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N A S A C O N T R A C T O R

R E P O R T

AEROSPACEPRESSUREVESSELDESIGN SYNTHESIS

by George Gerard

Prepared under Contract No. NA Sw-928 byAL L IED RESEARCH A SSOCIAT ES, I NC .Concord, Mass.

or

N AT I O N A LE R O N A U T I C SND SPACE AD MINISTR ATION WASHINGTON,D. C. 0 A U G U S T1 9 6 5

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TECHLIBRARYKAFB. NY

AEROSPACE PRESSURE VESSEL DESIGN SY NTHESIS

By George Gerard

Distribution of this report is provided in the i nterest ofinformation exchange. Responsibil i ty or he contentsresides i n the author or organization that prepared i t.

Prepared under Contract No. NASw-928 byALLIED RESEARCH ASSOCIA TES, INC.

Concord, M ass.

for

NATIONAL AERONA UTICS AND SPACE ADMINISTRATION

F o rs a l e b y t h e C l e a r i n g h o u s e f or F e d e r a lS c i e n t i f i ca n d Te c h n i c a l n f o r m a t i o nS p r i n g f i e l d ,Vi r g i n i a 22151 - P r i c e $3.00

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Summary

A governing structures-materials-design synthesis relationship i s deri ved forthe pri mary structural wei ght of membrane type pressure vessel s. T he structural

eff iciency i s associated wi th the configuration and the fail ure law characteri zi ng the

materi al used. Cl osed pressure vessel s of various shapes uti l izing monoli thic and

f i l amentary materi als are examined in some detai l to establi sh optimum desi gns.

T he structural strength/weight ratio has a prof ound inf l uence up.on the pressure

vessel eff iciency. V alues of this ratio real i zed currently i n monol i thic and f i l amentary

designs are evaluated, L ikewi se, the potential of ani sotropic metalo, f i l amentary-

monol i thic composi tes and whi sker composites s studied. The configuration and

materi al eff i ci enci es are then combined to investigate the comparative ef f i ci enci es

of pressure vessel s of vari ous shapes and materi als concepts.

i i i

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T able of Contents

SummarySymbols

1. I ntroducti on

2. Structures-M ateri als-D esign ynthesis

Governing EquationsM onoli thic M embranes - M aximum Shear L a wF i l amentary I sotensoid M embranesM onol i thi c M embranes - Octahedral Shear L aw

3 . Structural Configuration Efficiencies

B asic C onfi gurationsCyl i nders W i th Cl osuresSummary of ,R esul ts

4.E f fi ciencies of M ateri als

High Strength Sheet M etal sFilamentary CompositesM ateri al s for I nfl atable StructuresC omparati ve E f f i ci encies of M ateri al s

5. Potential of N ewer M ateri al s Concepts

A nisotropic M etalsT exture H ardeningM echanical A nisotropyF i l amentary-M onol i thi c C omposi tesW hisker C omposi tes

6. Overall Pressure V essel EfficienciesR eferences

i i ivi i

1

2

2578

10

10131518

19263132

34

3435394248

50

53

V

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a

A

b

C CC

d

e

h

kekL

P

rR

-

P

s

S

t

t

V

-

W

W

z(Y

c

P0

c+

Symbols

texture hardening coef f i ci ent, a = Z / C

surface area

mechanical ani sotropy coef f i ci ent, b = C / X

f i lament croBs-over coefficientstructural configuration eff iciency coefficient

di ameter

ductil i ty ratio

el l i psoi dal cl osure mi nor diameter

elastic stress concentration factor

pl asti c stress concentration f actor

overal l l ength of preseure vessel

pressure, psi

radial coordinateprincipal radius of curvature

ar c length

uniaxial structural strength, psi

thicknes s

average thi ckness

vol ume, n

weight penalty coefficient

weight, bs.

axial coordinatethickness coeffi cient

strain

density, pci

stress, psi

uniaxial strength, psi

angle

1 3

1 2

3

Subscripts

a anisotropic

C cylindere ellipsoidal

f filamentary

h hemi spheri cal

i isotropic

m monolithic

tuensionltimate

1,2,3 principal directions

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A EROSPACE PRESSURE V ESSEL DESIGN SY NTHESIS

1. I ntroduction

T he util i zation of pressure vessels i n aerospace appli cations i s manifold.

Consequently, it i s the objecti ve here to examine systemati cal l y those parameters

which have a major influence upon the weight of thin wall pressure vessel s under

speci f ied design conditi ons.

Si nce our i nterest here i s i n a broad design synthesis viewpoint which i s

generall y appli cabl e i n the preli minary design stage, we shal l be concerned wi th thepri mary structural wei ght associ ated wi th optimum membrane type pressure vessel s.

I t i s assumed that the secondary weight comprises the addi ti onal materi al associ ated

wi th nonoptimum membrane thicknesses, disconti nuiti es, oints, cutouts and f i ttings.

A ccordingly, Section 2 presents a general i zed treatment of the governing

primary weight equation which relates the structural conf iguration ef f i ci ency, the

materi al eff i ciency and the prescri bed design conditi ons for several dif ferent fai l ure

cri teri a. T he confi guration ef f i ciencies of var i ous pressure vessel shapes are treated

in Section 3 and encompasses both simple shapes as well as cyli ndri cal vessel s wi th

closures.

In Section 4 , the structural strength/weight ratios attai ned wi th current mono-

l i thi c metal l i ce , filamentary composites and inflatable structuree are evaluated.T he potential of newer materials concepts such as anisotropic materials, filamentary-

monol i thic composites, and whisker composites i s evaluated in Section 5.

T he conf iguration eff i ciencies and materi al eff i ciencies considered separately

in Sections 3- 5 ar e combined in Section 6 to treat the overall eff i ci ency of pressurevessels util i zi ng vari ous shapes and material s. T he resul ts of a comparative

eff i ci ency study are presented i n a design synthesi s chart whi ch summarizes themaj or resul ts of thi s investigation.

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2. Structures-M ateri al s-D esign Synthesis*

T he optimum design problem for pressure vessel s can be stated in the fol l ow-

i ng manner: f or prescri bed pressure (p) and vol ume (V ) determine the struc tural

configuration and material that resul ts in a minimum weight design. T he fol lowingdevelopment is based upon the simpli fyi ng assumption that the pressure vessel can

be treated as a membrane and,therefore, represents the primary structural weight

as defi ned i n the preceding secti on.

T his probl em has been consi dered n vari ous aspects by Schuerch , Hoffman ,4

1 2

Pipkin and Ri vl in3, and B rewer and J eppeson for f i l amentary sotensoids and al so

for monol i thic membranes. In the fol lowing, a systematic development of the design

synthesis equati on s presented for three cases: opti mum monoli thic membranes

that fail accordi ng to the maxi mum shear l aw, f i l amentary membranes, and mono-

l i thic membranes that fail according to the octahedral shear law. T he atter resul ts,

whi ch were not obtained in the above ci ted references, can represent an improvement

i n structural eff i ci ency as compared to the maxi mum shear case.

Governing Equations

In general 'form, we have the fol lowi ng relationships for the membrane of

revolution shown n F ig. 1. T he weight of an elemental ri ng of radi us, r , and width,

ds, s

dW = 2rrprt ds1)

T he equations of equil i bri um i n terms of pri nci pal stresses and radi i of curvature

ar e as f ollows:

u2 = pR1/2t

By substi tuting Eq. ( 3) into (2), we obtain

*T he contributions of C. L akshmikantham to Sections 2 and 3 are grateful l yacknowledged.

