for given

applied l o a d s ,

6U = 5(U3 + U^^ + Ug) = 0

l^ ^P

do u-c

^

) 2A,,T,6T, + 2.1,,(T,5S + S6T.) + 2A,, S6S ( dx.dy

J ) 1 1 1 1 1 3 1 1 53 (

ll

_{ry, / S 6s S'8S'}

2b / w w —

-w

do '

Fran Equation

(15)8-6T _ 2ir g _ (3/3-1) g

°-^1 c ^ 2 ^ ° 3X °

w w \ ,

+ — ^ r - j <3x +

\I

I 7 (T 5T + T'6T')dx

EA ^ w w W W '

f 2

^2 - x f ^S

V c / '^ ' ( c \ ' ^ \

6S =

5S = ^

_{w^ 3}

.(15)

5S» = - ^

Y/ 3

-9-And from Equation

(12):-6B = ^ (3,-1) 6 (^) . i f 6O3 )

8D

^d x} X 2.. /dC,

.(15)

,„ 2 /dC-\

= 2^c.5Cp+ii^6 -r-^J

2 X

\d x/

oontd,

The evaluation of the individual terms of Equation (14) is

given in Appendix 1, The resulting form of the equation

isS-6U = 0 =

[ F ^ ^ S C ^

.

F225 ( ^ ) ^ F^,6C3.F326

i^jj

dx..(lé)

^ ° ^ dc, dii;

where F». is a function of Cp, •^~^> and -r-r

'22

F3, • •

p I I

32

dCg dM^

' Tl^' °3' ^^1' d7' ^1

dC d x' 3 1 1 dC„ dC, dM. I I c — - — ^ — 1 L ""2» d X» d x' d x' 1 Integrating Equation (lé) by

partsi-6U = 0 F226C2-.F3,6C3

I

r^l

+ o >JO ~

[^21-

dx

^ V i V dx

J ^J

dx

and tip condition givesj - (C^) _, = (C,) , = 0 therefore (8C2)^j = (ÖC^)^^^ = 0

but otherwise arbitrary

.(17)

4« Results

Rewriting Equation (17) to give equations for Cp and C,

: 1 0 : -d^Cg dC, , „ dL. ^21 ' . dx C1L>_ -,— n r I ^

I *'^2°2-^'S3 d t =*'^24^*25 f •^•'26 d^ )

d^C dC2 ^ 31 ^ 2 33 3 32 d X 55 ^

and boundary conditions8- \ HS')

/dC \

^^21 {TÏl^^H3^''3^^o-^'^k^Vx^-'\5^''^^^^^ i

S i ldt)^^-^S2(°2)x=o = - ^ 5 ^^^^o

(°2)x=l = (S)x=l = °

T/here

A c 2 ^ 1 = - ^ 1 5 A l O / 3 + 3 ) + r % ( T - + T A - A Yc(3/?-l) •'^ V w w / • ^ ^ 1 l "

^ 2 = S ~

^ ^ 1 ^ ^1R 2 N ^ 1 ^ Mf^ 2 , \

^3 = - 3 r ( r ^5 -^^^-v^ ^2 = l i t ir^ ^^^v

4A 4A.., p

K^3 = - - 3 ^ (6^+1) K^2 = I 5 X (''5/5'^+é^+l)

3A (O A « ^ 2 4 ~ 4bXc 2 4 l^\o rr ££_ / 1 _ L.\ ^ 5 - 8Gc i t t ' ) TT - 33 ^ / j _ ^ 1 _ \ . 1 3 '

^ 6 " 8bc ,eGc'{K Kj sbcx

31 - ^ ^ 3 3 " U . 2 " 5 X - * - 7 •*• ^2

_{3x" -"^ y ^ X"}

S3 = - l ë ^^^"'^

^5^^^-^^) ^ l L

•^•^ 30bX 2bX

1 1

-EA A , . -EA A. _

/? = 1 1 Y = ^

\ = (1+/3) 0) = (1+3/9)

E q u a t i o n ( l 8 ) e n a b l e s t h e v a l u e s of Cp and C, t o b e found and hence c a n p l e t e s t h e s o l u t i o n ,

5 , Comments

Validity of Approximation

In order to derive Equation (l8) from the initial assumption of Equation (1) it has been found necessaiy to introduce the further asstiTTption of Equation (ll) that

2 ^

Y , 2 is zoro. It will be seen that this is correct T/hen Y is zero, that is when the box is vmsv/ept or vidien there are no spar boons. The values of the K. . for these two special cases are given in Appendix 2,

2 2 d B

For Y —Ö" """'O ^ s n e g l i g i b l e when t h e r e i s sv/eep and

^

dh

relatively large boom area, — r - must be small compared to dx

B as Y itself will not necessarily be small. The degree to v/hich this is sufficiently true is not immediately apparent, Accordingly tvro cases (ref, 3) have been analysed where Y is relatively large,

