The Diffusion of Loads in non-rigid Circular Frames

TECHNISCHE HOGESCHOOL Vin,- -v •'•: DE REPORT No. 33

12 Juli 1950

LIJCHT' '-n Itn Kluyverweg 1 - 2629 HS DELFT

THE COLLEGE OF AERONAUTICS

CRANFIELD

THE DIFFUSION OF LOADS IN

NON-RIGID CIRCULAR FRAMES

• b y

S. R. LEWIS, B.Sc.

TECHNISCHE HOGESCHOOL VLIEGTUiGBOUWKUNDE REPORT No. 33 F e b r u a r y , I 9 6 0 T H E C O L L E G E OF A E R O N A U T I C S . C R A N F I E L D

The Diffusion of Loads in non-rigid Circular Frames

-hy-S.R. Lewis, B.Sc,

of the Department of Aircraft Design.

— o O o —

SUMMARY

This report extends the work of W.J. Goodey) ' and gives n-omerical examples of the shear distribution around a frame subjected to a single concentrated rad-.ial load for variations in the parameters, such as,

frame stiffness, skin thickness, stringer spacing, etc, It also indicates when the beam theory dis-tribution of shear can be used with a reasonable degree of accuracy.

The report contains a number of curves, figs, 5-17 giving the shear distribution around a frame for a single concentrated radial load of 1000 lb. The parameters chosen are those common to aircraft design, and it is possible to obtain a reasonably accurate

shear distribution around a frame from the data supp-lied, vifithout doing the actual shear calculation,

The case chosen is that of a long cylinder with a closed end, or that of a long cylinder where the portion aft of the loaded frame has a restraining effect upon the forward section, see fig.1.

The appendix gives the method of obtaining the shear load due to a tangential load and moment from the radial load expressions,

L

R t rl.. j

L

V N

xe

m e c k X

-1-NOTATION The notation used is:

= the radius of the frame at the skin line. ins. = the actual skin thickness ins.

= the moment of inertia of the adjacent framesj, about an axis parallel to the skin line ins. = the moment of inertia of the loaded frame about

an axis parallel to the skin line ins.^ = frame spacing ins.

= Poissons ratio, value taken as 0.3'. - shear per ins in the skin.

= R L = 3 m^ f 2 k^(l + v ) - v/ = R^k

2 r

i t ^f = 6 m^ 0

ratio of equivalent skin thickness to actual skin thickness, see fig. (2)

Pig. (2)

If the skin is not buckled

_K-

(Ag + b t )

b t

If the skin i s bucMed and the

e f f e c t i v e width of skin i s bV k

_X (A3 + b ' t )

1

-2-INTRODUCTION

Goodey in his work considered two cases, namely (a) A long cylinder with the loaded frame at

the free end,

(b) A long cylinder with the loaded frame at a distance from the free end.

The most practical case is that of (b) above, v/hich was investigated by Goodey by considering a com-bination of the cases of a long cylinder loaded in the middle and a long cylinder loaded at its free end. See fig. (3). • —

W

J f f t

1

I

\ I

3'

i SOLUTIONS Pig. (3)

The solutions contained in this report are for'case (b) above the expression for the shear in .the skin at the loaded frame, due to a radially

app-lied load W is given by.

Nxe = 1

V r/^-2+^H^-l-^)^-(Al+^)('-2-^)^j n Sin n e ]

nirM 0.2+1) Oll-l)^(p2-^^-(^i+l)a^l)^(7+2-r) j

+ W Sin 6. (1)

2%R

or oo

^x0 = I > ia!a^Sin 2y+a(a^-2)SinV-a Sinvli Sin n 6

%R s~^-_{n=l jsinV'/2a^(a^+2a Cos "^-3)-^ (a^-6a^4-l)-2ra (l4-a^)CosK}

+ W Sin 0 27cR

(2)

The above expressions for N ^ are given by Goodey, where for the loaded frame K=l.

In expression (l) above

^-, and/"vp are obtained as shown below. Solve the equation

X^ + UX + V = 0

₍₃₎

where

U = /n^(n^-l)^ (n^-2e) - kl (U)

-3-V = 4 L ,n^(n^-l)^ f n ^ e ) | (5)

[.. c

The roots of equation'(3) are X-, and Xg Next solve the equations

X + 1 = X^ and A + 1 = Xg ~ ("6) The roots of equations (6) areA,-, and/\, , and^X^ and

^. respectively, whereA,= 1 a n d A 2 = 1 •

The values of A. used in the expression (l) for N „ are those less than unity.

If the roots X,and Xp of equation (3) are less than 2.0 then we use expression (2) for N ^ .

" a'' and ")/'' are obtained as shovirn below.

Y^-(V+i+)Y+ U^ = 0 (7) v/here U and V are given by expressions k and 5.