2

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- Z

Note: R2 is principal radiusof curvature of p ro f i l e r ( z)

Figure 1 Pressure V essel M embrane

3

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F or u to be posi tive (tension), the fol lowi ng condi tion i s imposed upon Eq. (4)1

T his condition i s necessary f or an i sotensoid f i l amentary membrane and s also

desi rable f or a monol i thic membrane to avoid buckl i ng. Furthermore, n the fol l ow-

ing development for monoli thic membranes i t i s convenient (although not essential )

that crl > cr2. F or thi s purpose we can mpose the more restri cti ve condi ti on on

Eq. ( 4 )

-

R2 for ul - 2u

In order to determine the minimum weight design for a prescribed pressure

and vol ume n a general manner, we can ntegrate E q. (1)

W / p = 2a / r t ds ( 7)

Now, i f represents the thi ckness averaged over the surf ace area, then E q. (7)

can be wri tten as

Note that Eq. (8) represents the vol ume of struc tural materi al rel ati ve to the encl osedvol ume and as such i s equivalent to the sol idity famil iarl y used i n the minimum weight

analysis of compression structures.

F or a given. shape wi th u1 = Zl , where Z1 represents the f ail ure strength of the

materi al , E q. ( 4 ) can be put n the following form,

-t = ap/C1

Substi tuting Eq. (9) into (8)

w = C(p/Z1)pV

where: C = aA/V

4

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I t can be observed that the structural configuration eff i ci ency f actor (C ) i s a non-

dimensional function of the membrane shape. In additi on to C , Eq. (10) contains the

materi al eff i ci ency parameter (p/Zl ) and the design index (pV ) representing the

prescribed design conditions.

In order to determine values for C, a f ail ure law descri ptive of the materi al

i s requi red to obtain the mini mum weight design. I n the fol l owi ng, three different

failure laws are examined in conjunction with the assumption that each point on the

surface i s subjected to the local fail ure strength and thus optimum thickness i s

achieved.

M onoli thic M embranes - M aximum Shear L awAs perhaps the si mplest example, we consi der first a monol i thic membrane

designed accordi ng to the maxi mum shear l aw as the f ai l ure cri teri on. T his cri ter-

i on can be used for yi eld or f racture strength accordi ng to the behavi or of the

material under consideration. Denoting the fail ure strength as X1 as ndicated n

Fig. 2, and assumi ng that each element on the membrane surf ace i s subject to Zl

simul taneously , then the optimum thickness i s obtained directly f rom Eq. ( 4 ) since

u1 = Zl.

By substituting Eq. (1 1) nto (1) and ntegrating, we obtain

Eq. (8) can conveniently be wri tten in the form of Eq. (1 0), where now

CV = H j R1 (2-R1/R2)dz13)

I n obtaining Eq. (13), the relati on r = Rl (dz/ds) was uti l i zed.

F rom Eq. (13) we can mmediately obtain the fol lowi ng results: for a long

cylnder R2-cO0 and C = 2; for a sphere, R1 = R2 and C = 3/2. F or other axi symmetri cshapes, it i s more convenient to uti l i ze the r-z coordinates.

R1 = r[l t (r ') ]

R , = - [

2 1 / 2

1 t (r ') 1 3/2

5

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Maximum Shear Lawf f / =c/

Figure 2 Failure L aws or M onolithic and Filamentary M embranes

6

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Differentials wi th respect to the z coordi nate are ndi cated by the pri mes. By suhsti-

tuting Eqs. ( 14) into ( 13)

C V = TT /[2r2 t 2r ( r l ) t r r "] dz2 3

I n an alternate form

cv = 21T r d e - a r2 ( r ' )2 z t l~ (d/dz)(r r' ) dz16)i l ' 30 0 0

F or membranes whi ch are closed and symmetri c wi th respect to the plane = 0,

Pi pkin and Ri vl in3 have shown that the last integral n Eq. (16) vanishes. As a conse-quence, E q. (16) reduces to the fol lowing form

E q. (17) appli es to a cl osed membrane of revolution sywmetri cal about the equatori al

plane, for which each point on the surface fails accordi ng to the maximum shear law.

F i l amentary I sotensoid M embranes

Under a combined tensi l e stress f i eld, where u1 and u2 ar e the pri ncipal stresses,an isotensoid f i l amentary network can be ori ented along the pri ncipal stress directi ons

or a speci f ic optimum angle wi th the u1 direction given by

F or these condi ti ons, the pri nci pal stresses are rel ated to the f ail ure strength of a

f i l amentary i sotensoid membrane by the fol l owi ng rel ati onship

T his f ai l ure aw s l l ustrated n F i g. 2.

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By adding Eqs. (3) and (4 ) and util i zi ng Eq. (19), the thickness required at

any point on the membrane i s

t = (pR1/2C1)(3-R1/R2)20)

Substituti ng Eq. (20) nto Eq. ( l ) , Eq. (10) i s obtained where now

A s discussed fol lowing Eq. ( 1 6 ) , the l ast i ntegral vani shes f or cl osed symmetri c

membranes and, therefore, Eq. (21) reduces si mpl y to

T hus, f or closed f i l amentary isotensoi d membranes of revolution, symmetrical about

the equatori al pl ane and designed for a prescri bed pressure and vol ume, the structural

confi guration eff iciency factor i s independent of shape and has a constant value of 3.

T his resul t i s i n contrast wi th that obtained for the monoli thic membrane.

M onoli thic M embranes - Octahedral Shear L awR eturning to the monoli thic membrane now, i t i s assumed that it i s designed

accordi ng to the octahedral shear law as the fail ure cri teri on rather than the maxi-

mum shear aw. We then have the nteresting si tuation that the fail ure strength n

general depends upon the location on the membrane surf ace si nce, as i ndi cated i n

Fig. 2, strength i s a function of u / r2 1'

A ccording to the octahedral shear l aw

F rom Eqs. (3) and ( 4 )

u2/u1 = ( 2 - R ~/ R ~) - '

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Consequently, r1 in Eq. (24) i s a function of the shape of the membrane and, i n

general , u1 = r,(z). T he optimum thickness requi red at any point i s

By substituting Eq. (23) nto Eq. ( l ) , Eq. (10) i s obtained where now

U sing Eqs. (23) and (24) n conjunction wi th Eq. (26), we can obtain the fol lowi ng

resul ts: f or a sphere C = 3/2, the same resul t obtained using the maxi mum shear

l aw, whereas f or a long closed cyl inder C = 1. 732, a signif icant reduction as com-

pared to the resul t obtained from using the maxi mum shear l aw.

F or other shapes, we util i ze the r-z coordinates i n conjuncti on wi th the fol low-

ing approxi mation for E q. (23)

Substituting Eq. (24) nto (27)

Z1/r1 = 1 - 0.6 (1 - R1/R2)(2 - R ~ / R ~ )

U til izing Eqs. (14), (26) and ( 2 7 ) , we obtain

CV = 'TTJ2r 2 t 2r2(r1)2 r r"1dz

- 0. 6a r [l t ( r l ) 1 (1 t r r t t ) (2 r r t t ) - ' dz2

Fol lowi ng the argument used wi th Eq. (16), E q. (29) reduces to

CV = 'TT 1 2r2 - r (r') 1 dz - 0. ~ ' T T [l t ( r l ) 1( 1 rrt t)(2 r r t t ) - ' dz2 2

I n comparing Eqs. (16) and (30). i t can be observed that octahedral shear val ues of Cwi l l always be lower than such values for the maximum shear case by vi rtue of the

negative value of the last i ntegral i n Eq. (30).

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3. Structural Configuration Efficiencies-

In Section 2, the fol lowi ng design synthesis relation, E q. (lo), was shown toapply to monoli thic and f i l amentary membranes of revolution.