The first of these tvro cases is a 45 swept box

having a boom area such that -r = 0,34» Considering the ca^e of loading by a normal sheco? force applied at the tip, with L. = 0, and assuming C, = 0, it was found from the theory

thatt-2 thatt-2 ,thatt-2

^^-2- . ^ = 0.04 XB, at the root (x=0)» dx

In this case Y = 0»75»

A 60 swept wing was the subject of the second example, and here, with /v^ct sz 0,266 it was found, for similar loading

thatj-2^2 ^2g

^ P , — s - = 0,006 XB, at the root» dx

TICHNISCHE HOGESCHOOL

VUEGTUIGBOUVi/KUNDE Kanaalstraat 10 - DELFT

-12-where Y = 0«60,

These exaciples thus indicate that Equation (ll) also holds for ^relatively large YJ ^nd in fact good agreement vd.th experimentally derived results was obtained in these cases (ref. 3 ) .

Unswept Case

Appendix 2 S2 shows that in the unswept case, all coupling terms between Cp and C, of Hguation (18) are zero, The equation for Cp is thus explicit and gives torsion cor-. straint plus the effect of \anequal webs. The equation for C_ gives shear lag effect,

Thus the method is of value in considering the effect of the distribution of end load carrying capacity in unswept boxes,

REFEIRENCES

1 • Hsrip, W,S, On the application of oblique coordinates to the problems of plane elasticity and swept v/ing structures,

R. and II, 2754 (Jan, 1950).

2, Henip, '\7,S, Constraint effects on a sv/ept box, Unpublished (l/Iay 1953).

3, Griffin, K,H,, Constraint effects in box beams, and Lecture to British Association, 1955« Howe, D, (To be published in Engineering).

1 3

-AEEENDIX 1

E v a l u a t i o n of t h e i n d i v i d u a l canponents of Equation (14)

''° R« (xa A\ / fW\ /'<3CoN , , 2 3 „ dC„ .•dCo\

. 2 g ! ^ 3 . . 1 ) ( ^ 1 ) c 3 s ( ^ > ^ ( M , . ^ ^

^,2^(3,.0(f-i:§.5C3

13-X * ; ; 2 j V S

* 2 . / ' ^ 2 \

ii°

-1fe*^-ëv(?i)^^^(l:"f}^2^(?^)

SS^1 ^ = - JbX ^ ^ 2 - ^ ^ (3X - 5) d i ^^2 + 12 bX ^

_{d Xy}

2\ dC.

„ 3 _ clLi„ / d C „ \ L. ,. . „ / . ÜV d c „

- ^ 9 . ^ - 1 ) ^ 6 ( ^ j . ^ ( ^ l) 5C3.2c^ ^ ^ . , . I , ] ^ 8C,

' ' c ^ . c . ^1°^ J " ^ 2 \ p 3 /;2 2^ 1 ^ ^ 2

S6S dy = - ^ ^ 6 f ^ ^ j + 2c^ ^/9 + ƒ + ^ j ^

dC. ^'dC2>

- c 2 ..

2bX 1^3X 5 ; U V * ^° (3^2 5X + 7] d X ^ \d x ; *

il

(T 8T + T«6T')dx = ^

^ W W T/ w ' 2

^l

(B6B + D6D)dx ,

i^o U o

- 1 2 ^ APEEHDIX 2 S p e c i a l Cases 1 , Boom a r e a z e r o , i , e , 00 = X = 1 /? = Y = O 3 A , , c K, ₂₁ K,

23

8 A ^ 5 K ' 3 = -3A. ^ 4 "" ~ 4bc

^^5 "" " 8Gc f •

2A c ^31 21 1_ 1 _ \ t " t< ) w w / Vi ^22 = " c 11 2A K32 = ^ 3 K' ^ 32 - 3 K' - - - ^ ^24 ~ ^30 K - ^ ^26 - 8bc / 1 1

h r + TT

A 16GC \^ W W _{: )} K

35

11

c

K

-til

'^35 ~ 24b 2 , Zero Sweep i . e . Y = A^3 = O

^21 = ^ (15^^10^+3) - | 4 ( f - P") K2,

U^^o,

K23 = K32 = K^3 = K^2 = ^24 = ^24 ^ °

^ 5 ~ 8Gc ( t t ' ) Tjr . / 00 2Ü3 1 ^

^31 =

^33°

( ^ " 5X ^ 7)

A (6^+1) xr =: —2-2 55 2 -'^ 3CfbX ^ 6 - 8bc . ^ „ 2 i t

, . - ^ i t ^ t ' j

_{loGc '. w w /}

^33 = - ^ ^ ^ ^ - ^ ^ ^

CLOSELY SPACED RIGID RIBS