Optimun design of a band reinforced pressurised cylinder

H CoA Note No. 116

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

CRANFIELD

OPTIMUM DESIGN OF A BAND REINFORCED

PRESSURISED CYLINDER

by

NOTE NO. 116 April, 1961.

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

Optimum Design of a Band Reinforced P r e s s u r i s e d Cylinder

b y

-D. S. Houghton, M.Sc.(Eng.), A . M . I . M e c h . E . , A . F . R . A e . S .

SUMMARY

The surface s t r e s s e s in band reinforced cylindrical p r e s s u r e vessels are examined, and an equivalent s t r e s s determined by using the M i s e s -Hencky criterion. By comparing the equivalent s t r e s s to the band s t r e s s , the efficiency of the structural material can be established, and by

equating these s t r e s s e s to their respective yield s t r e s s e s , the theoretical maximum strength of the structure can be found. Once the material

properties of the shell and the reinforcing bands have been specified, the optimum structural layout can be determined.

CONTENTS V Page Summary Notation 1. Introduction 1 2. The determination of the s t r e s s distribution 2

3 . The derivation of cr and c 4 1 2

4 . An exam.ination of the parameter ^ 5

5. Discussion of results 6

6. References 7 Figures 1 - 6

NOTATION A c r o s s s e c t i o n a l a r e a D f l e x u r a l r i g i d i t y of shell D ' f l e x u r a l r i g i d i t y of s h e l l - s t r i n g e r combination E Young's modulus F axial c o n a p r e s s i v e force E t ' - ^ k /9 4 D ' R ^ t length of s h e l l , d i s t a n c e between m i d - b a n d p o s i t i o n s

M bending m o m e n t p e r unit width M n o n - d i m e n s i o n a l bending m o m e n t N r a d i a l s h e a r f o r c e p e r unit width p i n t e r n a l p r e s s u r e R s h e l l r a d i u s t s h e l l t h i c k n e s s t ' , t effective s h e l l t h i c k n e s s e

T , T r e s u l t a n t longitudinal and c i r c u n a f e r e n t i a l f o r c e p e r unit width w r a d i a l d i s p l a c e m e n t w <i p R' 2 yield s t r e s s F vrp R^ 2t k A^ b y y defined by equation (8) 1 2

Notation (Continued)

e , e„ longitudinal and circumferential strains k*

V P o i s s o n ' s r a t i o (taken a s 0.3)

cr o' s h e l l s u r f a c e s t r e s s e s in longitudinal and c i r c u m f e r e n t i a l ' ^ d i r e c t i o n s

cr equivalent s t r e s s defined by equation (13)

<t> 1 + i y(a - 1 ) ^

e Subscripts

s refers to reinforcing stringers b refers to reinforcing band

1

-1, Introduction

In the design of thin shell s t r u c t u r e s , internal p r e s s u r e can be considered as an effective means of stabilising the body shell against collapse which might otherwise occur due to axial compressive forces. These internal p r e s s u r e forces can be sufficiently large to cause

bursting of the structure and a limiting allowable hoop s t r e s s is generally specified, which in turn limits the stabilising p r e s s u r e which can be used.

It is well known that longitudinal stringers can be introduced in order to assist the shell to c a r r y the axial forces, in much the same way as is used on a conventional aircraft structure. However, unless the axial forces a r e very high, this form of construction is structurally l e s s efficient than the p r e s s u r e stabilised shell^^^. A combined form of stabilisation may be considered, by using both pressurisation and longitudinal stiffeners, but the effect of the stiffeners on the maximum hoop can be shown to be small, so that a comparatively heavy structure would result.

One form of construction which could be used, might be in the form of radial bands, which are spaced sufficiently close together, so that the hoop s t r e s s in the shell is considerably reduced. This could mean that the internal p r e s s u r e might be increased to sustain a greater axial load, and the bands would help to stabilise the shell during manufacture, and so prevent large shell distortions,

An experimental investigation into the s t r e s s distribution in a band reinforced p r e s s u r e vessel has been made by Mantle, Marshall and Palmer^^', where in this case very closely spaced bands were used. It was found that this form of construction enabled an efficient shell structure to be designed.

The following analysis investigates the s t r e s s distribution in a band reinforced cylindrical shell, which is subjected to combined axial load and internal p r e s s u r e .

By comparing the maximum s t r e s s in the shell and reinforcing bands, the optimum geometry can be established.

2 . T h e d e t e r m i n a t i o n of the s t r e s s d i s t r i b u t i o n

If the c y l i n d r i c a l s h e l l of r a d i u s R i s subjected to a c o n a p r e s s i v e a x i a l f o r c e F , and i n t e r n a l p r e s s u r e p , then the r e s u l t a n t longitudinal f o r c e / i n T b e c o m e s

o r T = i p R d - a ) . (1) If the s h e l l h a s longitudinal s t r i n g e r s , then t h i s f o r c e can be e x p r e s s e d

in t e r m s of the s t r a i n c o m p o n e n t s a s E t

T = E t e + (e + ve^) , (2) i s s x j ^ _ j ^ 2 X 6 '

w h e r e t i s the effective t h i c k n e s s of the s t r i n g e r s . The f i r s t t e r m on the right hand side will of c o u r s e d i s a p p e a r if only band r e i n f o r c e -m e n t i s u s e d .