In a stri ct sense, the structural confi gurati on eff i ci ency coeff i ci ent, C i s a function

of the fail ure aw as wel l as the shape. H owever, since eff ects of biaxial i ty upon

the f ail ure l aw as represented by u2/u1 can be directly rel ated to the configuration,

i t s convenient to ncorporate them di rectly n C. T hus, the coef f i cient C represents

all shape ef fects and the strength Z1 i n Eq. (31) represents the uniaxi al tensil e

strength in al l cases.

V alues of C for optimum thickness spheres and cyl i nders were given in Secti on 2as l l ustrative exampl es. H ere, we consider n some detai l the confi guration eff i ciency

of other pressure vessel shapes of i nterest. I n addition, cl osures of vari ous shapes for

cyl i nders of di f f erent l engths are consi dered i n terms of thei r comparative eff i ci enci es.

Basic Configurations

By util i zi ng Eq. (17) and Eq. (26) or thei r equivalents n r-z coordinates, confi g-

uration ef f iciency coeff icients were computed for ong cyli nders, spheres and el l ipsoids.

I t i s noted that J ohnston has previously treated the el l i psoid f or the maxi mum shear

case. The formulas for C are presented n T able 1 and numerical results for optimum

thi ckness membranes are presented n Fi g. 3 i n terms of the parameter L /d. For all

cl osed f i l amentary membranes of optimum thickness C = 3. n connection wi th Fig. 3

i t is to be noted that because of the equal vol ume requirement associated wi th E q. (31) ,

comparative values of C at the same L /d do not necessari l y ref l ect relati ve eff i ci enci es.

5

A lso given in T able 1 ar e C values based upon the maxi mum rather than the opti-

mum thi ckness. Such resul ts are of i nterest when opti mum taperi ng may not be prac-

ti cal . Both resul ts are obvi ously dentical for ong cyl i nders and spheres but not for

other shapes. For the atter, the maxi mum thi ckness i s determined from Eqs. ( 11)and (25). In conjunction with Eq. ( 9 ) , where nowT = the value for C i s deter-

mined as ndicated n E q. (10).tmax

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Table 1

Confi guration Ef fi ciency Coeffici ents for M onolithic M embranes

Configuration Maximum Shear L aw

~ ~

Octahedral Shear Law

Long Cylinder: t and tmax0

2

Sphere: to and t ma x 1. 5

Ellipsoids:

0 - (d/L ) - 1; t 2 - (1/2) (d/L I 2

0.707 - (L /d) 5 1; tmax 1 t (1/2) (d/L j2

0

0 - (d/L ) - 1; tmax (3/4) [2 - (d/L)21 [(d/L ) t (1/A) sin X ]-1

0.707 < (L /d) 5 1; tmax (3d/4L ) (d/L ) t (1/2X) log ( d/ L ) t A 1

(d/L ) - A-

1. 732\

1.5

d/ L 0 0.2 0.4 0.6 0.8 1.0

C 1.732 .711 1.665 1.600 .534 1.500

L /d

1. 530 1.518 .611 1.850

1.0 0.9 0.8 0.707

(3/4) [3 - 3(d/L )2 (d/L )4f/2 [(d/L ) t ( 1 / X ) sin-lh]

Same as Max i mum Shear Case

where: A = (1 - (d/L)2

c

c

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C

3. 0

d

2. 0

I .9

1.8

I.7

I.6

1.54

I.8

I .7

I.6

1.5

I.4

I I I

Maximum Shear Monolithics

Closed Cylinders

I I \ \ \ \ I

Octahedral Shear Monoli th

0.2

Long Cylinder!i

0.4 0. 6 0. 8 I .o1

Sphered /L

I . 2 1.4

Figure 3 Configuration Ef fi ciencies or Optimum Thickness M embranes

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The weight penalty for using the maxi mum thickness relative to the optimum

thickness i s given by the coefficient

w t = C(for tmax)/C (for to)

Resul ts f or the el l i psoid are given i n Fi g. 4.

It i s i nteresting to observe f rom Eq. (31), that f or a given shape, no weightpenal ty i s incurred i f the vol ume s divi ded among several pressure vessel s. Thisfact may be useful in certain design si tuations where space l imitations may be of

importance. In this connection, it i s possibl e to uae a seri es of spheres n pl ace of

a cylinder and obtain the inherently greater efficiency associated with the sphere.

T hi s i s the l i miti ng case for a segmented sphere design.

Cylinders with Closures

The minimum weight design of monol i thi c membrane end cl osures for cyl i ndri -

cal pressure vessel s has been consi dered i n some general i ty by Hoff man6 and Bert ,among others. T he design probl ems that they consi dered Cali be stated n several

different ways :

7

a) F i nd the mi nimum weight cl osure for prescribed pressure

and di ameter.

b) F i nd the mi nimum weight cl osure for prescribed pressure

and volume.

c) F ind the minimum weight design of cl osures and cyl i nders

for prescribed pressure and volume.

Hoffman and Bert have considered (a) and (b) and an extension of (c) which

includes consideration of minimum ski rt ength. Because of our i nterest i n the

compl ete pressure vessel i n terms of the conf iguration eff iciency coeff icient, design

problem (c) is the most meaningful here and accordingly s used in the fol lowing.

F or our purposes we shal l restri ct our attention to hemispheri cal and ell i psoidal

cl osures of optimum design. Other closure shapes may be sli ghtly more eff icient

than the el l i psoid but are general l y more complex to treat anal yti call y.T he configuration eff iciency coeffi cients for cyl inders wi th hemispheri cal

and ell i psoidal cl osures of vari ous overal l L /d ratios can be determined by summing

the respective C V values for the cyl i nder and closure and divi ding by the total vol ume.

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I. 3

I .2

1.1

1.0

1.3

I 2

I.

I.o

Maximum Shear Monolithies I

,I

I

0 0. 2 0.4 0.6 0.8 Io I.2 I.4

d / L

Figure 4 Weight Penalty f or M aximum T hickness Ellipsoidal Closures

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In Eq. (33), Ce is the el l i psoidal closure value and C i s the cyl inder value as given

in T able 1. N umeri cal val ues of C for cl osed cyl i nders are given n T able 2 and

are l l ustrated n F i g. 3. A l s o shown n F ig. 4 are the weight penalti es associ ated

wi th using a constant rather than opti mum thi ckness closure.

C

A di rect compari son of the relative eff i ci enci es of the hemi spheri cal and

el l i psoidal closures cannot be obtained f rom F ig. 3 si nce the L /d rati os are sl i ghtl y

different or the same volume. For the atter condition

In Eq. (34), the subscri pts e and h represent ell i psoidal and hemispheri cal cl osures,

respectively. F or he same di ameter, d,

( L / dI h = ( L / d) t (1/3)(1 - h/d)35)

T he weight penalty w associ ated wi th an el l i psoidal as compared to an hemi-espheri cal optimum cl osure i s obtained by using the C values given in Table 2 for these

cases n conjunction wi th Eq. (35). N umerical resul ts for both the maxi mum shear

and octahedral shear cases are shown n F i g. 5. I t can be observed that the hemi-

spheri cal cl osure resul ts in the most ef f i ci ent pressure vessel f or a prescribed

volume.

Summary of R esults

Of the monoli thi c structures consi dered, the sphere i s the most eff ici ent by

vi rtue of the l east surf ace area per uni t vol ume and favorable thickness di stri bution.

Other monoli thic shapes such as cl osed cyl i nders and el l i psoids are somewhat less

eff icient depending upon thei r L /d. ratio and the fai lure law characterizi ng their be-

havior.