The roots of equation (7) are Y, and Yp vi/here one root is greater t h a n ^ . 0 and one less than 4.0.

Let Y, be the root greater than k>0 and Yg the root less than k.Os we then solve the equations

(a + 1) = +/Y. and (2 Cos'V)^ = Yg — ( 8 ) a

The values of " a" found are a, and ao where a-. =1 ap and the value of " a " used in the expression (2) for N f. is that which is less than unity.

The value of '•L/'" used in the exuression (2) for N ^ is the value given by Yp which is of opposite sign to " U" in expression (k). This is so, because

- U = 2(a+l) Cos V a

the term (a-t-^) is alv/ays +ve hence Cos'y'must be -ve, i. e, the sign of " y" must be opposite to that of " U" The expression for N ^ can be written in the form

^ = W \~'P(ATAok,r) n Sin n0 + W Sin 0 (9)

xo ^-^ ^.^ 1 ^ 27^R

or

N^n.= I y P(a,>,k,r) n Sin n0 + I Sin 0 (10)

^" 7cR -•-- 2%-R

n=l

In both expressions 9 and 10 the value of F( ) n Sin n0 for n=l is equal to Sin 0

2

For practj^cal purposes it is sufficiently accurate to take the > term UTD- to a value of n=6, hence the

n=l

.6,

-'^-N^Q = W Sine + W \ 'F( )n Sin n0 (11) TcR %'R "C-J

n=2

The curves of figs. 5-20 were obtained by putting

numerical values in the expression (11) above. The parameters for each particular case are given in the accompanying curves.

I am indebted to Dr. Kirkby and the com-puting section of the Aerodynamics Department for the valuable aid given in the computation of the numerical examples.

Conclusions

1) The distribution of shear around the frame is not sensitive to variations in values of " k" the results shov/ a maxim\im increase of 7/o in value of N ^ at 30 (i.e. maximum value) for a 100/6 increase in value of k , see figs, I5, I6 and I7.

2) The distribution of shear around the frame is not sensitive to variations in skin thickness " t" , In aircraft structures going from one gauge to the next is approximately an increase of 30/6 and for this increase in skin thickness the average percentage decrease in maximum value of N ^ is k/o. This is assuming that k remains constant.

3) The distribution of shear is not sensitive for reasonably large variations in the moment of inertia of the adjacent frames, and to take a mean value of I^ is sufficiently accurate for the results. The effect can be obtained from fig. (5) where it is seen that increasing the stiffness of the adjacent frames, the loaded frame remaining constant increases the maximum value of N Q. For the parameters chosen it

is seen that the average nercentage increase in value ,9 at 30° - - -^'

of I

of N Q a t 30 i s 0,12/0 for a 1.0/5 increase in value

^f*

k) The distribution of shear is not sensitive for reasonably large variations in the moment of inertia of the loaded frame, and a mean value is sufficiently accurate for the results. This effect can be obtain-ed from figs. 12, 13, 1/4, 18, 19 and 20.

5) The distribution of shear is not sensitive for moderate variations in frame spacing. Por a frame radius up to kO" the average increase in value of N g at 30 is 0,k% for a l.O^o increase in value of

"m" and for a radius of 60" the increase in value of ^x0 ^* ^^ ^^ 0.22% for a 1.0% increase in 'hi" . This effect can be obtained from figs. I8, 19, and 20. 6) The beam theory distribution of shear is suffic-iently accurate if the value of rl^ lies above the line of fig.21, • This curve is obtained by extra-polation of the curves figs.12, 13 and 14, and is intended to give the lowest value of rio for which the beam theory is a reasonable approximation.

-5-7) The shear loading on frames adjacent to the loaded frame can be obtained with sufficient accuracy for practical purposes by interpolation. The shear dis-tribution on a frame ^ frame spacings distant from the loaded frame can be taken as that given by the beam theory and intermediate frames can be obtained by a straight line interpolation between the frames, see

i ^Loaded 4^/jmiL W

Pig. (k)

Reference

(1) Royal Aeronautical Society Journal Nov.1946. " The stresses in a circular fuselage" by W.J,Goodey.

2 4. Nov 1950

Page 6, Expression for TR should read

r>2% p

-TR ^ (N^Q ) R d0.

The -ve sign for (TR) should be put throughout the expressions and on page 8, express ion (14) becomes

6

(^V^eU

^

^ I {-^ - ° ° ^ Q - E F(—-)c°^ nej—(1

- At the bottom of page 8, expression becomes

APPENDIX A

Derivation of the shear load due to tangential load and moment from the radial load expression.