The hoop f o r c e / i n b e c o m e s

T, = ^ ^ , (e +ve ) , (3) 1 _ y2 e X

E q u a t i o n s (1), (2) and (3) can be combined to give

T^ = E t ' l + i p R ( l -a)v^- , (4) e

^ ^ ^ ^ ^ t = t + t (1 - . ^ ) ^

e s E

1 -v^ *e

T h e e q u i l i b r i u m conditions of an e l e m e n t of the s h e l l F i g . 1 a r e given a s

dN ^ '^2 , -, dM . . . - + p = - , and N = - - , (5) w h e r e M and N a r e the bending m o m e n t and r a d i a l f o r c e / i n .

The r e l a t i o n s h i p between the bending m o m e n t and the r a d i a l d i s p l a c e m e n t w i s

3

-w h e r e D ' i s an effective bending r i g i d i t y of the s t r i n g e r - s h e l l combination. Substitution of equations (5) (6) into (4) gives a f o r m of the well known differential equation d'^w . . , 4 p 0 ,„x + 4 k w = ^ , (7) dx" D , t Et w h e r e 6 = l+iv(a-l)- , and 4 k* = . *e D ' l f

If the o r i g i n i s taken about a point on the s h e l l which i s midway between t h e band r e i n f o r c e m e n t s , then the r a d i a l d i s p l a c e m e n t b e c o m e s

w = w + C, cos kx cosh kx + C„ sin kx sinh k x ,

o ^ 2

w h e r e w = 6 i ^—r o ^ \ E t ,

dw •^

T h e conditions of z e r o slope at the f r a m e s ( i . e . -r— = 0 at x = + — ), and e q u i l i b r i u m between the f o r c e s in the band, and the s h e l l g i v e s

W i = - 2 N R , a t x - ± | ,

w h e r e _ / d ^ w dx^

T h e r e a r e two conditions which give C = - w y and C = - w y ,

" 1 o 1 a 0 2 '

, sin 77 cosh 77 + cos 77 sinh 77

^^^""^ \ ~- rT~~:—•. „ . . 2t^—B

k A^ E, b b sin 77 cosh 77 - cos 77 sinh 7?

^^ ^ T T ~ ~ I ' . , „ . . 2t' Ë

(8)

2

and

I ( s i n 2 77 + sinh 2 77 ) + —-r- = - (sin^ 77 + sinh^r7)

k^ ^ ^

Hence w = w (1 - y cos kx cosh kx = y sin kx sinh kx) . (9)

o 1 2

T h i s equation d e s c r i b e s the a x i - s y m m e t r i c r a d i a l d i s p l a c e m e n t s of the s h e l l along i t s length. In t h i s a n a l y s i s , an effective s t r i n g e r skin w a s u s e d , so that t h i s equation i s incapable of r e c o g n i s i n g the fact that s h e l l quilting between s t r i n g e r s might o c c u r for l a r g e v a l u e s of R / t .

4

-If cr, is the band s t r e s s , then

D

^b '- \ l ' ^henx=±| .

Hence from equation (9),

cr = E, ~ 11 - y^ cos 77 cosh 77 - y ^ sin 77 sinh 77 J (10) Having established the radial displacement of the shell, it is a simple

matter to establish the distribution of the shear force and bending moment from equations (5) and (6).

These s t r e s s e s together with the membrane s t r e s s e s arising from equatione (1) and (4) constitute the s t r e s s distribution in the shell.

If no longitudinal s t r i n g e r s exist, then the surface s t r e s s e s in the shell become

6D d^w , 1 pR n . n i l or = —r + 2 T" (1 - a ) (11)

' t dx t

and , Ew . , „ . The other s t r e s s components a r e all z e r o .

3 The design condition for the shell, is that the equivalent s t r e s s is given by the equation

o^ + G^ - (T cr = "? (13) 1 2 1 2

3. The derivation of cr and cr

1 2

The bending moment in the shell M, can be found from equations (6) and (9) as

— = M' = y sin kx sinh kx - y cos kx cosh kx ,

9 P 1 2

where y and y a r e given in equation (8). 1 2

5

-Since i> = 1 + 51; ( a - 1), when no s t r i n g e r s a r e p r e s e n t , then „

- ( 0 - 1) = a - 1,

and equation (11) can be w r i t t e n a s

1 k^ t^ yt

F o r an u n r e i n f o r c e d s h e l l , k^ = —•=- , and cr b e c o m e s Kt 1

cr l~3 (1 _ ffl) —ir = 0 ( y sin kx sinh kx y cos kx cosh kx)