I n t erms of the confi guration ef f i ci ency coef f i ci ent, f i l amentary shapes are

considerably less eff i ci ent than corresponding monol i thic shapes by a f actor as high

as 2 for the sphere. This signif i cant di f ference i s attri butable to the f act that f or a

bi axi al stress f i eld, two separate sets of f i l aments are requi red, whereas n a mono-

l i thic membrane the minor pri ncipal stress i s carried without any additional thickness

requi rement.

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Table 2

Confi guration Ef fi ciency C oeff ici ents for M onoli thic Cyli ndersWith El li psoidal C losures

~ - . " .d / L =

Case h/ d 0 0. 2 0. 4 0. 6 0. 8 1. 0-~

._ _ _ _ ~ ~ ~

M ax. ShearL

C0

Oct. Shear

to

M ax. ShearI

Lmax

Oct. Shear

0.707

0. 8

0. 9

1.0

0.707

0. 8

0. 9

1. 0

0. 707

0. 8

0. 9

1. 0

0.707

0. 8

0 . 9

1. 0

2.000 2.000

2.000 1.975

2.000 1.950

2.000 1.930

1.732 1.741

1.732 1.719

1.732 1.705

1.732 1.702

2.000 2. 043

2.000 2. 004

2.000 1.965

2.000 1.930

1.732 1. 802

1.732 1.767

1.732 1.732

1.732 1.702

2.000 2.000

1.945 1.915

1.910 1.830

1.845 1.750

1.760 1.769

1.705 1.688

1.674 1.638

1.661 1.618

2.090 2. 143

2.009 2. 015

1.926 1.881

1.845 1.750

1. 879 1.964

1. 805 1. 849

1.731 1.731

1.661 1. 618

2. 000 2.000

1. 880 1.840

1.760 1. 670

1. 635 1. 500

1.788 1.806

1.670 1.648

1. 597 1.549

1. 562 1. 500

2.203 2.268

2.020 2.028

1. 829 1.768

1.635 1. 500

2. 059 2. 166

1. 898 1.955

1.731 1.730

1. 562 1. 500

16

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1.3 I

12

1 I

Ahxihum Shear Monolifhics

c

-

I

%I.0

Ocfohedrol Shear M m l i f h h1.2 I

/

1.I

I- 0

-

0 0. 2 0.4 0.6 0.8

Figure 5 Weight Penalty f or Cylinder with Ontimum Ell ipsoidal C losures as Comparedto an Equal Volume and Diameter Cylinder with H emispher ical Closures

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" . . . . . _. . ." . .

4. Eff iciencies of M aterial s

In the preceding section, the eff iciencies of vari ous pressure vessel conf i gura-

tions were investigated and it was shown that the overall weight can be affected by a

factor as large as 2 . A s indicated by Eq. (31), the only other f actor affecting the

weight for prescri bed design conditi ons (pV ) i s the materi al eff ici ency parameter

(p/Zcl). T his factor obviously has a most profound effect upon the overal l efficiency.

Consequently, we shal l examine i n some detai l vari ous aspects of weight/ strength

l evel s that can be achieved wi th materi als characteristi call y used in pressure vessel

applications.

A t the outset, i t i s important to recogni ze that there can be signif i cant dif fer-

ences between the tensil e strength of materi als and the structural strength l evels

achieved i n pressure vessels, parti cul arl y when high strength materi als are used.

A ccordingly , we shall be concerned in this section wi th structural strength l evels.H owever, for ref erence purposes, Tabl e 3 l i sts representati ve val ues of room

temperature materi al strength/weight rati os f or vari ous cl asses of materi al s as an

indication of their potential.

T able 3

Representative Strength/W eip;ht L evels of M ateri als

A t R o o m T emperature

~

M ateri al Type~ ~ - ." ~

"".

metal s x 10

monofilaments 5-8 filamentary

films 0. 5 monolithic

fabrics 2. 5 f i l amentary

6monolithic

T he appropriate strength/wei ght rati o to be used in the design synthesis rela-

tion, Eq. (31) s the uniaxial value since any effects of biaxial i ty have been i ncorpor-

ated nto the confi gurati on ef f i ciency coef f i cient. Furthermore, thi s ratio should be

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the structural strength/weight ( s / p ) rather than that associ ated wi th the material

strength/weight (C 1/p) such as given n T able 3. In general , S / p i s ess than Zl /pand we shal l evaluate i n the fol l owi ng,structural strength l evel s achievable in mono-

l i thic and f i l amentary structures.

High Strength Sheet M etal s

One of the maj or f actors l i miti ng the use of high strength sheet metal s in

pressure vessel appl i cati ons i s their loss of ductility as the strength level i ncreases.

Ductility i s required to reduce by plastic behavi or the stress concentrations resulti ng

f rom geometri c di scontinui ties and fabri cati on processes and thus permit the struc-

tural strength to approach the strength of the material used. T he probl em i s reason-

abl y w ell recogni zed and terms such as f racture mechanics, notch toughness, f rac-ture ini ti ati on and fracture propagation are associated wi th vari ous aspects of thi s

probl em. We shall be concerned here wi th the f racture ni ti ation phase since thi s

appears to be the governing factor i n achieving satisf actory structural strength

levels.

T he simplest representation of a tensil e structure i s a fl at stri p similar to the

smooth tensi l e specimen used to obtain the strength of a materi al, but containing a

sui table stress concentration. By testing to fai lure specimens containing a range

of elasti c stress concentration factors, the pl asti c stress concentration factor can

be determined. A s shown n Refs. 8 and 9, these data can be plotted n a form whichyi elds the ductil i ty ratio, a quantity whi ch can be l ooked upon as a basic mechanicalproperty that provides a meaningf ul measure of ductil i ty n a structural sense. T he

ductil i ty ratio has a value of uni ty for a completely bri ttl e materi al and a value ofzero for a compl etely ductil e material . In general, the ductil i ty rati o

-e =

I n Eq. (36), cb = C tU/E and i s the "bri ttle materi al " strain whi l e E i s the l ocal strain

or zero gage l ength strain at fr acture.f

Ductil i ty ratio data obtained from such tests on vari ous steels, ti tanium all oys

and beryl l i um are shown i n F i g. 6 i n terms of the materi al sfrength/weight rati o.

T he data tend to fol low the li ne shown in the fi gure wi thin ten percent li mits and thus

ref l ect the fol lowing convenient s t r engt h/ wei ght - duc t i l i t y ratio Irlawft hat hardly

coul d have been antici pated.

1 / 6C / p = 1.6 x lo6 etu

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A lso shown in Fi g. 6 i s an esti mate of the improvement i n ducti l i ty ratio that may be

associ ated wi th the more recent hot-work and maraging ultrahigh strength steels.

By use of such data, i t i s possibl e to estimate the i nfl uence of ductil i ty and

stress concentrati ons upon structural strength. F or such purposes, we util i ze the

following development of Ref. 8. T he structural strength

S = Z / ktu P

T he pl astic stress concentration f actor ( k ) and elasti c stress concentrationP

.f actor (k ) are rel ated by the ductil i ty ratio as f O l l O W 6 :e

k = 1 t (ke - 1);P

By utilizing Eqs. ( 3 7 ) and (39), Eq. ( 38) becomes

(39)

T he resul ts presented i n Fi g. 7 ar e obtained from Eq. (40) where structural

strength/weight i s pl otted as a f unction of materi al etrength/weight for various

reference values of the elasti c stress concentration f actor, ke. It i s most nterest-ing to observe that for each value of ke, the structural strength reaches a maxi mumand then decl i nes wi th further i ncreases i n the materi al strengthlweight ratio. T his

result i s associated with the reduced ductil i ty as the strength level of the metal i si ncreased. T he resul ts shown n F ig. 7 i ndicate that there s an optimum Ztu/p for

each elastic stress concentration at which S / p has a maximum val ue. Departures

to either si de of thi s strength l evel resul t i n a decrease i n structural strength.