Goodey in his work indicates a method by which the shear load for a radial load and moment

may be obtained from the expression for a tangentially at)plied load. Here the complete solution is given for obtaining the shear load due to a tangential load and moment from the radial load exuressions,

Let (N Q ) = shear load due to a tangentially applied load T

This will be a function of T and 0. (^xeV = -Tf(0-a) = -Tf(0) + Taf'(0) when a is small

Ta = W and T = T Cos a.

•''^-^^XQ\-= Taf'(0) = W f ( 0 ) = (N^Q)^^

6=0 '• Jo

(12)

where the term \(jj ) j is a constant of integration and can be evaluated by considering the equilibrium of the frame. r2%

TR =1

JO (Nxö) R d6. )=o ^ 2 2^ 2%R^ + T R W 2% -d u O J o

i (Nx0)^(^)^^ ^^'

2 r 2 '•'^'^

-7-2 1 } -7-2 • '^'^

V. -'ö=0 .J o

f ) • [2^

•'* ^^^""^9^/0=0 ^2iR " 2!^ : ^ (\0)w(^)^^ ^^i^^ gi^^^ ' ü o

the constant of integration for (12) above.

ü" to Por the case considered, of a loaded frame along a

cylinder.

(N^a)™ = ^^ '^'F( )n Sin n0 + W Sin 0

= W Sin 0 + W v; p( )n Sin n0

%R %R ^-'2

r2x ,-27. / 6

• ' • I 1 ^(^^J^Jd0 = 1 ' Ö /w Sin 0 + W C F ( )n Sin no; d0

^\lo ^^ '' '^ ;o (^R ^R ^ 2

r^^ • ' e

f 2 ^

= 1 ? I 0 Sin 0 d0 + I E > P( )1 0 n Sin n 0 d0

.' o n = 2 '.J o

2% j6_,,_ ^ .,2%

~o-2-o 'Sin 0-0 Cos 0 j "^0-2^ x-",F( )n/Sin n0-n0 Cos .n0 |

° n=2 . ;_ n -^o 27C^R I I 27C2R ^ ^ ^ ^ -'So ;" "^^^ n=<i n*^ -6

= - T - T j r p ( . . „ ) .

TcR TtR ^ - - ^ ^ ^

s u b s t i t u t i n g in equation (13)

o 6 f' -.0 = - I - T ). p ( ) + T.'I W S i n 0 d0 27i:R ^cR ' ^ - ^ Wi I _TCR + W > ' P ( ) ; n S i n n0 d0;

^^

È^'^

J o . j

8 -27.R 7.R ^i^ ^ w ^R - ^R ^^.^ ' ïï L J = _ T - T v p ( ) _ T Cos 0 27cR TcR ^^^ ^R ' i ' i i " ( - — ^ - I f F ( ) c o s n0 = T - T Cos 0 - T é-F( ) Cos n 0 . 27iR %R xR '"-U

^'" A 1

= ï ! l - Cos 0 - > P ( ) Cos n0 ( 1 4 ) ^R (^2 n=2 .. O L e t (N p^) = s h e a r l o a d due t o a n a p - p l i e d moment. X Ö i •}

(N^el =M jf(0).,., + a f f i M r /

d0 -T/here f (0) i s t h e s h e a r due t o u n i t t a n g e n t i a l l o a d . T ^ {Q\ = 1 j i - Cos 0 - >'' P ( )Cos n df s e e ( 1 4 ) a b o v e TtR | 2 n='^

f ' ( e ) ^ = 1 I s i n 0 + 2 1 5 ^ P( )n Sin Q0>

TCR i n=2 ^ j LJ(Q)r = 1 / c o s 0 + ) " ' ' p ( ) n ^ C o s n0!'

d2e ^R I n=2 )

^^^^- = i k - ^ ^ - i i/(--)Oos n0

i n=2 + Cos 0 + 1 \ f ^ F ( ) n ^ C o s n0j> %'R %R '••' -o n = ^ ,2 : '1 - y {l-n)F( ) Cos n 0) xR^ ( 2 ri^^ .oooOooo,

n

n

c ^ o 01

3 0

VALUES OF S H E A R / I N S FOR VARVINQ RADII O F F R A M E SP^WCINO CONSTANT AT 15 INS.

RADIAL A P P L I E D L O A D O F lOOO lb. FRAMES. TO m

z

I .

"is

g

8

5 0 4 0 5 0 lb/INS 20 lO

Tl

O

/ \

r:

w

1

\ -I

VALUES O F SHEAR/ INS FOR F R A M E Sf=WOtNG. RAOIAL A P P U E D LOAD OF I 1 1 L - I 5 0 INS ! ƒ B OS I N S * K« s l O k x « l'S3 fc s - 0 2 © INS X ' ^ ^

r7

VARYING RADII O F FRAMES A N D CONSTANT

OOO lb. • R s 1-62 INS m s I 0 3 + R » 4 0 IMS m = 2-67 O R s 6 0 INS in - 4 " 0

xT

^

88

Sc

^8

So

01 ^

n

Q

5

C H

8

2 0 6 0 JOO

iSZ'

i4o I 8 0

COLLEGE OF AERONAUTICS REPORT N a . 3 3 .