-pR \ 1 1 _y2 1 2 ^

t (14) S i m i l a r l y from equation (12) cr b e c o m e s

2 cr^

fs

—|r = ï^, \z 2 M ' 0 - 0 ( y COS kx cosh kx +y sin kx sinh kx) + 1,

p R N l - y ^ ^ If w ^ /g / ,

t

cr To"—

—7: = y ci(y sin kx sinh kx - y c o s kx c o s h kx) £R \\l _i;2 ^ 2

t

- i> {y cos kx cosh kx + y sin kx sinh kx) + 1 . (15)

o r 4 . An e x a m i n a t i o n of the p a r a m e t e r ^ F o r a s h e l l having no longitudinal s t i f f e n e r s , 0 = 1 + J (ffl. - 1) , w h e r e a = TrpR^

If no a x i a l c o m p r e s s i v e force F e x i s t s , and the s h e l l i s subjected to the i n t e r n a l p r e s s u r e only, then

0 = 0.85 if V = 0.3 (16) H o w e v e r , for the c a s e of a p r e s s u r i s e d s h e l l having an axial c o m p r e s s i v e

f o r c e , then a and h e n c e 0 , i s a function of that f o r c e , in a c c o r d a n c e with the above e q u a t i o n s .

6

-F o r the ballistic missile application, if one introduces the stabilising p r e s s u r e in order to produce zero longitudinal s t r e s s in the shell,^ then

a = 1 and 0 = 1 (17) Hence the limiting conditions for ^ , a r e that it must be between the

values given in equations (16) and (17) i . e . .85 < 0 < 1,0 .

In the following work, and the figures which are presented, only the lower value of 96 is considered.

5. Discussion of Results

The band s t r e s s given by equation (10) is presented in Fig. 2 for various shell geometries. The analysis assunaes that the band depth is small conapared with the radius, and if this is not the case, it would be better to replace the shell radius with the centroidal radius of the band. The only s t r e s s which is considered in the band is the hoop

s t r e s s , as this will be the only significant s t r e s s , unless the band width is large compared with its thickness.

The Fig. 2 suggests that a maxinaum value of the band s t r e s s is developed for a shell parameter 77 = 1 . 5 , but for most missile applications the value of 77 will generally exceed this value.

2

F o r the commercial application referred to e a r l i e r , where conaparatively large shell thicknesses a r e used, the value of n will be small, and hence the s t r e s s developed in the bands will be small conapared with the nominal hoop s t r e s s in the shell.

Examination of equations (14) and (15) together with equation (13) suggests that the s t r e s s conditions at only two points in the shell need be examined. These a r e at the ends of the shell (when x = ± £) and at

2

the mid-shell position (x = 0). The equivalent shell s t r e s s cr , is shown as a ratio of the nominal hoop s t r e s s at these two positions in F i g s . 3 and 4, for various shell geometries, and the equivalent shell s t r e s s to frame s t r e s s ratio a is presented in F i g s . 5 and 6.

7

-The theoretical maximum strength of the band reinforced shell will be obtained when simultaneous yielding occurs in both the band and the cylinder. Hence for any shell having a known geometry, and hence 77 and /?, simultaneous yielding will occur when a = Y and cr, = Y , and the ratio Y_ will be given by Figs. 5 and 6. Clearly

b

Y < Y so that a lower grade of material can be efficiently used for the bands.

Once the properties Y and Y, are known, and the values of R, t and I given, then the value of 77 is known. Hence the value of /?, can be found, and the optimum value of A^ determined.

6. References

1. Sandorff, P . E . Structures considerations in design

for space boosters.

A . R . S . Journal, vol.30. No. 11, November 1960, pp 999.

2. Mantle, K.G. ,

Marshall, N . , P a l m e r , P . J .

Experimental investigation into the s t r e s s distribution in a band -reinforced p r e s s u r e vessel.

P r o c . Inst. M e c h . E n g s . , vol.173, 1959, pp 123.

3. Hill, R. The mathematical theory of plasticity.

Clarendon P r e s s , Oxford. 1950.

4. Houghton, D . S . ,

Johns, D . J .

An analysis of an unstiffened cylindrical shell subjected to internal p r e s s u r e and axial loading.

4X

FIG. 1. Forces and moments on an element of the shell and geometry of reinforced cylinder oe SI ^1 0-2 1. M^

I T - + + " ^..

J

1

1

f

-•"'••-4"f;

.J

1

7

i

1

/5 IS THE BAND FLEXIBILITY PARAMETER

WHERE ^ ^ = - ^ OS l O F 1-5 2 0 6 4 3 / 2 5 3 0

FIG. 2. The s t r e s s in reinforcing bands for various shell geometries and band flexibilities

FIG. 3. The equivalent surface s t r e s s in the shell at its end (x = * )

cr o-s 2 0 < 5-2 !• C 3 ^ . / / • / ^

7

f

f

/5 IS THE BAND FLEXrBILITY PARAMETER

WHERE / ) ^ . - ^

os l'O 1-5 2 0 2S 3 0

FIG. 4. The equivalent surface s t r e s s in the shell at its centre (x = 0)

3-5 Ob y ^ /

Y

^1^

y^

y

FIG. 5. The ratio of the equivalent s t r e s s to batul hoop s t r e s s at the end of the Shell (x = + i )

2

FIG. 6. The ratio of the equivalent s t r e s s to band hoop s t r e s s at the centre of the

Shell (x = 0)

IS

2