In order to conf i rm these predicti ons, burst pressure test data on welded steel

cyl i nders heat treated to vari ous strength evel s from Ref. 10 ar e shown n Fig. 8.

A lso shown s the predicted trend based on the use of Fig. 6 and E q. (40 ) . I t can beobserved that a maxi mum structural strength ( S) and optimum Ztu ar e indeed obtained.

T he resul ts presented in F ig. 7 can be synthesized to p.rovide some approxi -

mate guideli nes for the use of high strength metal s in pressure vessel appli cations.

By using the elasti c stress concentrations factor as a reference val ue whi ch character-

i zes the eff i ci ency of the structural design and its fabricati on, the resul ts shown

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SIPp s i / pc i

1.2 x IO6

I.o

0. 8

0. 6

0.4

0.2

0 0.2 0.4 0. 6 0. 8

Ct" /p -ps i /pc i

I .o 1.2 I . 4

Figure 7 Structural StrengthIW eight as a Function of M aterial StrengthIW eight orV arious El astic S tress Concentration F actors

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S

ks i

300

200

1 0 0

0

0

0

IO0 200

Z t u k s i

300

Figure 8 Structural and M ateri al Strengths of Welded Cylinders Fabricated

f rom High Strength Steels. T est Data rom Ref. 10.

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in Fi g. 9 are obtained. On the eft scale, the opti mum materi al strengthlwei ght

ratios and the associated maximum attai nabl e structural strengthlweight l evel s are

slmwn. On the ri ght scale, the mini mum required ducti l i tyfor a given elasti c

stress concentration factor s shown. t i s to be noted that as shown n Ref; 9, the

ductility i s associ ated wi th the zero gage l ength fracture strai n.

T he results shown n Fig. 9 are presented i n terms of the elastic stress con-centration factor, ke, because t s bel i eved that this factor can provi de a meaning-

ful characteri zation of the eff iciency of the structural design and fabri cation. For

example, the maximum k resul ting f rom geometri c discontinui ties n the structure

can be establ i shed analyti call y or by experimental techniques such as photoelasticity,

strain gages or coatings. The stress concentrati ons ari si ng from fabri cation such

as tol erance mismatches or the mini mum detectable f l aw si ze can also be r epresented

i n terms of an eff ective elasti c stress concentration actor. T hus, k can be used as

e

ea

basic design parameter to characterize the eff i ci ency or qual i ty of the structur adesign and fabricati on.

I t i s for this reason that the hori zontal scale of Fig. 9 i s somewhat arbi trar

divided nto three "quali ty" regions as follows:

Region k p Rangeequirements

Quality A 1-3

Quality B 3- 8

Quality C > 8

meticulous design and fabricati on

careful design and fabri cation

routine design and fabrication

1

i ly

T hese regi ons are to be ooked upon as conceptual rather than quantitati ve at this

stage of development and were selected pri mari l y for the purpose of provi ding some

guidel i nes as to the mini mum ducti l i ty that i s required i n each of these regions.

I n the Quali ty C region which i s associ ated wi th stress concentrati on f actors

greater than 8, structural strengthlweight evel s of approxi mately 0. 7 x 10 psi/pci

can be real ized using 0. 85 x 10 psi /pci strengthlwei ght metals of ade'quate ductil i ty.

A rough estimate of the mini mum required zero gage l ength ducti l i ty i s approximately30 percent as i ndicated n F ig. 9. F or thi s region, t i s antici pated that rather

routi ne aerospace design and fabri cation techni ques can be employed because of the

relatively l arge ducti l i ty requirements.

66

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1.1.

ID '

0

g 0.92

0w

0.8to

0.7

0.6

ELASTICSTRESS CONCENTRATIONFACTOR -k e

Figure 9 Opti mum Strength L evels and Requi red Ductil i ty for V arious E lastic StressConcentrations

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T he Qual i ty B region r equi res rather careful design and f abri cati on techniques

to achi eve elasti c stress concentration factors n the 3 to 8 range. F or k = 3 and10 percent zero gage ength ducti l i ty, 0. 9 x 10 psi /pci structural strengthlweight

l evels appear to be attai nabl e wi th 1. 1 x 10 psi /pci ul timate tensil e strengthlweight

metal s.

6 e

6

M eticulous design and fabrication techniques ar e required to operate i n theQuality A region because of the relatively low stress concentration factors associ ated

wi th this region. T he requi red ductil i ty values become quite low and the structural

strength l evel refl ects a dangerous sensiti vi ty to small changes in stress concentrati on.

T hese tentative concl usi ons are based upon the main trend ine shown n F ig. 6 . I t i sbeli eved that a signif icant improvement in thi s picture can be real ized with the newer

hot-work and maraging steels for whi ch an estir r,ate of improvement in ducti l i ty i s

shown n F ig. 6.

With regard to the further development of metal l i c materi al s, i t i s qui te apparent

that improvements in the zero gage l ength ductil i ty parti cularly i n the Qual i ty A and B

regi ons are most desi rable. M ore mportant, perhaps, s the concept that opti mum

heat treatment procedures should not be based upon achieving the highest tensil e

strength of the materi al , but upon achieving the highest structural strength for an

elasti c stress concentration representative of the qual i ty region of i nterest. T hi s

concept, which i s i l lustrated n Fi g. 10, accounts for ductil i ty and i ts effect upon

stress concentrations and could lead to an ef fective increase i n the structural strength

l evel of existing high strength sheet metals.

In summary, t s quite obvious f rom F igs. 7, 8 and 9 that the structural desi gnermust stri ve to reduce stress concentrations i n order to achi eve maximum structural

strength evels compati bl e wi th the material selected. If relatively low stress concen-

tration factors cannot be achi eved there i s obviously no point in using ultrahigh strength

materi al s. In f act, their use coul d ead to ower structural strength than by use of a

l ower strength, more ducti l e materi al. D ata such as presented n F igs. 7 , 8 and 9

may be used to provi de estimates of the appropri ate val ues of S / p to be used in con-

junction with Eq. (31).

F i l amentary Composi tesI t i s a well known f act that f i l amentary composi tes, such as the glass-epoxy

composi tes currentl y used i n pressure vessel appl i cati ons, real i ze only a fraction

of thei r monofi lament strength potential . In a manner somewhat aki n to stress con-

2 6

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STRENGTH

AGEINGOR TEMPERINGTEMPERATURE

F igure 10 Schematic l lustration that Heat Treatment Should be S elected toProvi de Smax Rather T han

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centrati ons in metal l i c materi als, the formati on of f i l aments i nto strands and rovings

and the cross- over of the rovings in the composite act to reduce the useable structural

strength. T he composi te becomes a structural materi al f or pressure vessel appl i ca-

ti ons by vi rtue of the fact that the f i laments provi de the load carryi ng function while

the matrix basicall y provides the contouring and seali ng functi ons. T hus the degrada-

tion of the monof i lament strength i s to some extent associated wi th the structuralfunctions required of the composite.

A lthough it is not now possi bl e to analyze the structural strength of f i l amentary

composi tes in a manner simil ar to that used for monol i thic metal l i cs, i t i s possible

to obtain an insight into the factors whi ch tend to aff ect the structural strength of

composites. F or thi s purpose, we shal l util i ze the data presented by M orri s" n

hi s rather comprehensive survey of cyl i ndri cal gl ass-epoxy composi te pressure

vessel s.

In this evaluation, i t i s i mportant to recognize that there are several di f f erentstrength evel s that are si gnif i cant; monofi l ament strength, rov i ng strength, uni axial

composi te strength and biaxial composi te strength. T he uni axial composi te strength

i s the proper structural strength value to be used i n conjunction wi th the design

synthesis elation, E q. (31).