So

^f

5o

zo

l« Tl Olm • n

o

z

>

c

H O ISO

/ ,

I

n

^ / ^ • ^ VALUES C R A D I A L M » 2 0 k j c = 1-53 t « - 0 2 8 DF S H E A R A P P L I E D INS ^ ^ ~ ^ / I N S F O R L O A D Of -^ ^ VARVING - 1 0 0 0 l b . R A D I I O F _{F R A M E S .}

1

. X f - 1946. INS"*" Q a 4 0 INS. + I f = ' 1 9 4 6 . I N S ^ R « 6 0 INS. ^ \ ' ^

k.

. 2 0 S O 100 I 4 0 B O

e*

S8

K.O • ni

B

Z > c H O

1 \ 1

VALUES OF SHEAR/ INS FOR VARVING STIFFNESS OF LOADED FRAME: FRAME RADIUS CONSTAINT

RADIAL APPLIED LOAD OF lOOO lb. -4-I f » 'OS -4-INS R • 4 0 INS. in s SO ka^s 1-55 t s OPSINS. • r « 2 5 . I^If « 1-25. IN5^ 4-1* « 50. r'lf s zs © »* s lOO r i f s 5 ' 0 A = B E A M T H E O R V

30 15

^ . ^

y

\ \ >

1 1 1 1 I I

_{1 1 ' 1 1 i} i ' ' • VALUES O F S H E A R / I N S F O R V A R V I N G S T I F F N E S S O F L O A D E D F R A M E ^ F Q A M E R A D I U S C O N S T A N T . 1 R A D I A L A P P L I E O L O A D O F lOOO I b . I I I 1 1 I ƒ s OS INS. •*• R » 6 0 INS. m. s 2 0 k x . « I-S3 t s 0 2 8 INS. - f - * > ^ •

i_-4---^

1 1

\'y- a 2 5 r*If SS I-2S iNs:*- 1 1

L - h = 5 0 h i f s 2 5 •• O»* « l o o h i f » 5 o " j _ | A » B E A M T H E O R V |_J < > ^ ^ ^ 2 0 8 0 lOO l 4 0 1 8 0

COLLEGE OF AERONAUTICS R E P O R T Nft 3 3 . '

ÜI

1

z

tf) 7

i .

is •

^B

0 ? «

VALUE S RADIA L OATM l le t 1 •• tf) 0 P 0 - w « H j ;

u ^ J.

«5 tf) $

z Z 2

tf) 10 tfj «jl <vi W ^ ^ M -11 n u

a a a

, ® +

't.

) tf) tf)

: Z I

1 < ^ «0 Q) <D . 0 4 (OW IN ? i b 9 II ii II H 1 m • ! > ^ « ^ " ^ ^ ^ ^ •

y

• '. : ^ *^ / /

t

₁

J

/ .

^s=^

' 0

I

0 0 0

o

0 tf) 0 QJ Z :9 R G . I S .

VALUES OF S H E A R / I N S FOR VARIATIONS IN VALUES O F U j RADIAL A P P L I E D L O A D O F lOOO l b . RADIUS s 4 0 I N S

J.

m

I f

h t

* so

s 1 9 4 6 INS = 3 0 9 2 « ' 0 2 S (NS. . R = 4 0 INS. e R a 4 0 -+ R e 4 0 " k a t » O S k x » l O k x » &0

&0

_{e o}

_too

i 4 0

_leo

1 •

k

r^

{

\ \ VALUES O F SHBAC R A D I A L A P P L I E D

1

RADIUS = S O INS.

If

1*

t

^ » 20 a '1946 IN s» 3 S S 2 = O S S IN ; ^ ^ / I N S F O R L O A D Of

s:*

-1 -1

VARIATIONS IN VALUES O F k - l O O O I b . . R a S O INS k x «s - 5 O R a 6 0 " k x s!lO . •f R a 6 0 « k x « 2 0 J ^ • - ^ X .

2 0

_êo

lOO I 4 0

_leo

C O L L E G E O F A E R O N A U T I C S R E P O R T N 2 . 3 3 . VARIATIONS IN V A L U E S O F N » * A T S O * 5 0 , J- p Q p VARVING V A L U E S O F ^ I f A N D m." L\\ R A D I A L A P P L I E D L O A D O F l O O O l b . 5 0 i r l | INS VARIATIONS IN VALUES O F N x o AT S O * F O R V A R V I N G V A L U E S O F Y^Ir^y/l. g, I r . R A D I A L A P P L I E D L O A D O F l O O O I b .

C O L L E G E O F AERONAUTICS