F rom data presented by M orr i s for E gl ass and S - 9 9 4 glass-epoxy composi testhe nformation presented n T able 4 has been assembled. An evaluation of thesedata indicate the fol lowing:

a) T he average roving strength i s 0.7 of the average mono-f i l ament strength. T his reduction i s probably associ ated

wi th l ocal contact stresses among f i l aments.

b) T he uniaxial composite strengthiweight ratio (smal l scale)

i s approxi mately 0. 73 of the average roving strengthiweight

ratio. F or a 67 vol. yoglass - 33 vol. '70 epoxy compositethis value should be 0. 8 f or a non-load carrying matrix,

thereby indicating some loss probably associated with

cross-over of the gl ass f i l aments.

c ) T he biaxial composi te strength s approxi mately 2/3 of the

uniaxial composite strength. T his actor corresponds

directly wi th that predi cted by Eq. (19) for a cylinder

(u2/u1 = 1/21.

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T able 4

T est Data on Glass-Epoxy Composites

A t Room Temperature (R ef . 11)

"E-glass-994lass

Property S I P * S I P *. .

density 0. 088 pci

monofi l ament average C 500 si .45 x 10n. 650 ksi 7.38 x 10 n.6tu

350 ksi 3. 1 x 10 6 450 si . 1 x 10 6rovi ng average Ctu

compositeensi ty 0. 076 pci 0. 073 pci

220 ksi 2. 90 x 10niaxi al composite strength 6 260 ksi 3. 56 x 10 6

(smal l scale)

biaxial compositetrength20 - 1.8 - 170 - 2.33 -150 ksi 1. 97 x 10cylinders-small scale) 6 180 ksi .47 x 10

biaxial composite trength 25si. 64 X 10 6 155 ksi.47 x 10

6

6. - "

. - .~ ~ _ _ - " ~~

(cylinders -full scale)

187 ksi 2.46 x 10 uniaxial composite strength232 ksi . 8 x 10 6

(calculated)

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d) The ull scal e cyl i nder biaxi al composi te strength data

shown in Table 4 are somewhat lower than the smallscale data. T he corresponding uniaxial composite

strength was cal cul ated by multiplying the biaxial data

by 3 1 2 . On this basis, the full scal e uniaxial composite

strengthlweight i s approximately 0.63 of the averagerovi ng strengthlweight.

To summari ze these data on f ul l scale gl asa-epoxy composi te pressure vessel s,

the fol l owi ng uniaxial structural strengthlweight ratios are representative of cur rent

practice:

S l p = 0 . 4 4 C1/p (monof i l amente) ( 4 1

S l p = 0. 63 C1/p (rovi nga) ( 4 2

T he appropri ate value to be used depends upon what one conriders to be the r awmaterial, Eq. (41 ) i s representati ve of the potential of the materi al wheraar

Eq. (42 ) i s real i sti c i n terms of the materi al currentl y uead in the fabricationprocess.

T here are Sther f i l amentary matsriala such a @ high atrength rnetaJ l is wireathat can be util i zed for preeaure veeael appl i cationa parti cul arly in the form o ff i lamentary-monol i thic composi tes, A l though we ahal l conrider the efficisnciea o f

such compoeite in the next section, i t i s advantageour to conri der the rtructurr rlstrength/weight of the f i l amentary materi als here,

F or thi s purpoae, the rspreeentati ve mater ial teneile strengtha (Zl ) gi ven inTable 5 were assembl ed f rom avai l abl e i terature (Ref r. 1 , 11, 12). Also given inTable 5 are the materi al strength/weight ratior (Z, /p) and the theoreti cal uni axial

compoei te strength/wcight ratios (C 1/p), based upon a 6 7 vo1.70 f i l ament -33 vol. '70epoxy composite. T he aet col umn l ie te the S / p val ues based upon 9070of the theo-retical uniaxial composi te etrength/weight ratio. T his number wae cited bySchuerch as that typicall y obtained n f i lament wound structures uti l i zi ng hoop

windings only.

1

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T able 5

Filamentary" M ateri als and Composi tes at R oom Temperature

Type zl( ksi )

~ - . ~._ _ _

beryllium wire 5mil s) 200

S-994lassoving50

boronnt

titanium i re ( 5 mils) 280

steele 57 5

P x+ P F1lP)c S I P(PCi) (psi/pci)

~

0.066 3. 1 x 10 2 . 3 ~ 0 2 . 1 x 1 00. 088 5. 1 4.1 3.7

0.090 5. 6 4. 5 4.00.174 1.6 1.4 1.3

0. 278 2. 1 1.9 1.7

6

I t can be observed f rom T ables 4 and 5 that f i l amentary composi tes as a class

have a considerably higher S / p potential than monoli thic metal l i cs for certain pres-

sure vessel appli cations. T hi s observati on s based upon room temperature and

short time load appli cati ons for which the data given herein apply. I t i s to be noted,

however, that an evaluation of the relative eff i ci enci es of monoli thic and f i lamentary

materi als f or pressure vessel s cannot be obtai ned f rom a di rect compari son of thei r

respective S / p values since the confi guration ef f ici ency coeffici ents must al so be

considered.

M ateri al s for I nfl atable Structures

For nfl atabl e structures appli cations where packaging requirements are

important, monoli thic plastic films and f i lamentary fabri cs have been employed.

B ecause of the fact that pressuri zation i s used to expand the packaged structure

and then maintain the expanded shape, inf l atable structures are essenti all y pressure

vessels. n this case, the vari ous structural f unctions are perf ormed as follows:

" Function ~ Fi lm Fabricload carrying f i lm cloth

contouring pressuri zation pressuri zation

sealing f i lm seal ant

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B ecause of these functional requirements, the overall ef f i ci ency of i nfl atable

structures when consi dered as pressure vessel s should i nclude the weight of the

pressuri zation equi pment and that associ ated wi th the sealant requi red for f abri cs,

I n additi on, there is an i nherent penal ty on the confi guration eff ici ency coeffi cient

since maxi mum rather than opti mum thi ckness structures may be requi red when

using films and fabri c. F rom a materi als standpoint, oining of f i lm and f abri csegments to achi eve the desi red shape causes a signi f icant degradation of the struc-

tural strengthlweight as compared to the material strengthlweight when the weight

penal ty associated wi th seams i s considered.

Brewer and J eppeson4 have considered these factors in considerable detai l .

B ecause of the form in whi ch they present thei r data, i t i s not possi bl e to ascertai n

structural strengthlwei ght level s for films and f abri cs i n the sense used herein.

Consequently, f urther consideration of the efficiency of i nf l atabl e structures i s

reserved f or di scussi on i n a subsequent section.

Comparative E ffi ciencies of M ateri al s

F ig. 11 has been prepared to summari ze the evaluation presented n thi s sec-

tion. On the hori zontal scale, the uniaxi al materi al tensi l e strengthlweight ratios

are ndicated. F or glass f i l aments, this ratio i s based upon the strength of the

rovi ngs. T he verti cal scale represents the uniaxi al structural strengthlweight

ratios whi ch can be achieved by application of the best current technology. Al though

f i l amentary composi tes appear to be superi or to monol i thic constructi on, i t must

be noted that the configuration eff iciency coefficient must also be considered when

evaluating overall pressure vessel efficiencies. Consequently, Fig. 11 does not

permit a di rect compari son of the relative ef f i ci encies of materi al s as used i n

pressure vessel s.

I t i s al so to be noted that F i g . 11 i s based upon short time l oad appli cati ons at

room temperature. C onsi deration of cryogenic and elevated temperatures and other

envi ronmental factors can change the relative eff i ci enci es of monoli thic and fila-

mentary materials substantiall y.

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5

4

3

2

I

/Fi'/amentary

Composites

Glass BaronRovings, f i l a p n t s

ww

0

Figure 11

I 2 3 4 5

X, p lo6 psi/pciComparative Structural E ffi ciencies of V ari ous M ater ials n Pressure V esselApplications at Room Temperature

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5. Potential of Newer M ateri al s Concents

M aterial s characteri stical l y employed in aerospace pressure vessel appl i ca-

ti ons were consi dered n the previ ous section. H ere, we shall be concerned wi th

the potential eff iciencies of certain newer materi al concepts f or such appli cations:

ani sotropic metal s for monol i thic construction, combined monol i thic and f i l amentary

designs, and whi sker composites.

Obviously, there may be many probl ems in the appli cation of these concepts to

the production of pressure vessel s and many of the f actors whi ch result i n a reduction

of the materi al strength to the structural strength l evels di scussed i n the precedi ng

section w i l l operate here also. A l though the ful l potential represented by the materi al

strengthlweight ratio may not be reali zable, these newer concepts coul d resul t n

signif i cant increases in structural strengthlweight l evel s in the future.

A nisotropic M etal s

A lthough theori es of yi eldi ng and plastic flow of anisotropi c metals have been

avai l abl e for some time, B ackofen et all 3 appear to have been the fi rst to observe

that signif i cant strengthening eff ects are predi cted by such theori es for combined

loadings typical .of pressure vessel s. A ni sotropy of mechanical properti es i s in-

herent in metal l i c materi als as a result of thei r basi c crystall i ne f orm and al so as

a resul t of differences in deformation along various rol l ing axes in processing the

materi al nto sheet f orm. In fact, metal producers expend consi derabl e eff ort to

achieve as nearly sotropic a product as practical. Conversely, t should be possibleto produce sheets wi th control l ed ani sotropy f or pressure vessel appli cati ons.

A nisotropy due to a preferred ori entati on or texture of the cry stal structure

was suggested by Backofen13 as a method of i ncreasing the yi eld strength i n the

thickness di rection of sheet. P articul arly f or hexagonal cl ose-packed metals such

as ti tanium and beryl l i um, the sl i p systems can be so ori ented as to resul t i n a

signif i cant ncrease n yi eld strength n the thick ness di rection. Other mportant

f or ms of anisotropy can be obtained by unidi rectional plastic worki ng of the sheet

as a resul t of rol l ing o r stretching.

H i d 4 has presented a general i zation of the octahedral shear law for ani so-

tropic behavi or. In terms of the pri nci pal stresses, hi s relation reduces to the

fol lowi ng for plane stress:

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- = l 2 l - ( l t 7 12 - Zl 2 2 Z12 2 22 7) - 2 -1

1 Z2 Z3 1 Z2 1(43)

In Eq. (43), Zl , Z2 and Z represent the uniaxial strengths n the pri nci pal stress

and thickness directions, respectively.3

T exture Hardening

T o represent texture hardening, we can et C 1/Z2 = 1 and Z1/Z3 = a. ThusEq. (43) becomes

= 1 h = [l - ( 2 - a ) (u2/u1) t (U2/U1) I1/ 2

(44)

T ensi l e strength surf aces f or vari ous values of a are l l ustrated n F i g. 12. Note

that strengthening occurs f or a1 weakening occurs and, n fact, for

a = 2, the f i l amentary strength aw, Eq. (19) i s obtained. By use of Eq. (26) nconjunction wi th Eq. (44) , we can obtain the fol lowi ng results for the configurationeffi ciency coeff icient of ani sotropic monol i thic shapes:

Sphere: C = 1. 5a

L ong Cyl inder : C = ( 1 t 2 a )1/2

(45

(46

T hese resul ts are i l l ustrated in Figs. 13 and 14 and it can be observed that

signif icant improvements in eff ici ency can be real ized by rai sing Z3 rel ative to Zl .Backof en13 has discussed the degree of texture hardening associ ated wi th vari ous

crystal l ographic structures and hi s estimates are i ndi cated i n these f i gures f or

HC P, BCC and F C C metal s. A weight saving potential of roughly 50 percent seems

possibl e for a sphere of properl y textured HC P metal . For the l ong cyl i nder the

weight savi ng potential i s considerably ess although still attracti ve. A hemispheri c-

ally closed cylinder would lie between these two limiting cases.

I t i s important to note that Sl iney et a l l 5 have conducted tests on two cylinders

fabricated of Ti -5A1-2. 5Sn ti tanium al loy sheet which has a HCP structure. F romauxi l i ary tensil e tests, i t was establ i shed that the anisotropy coef f i ci ent (a) had an

average val ue of 2/3. By use of Fig. 14, a weight saving potential of 2070is obtained

f or a = 2/ 3 as compared to an i sotropi c materi al (a = 1). A lthough n the two cyl inder

tests fai l ure occurred at l ongitudinal welds, the burst strengths (S) were signif i cantly

higher than the uniaxial tensil e strength (X1) as i ndicated in T able 6.

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C

Figure 12 Strength Surfaces Representative of T exture Hardening

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C

2. 0

I 5

1.0

0. 5

0

I

1

02 0.4 0. 6

I

"t-I I

I II I

I I

0. 8 1.o

Figure 13 Configuration Efficiency Coefficients or T exture Hardened M etals

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1.0

0. 8

0.6

C a / C i

0.4

0.2

0

' I I I

I I

I I I

HCP ecc f cc"' I I

I ' I

0. 2 0.4 0.6 0. 8 I oa=Xl/ C3

Figure 14 Weight Saving Potential of T exture H ardened M etal s(F or Same Xl as I sotropic M etal )

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Table 6

T est Data on A nisotropic T i tanium A l loy Cyl inders

A t R oom T emperature. .

M ate rial Z (ksi)';' S( ksi )* */Z l a Reference

Ti-5A 1-2. 5Sn 132.5 156.8 0. 67 15

132.553. 16 0. 67 15

Ti -6A 1-4V 14795 1.33 "14795 1.33 "14798 1.35 --

16

16

16

::cUniaxial T ensil e trength ':"Hoop Stress turst

A dditional test data by M artin et a l l 6 on Ti -6A 1-4V cyli ndrical pressure

vessel s are presented i n T able 6. A lthough the value of the anisotropy coeff ici ent

(a) s uncertain, the burst strengths (S) are signif i cantly higher than the uniaxi al

tensi l e strength. T hese resul ts are parti cularly encouraging n vi ew of the fact

that they are based on f ail ure rather than yi eld strength.

M echanical A nisotropy

A nother technicall y interesting form of anisotropy i s that obtained by mech-

anical unidi rectional plastic worki ng of the sheet by rol l i ng or stretching. H ere

the tensil e strengths i n the pl ane of the sheet are i ntentional l y dif ferent so that

Z1 > Z: If i t i s assumed that Zl /Z 3 = 1 and Zl /Z 2 = b where b > 1, then Eq. (43)

becomes for thi s case2' -

=1 'l = [1 - b (u2/u1) 1 - u,/u,)]1 2

(47)

T ensi l e strength surfaces f or vari ous values of b ar e shown n Fig. 15.

I t i s important to note that mechanical anisotropy can improve eff ici ency by

two di f ferent mechani sms. T he f i r st i s the biaxi al i ty effect displayed n F ig. 15.

T he second i s the i ncr ease in Z l over that for an isotropi c material whi ch presumabl y

can be attai ned as a resul t of the decrease i n Zz. T hi s mechanism i s not shown by

the presentation of Fig. 15.

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I c

OX

0.6

0I

t

10 1.2 14 1.6 1.8 2.0

=0. 5

u 1

Figure 15 Strength Surfaces R epresentative of M echanical A nisotropy

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By use of E q. ( 2 6 ) n conjunction wi th Eq. ( 4 7 ) , the following configuration

eff ici ency coef fici ents are obtained for monoli thic shapes of mechanicall y ani so-

tropic materi al s:

Sphere: c = 1. 5L ong Cyl inder: C = ( 4 - b ) 112

I t can be observed that the biaxial i ty eff ects of mechanical anisotropy do not result

i n any mprovement n ef f i ci ency for the sphere. For a long cyli nder, on the other

hand, signif icant mprovements n eff iciency can be obtained. In f act, as b approaches

2, dramati c mprovements are predi cted.

In compari ng the resul ts obtained for the texture hardening and mechanical

ani sotropy cases for the sphere, signif i cant mprovements n eff i ci ency are pre-

dicted for texture hardening only. On the other hand, comparable results obtainedfor the long cyl i nder i ndicate that large i mprovements are predicted for mechani cal

anisotropy. T hus, the type of anisotropy that may be opti mum for a given shape

depends specificall y upon the configuration.

T o account for the possibl e i ncrease i n E of the mechanicall y ani sotropi c1materi al as compared to that of the sotropi c materi al ( X ) the fol lowi ng assump-

tion i s made which appears reasonable for a smal l degree of anisotropy:1 i '

Zl t z2 = 2(Z14

Since Zl/Z2 = b,

b = [2(Zl)i/C1 - 11" (51)

By i ncorporati ng he ncrease n ensi l e strength as indicated by Eq. (50), he

corresponding configuration eff iciency coefficients become:1

L ong Cyl inder: C = (4 - [z(Zl)i/Zl - 1 r 2 } (53)

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N umeri cal resul ts based upon Eqs. (52) and (53 ) ar e shown n Figs. 16 and 17.

I t would appear that real ly signif icant improvements in eff ici ency are predicted for

long cyl i nders for moderate degrees of mechani cal ani sotropy. F or spheres, t i s

apparent that texture hardening has the greater potential for weight savi ng. T hus,

for a hemi spheri call y cl osed cyl i nder, mechanicall y ani sotropi c materi als are

indicated f or the cyl i ndrical portion and texture hardened materi al s f or the hemi -spheri cal cl osures.

F i l amentarv-M onoli thi c ComDosites

A l though the confi guration eff iciency coef ficient is more favorable for mono-

lithic as compared to f i l amentary shapes, the atter have a greater overall eff i ci ency

because of the use of hi gher strengthlwei ght materi als. I t s of i nterest, theref ore,

to consider f i l amentary- monoli thic composites whi ch woul d use to advantage the

greater confi guration eff ici ency i nherent in monoli thics wi th the greater materi al

eff iciency of the f i l amentari es.

Of the practi cal pressure vessel shapes of i nterest, the cyl i ndri cal porti on of

a cl osed cyl i nder appears to have the most interesti ng potential as a filamentary-

monol i thic composi te. T he monol i thi c porti on of the cyl i nder forms the nside shel l

to whi ch the cl osures are attached. T hi s shell provi des the contouri ng and seali ng

functions, and i t i s designed to carry the end loads and one-hal f the ci rcumferential

l oads. T he f i l aments are wound on the cyl i ndrical positi on n the hoop direction only

and carry the other half of the ci rcumferential oads. A s such, the f i l aments act n

auniaxial stress f i eld and should not be degraded by the f i l ament cross- over asso-

ci ated with biaxi al stress f i elds. The f i l aments provi de only a unidirectional oad-

carryi ng function n the composite. I t i s assumed that the el astic modulus mismatch

between the monol i thi c and f i l amentary materi als can be accomodated y yielding of

the monoli thic materi al.

The weight of the composite cyl i nder minus the end closure weight (approxi -

mately equal to a long cyl i nder) i s given by

W = 2rRL pmtm f pftf) (54)

H ere, the subscri pts m and f refer to monoli thic and f i l amentary, respectively, In

both cases, t = pR/2S, and therefore, Eq. (54) can be wri tten as

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2. 0

I 5

C I .o

0. 5

0

1phere

I.o I. I I 4 I 5

Figure 16 Configuration Efficiency Coefficients or M echanical ly A nisotropic M etal

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I o

0. 8

0. 6

C a / C i

0.4,

0. 2

01.0 I. I 2 I 3 I 4 I 5

Figure 17 Weight Saving Potential of M echanical ly A nisotropic M etals(For Same as I sotropic M etal )

4 4

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In Eq. (56), the coeff icient a i s the anisotropy coeff ici ent for texture hardened mono-

l i thic metal s. For an sotropic materi al (a = 1) note that C = 2 rather than C = 1. 732in the i miting case when ( P / S ) ~ (p/S), because the biaxiali ty effect i s u2/ul = 1 f or

the composi te as compared to u2/u1 = 1/2 n the monoli thic design.

N umeri cal resul ts based upon Eq. (56) ar e shown n F ig. 18 together wi th

appropri ate strength/weight ratios of f i l amentary composi tes given n T abl e 5. T he

potential increases in eff iciency for both i sotropic and anisotropic materi als are

indeed attractive particularly for rhe glass filaments.

In order to indi cate the overall weight savi ng potential of the composi te, i t i s

of i nterest to compare the weight of the composite wi th a fi lament wound cyl inder.

F or the f i l amentary- monoli thic cyl i nder, E qs. (55) and (56) are used whil e the

weight of a filament wound cylindrical pressure vessel i s taken as

Si nce (P /S )~ represents the uni axial tensil e composite strength/weight ratio i n both

Eqs. (56) and (57), the factor cc i s introduced n Eq. (57) to account for the degra-dation in strength associated wi th fi l ament cross-over required for a biaxi al stress

field, T he weight ratio i s thus

N umeri cal resul ts based upon Eq. (58) are shown n Fig . 19 for sotropic and

anisotropic metals. Wi th glass rovings or which c = 1. 15 (S /p for 5-994 n

T ables 4 and 5) i s representativ e of curr ent practi ce, i t can be observed that the

f i l amentary- monoli thic composi te i s more eff i ci ent than the f i l amentary composi te

only when ani sotropic metals (a = 0. 5) are uti l i zed. F or sotropic metal s the re-

sults are suff i ci entl y cl ose that other considerations may govern the choice of

construction.

C

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C

2 o

1.5

I o

0. 5

0. 2 0.4 0. 6 0. 8 I o

Figure 18 Configuration Efficiency Coeff icients or Filamentary HoopWindings on a L ong M onolithic Cylinder ( S / p ) = 10 psi /pci

m

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2 o

I 5

I.o

0. 5

-\ Isotropic Metals, \ / a= / .O

Aniso tropic Metalsa =0. 5

I

0 0.2 0.4 0. 6 0. 8 I o

F i gure 19 Overall E f f i ci encies of F i lamentary-M onoli thic CompositesCompared g Filamentary Composites. L ong Cyl inder

(S/p), = 10 psl/pci

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W hisker Composi tes

T he f i l amentary composites considered up to this point al l utilize continuous

reinforcements n the form of rovi ngs, monof i l aments or i ne wi res. H offman has

di scussed, at some ength, the nteresti ng potential for pressure vessel appl i cations

of composi tes wi th discontinuous reinforcements n the form of whi skers. A l though

the curr ent research work i n thi s area i s exploratory i n many respects, the achieve-

ments are suff i ci ently encouraging to warrant seri ous consideration of whi sker

composi tes as a potential pressure vessel materi al. T abl e 7 l i sts some representa-

ti ve strength l evels of the best whi skers tested.

2

Table 7

Reoresentative Strength Data on the B est W hiskers

A t R oom Temperature

=1 P =I 1 P S I PM ateri alksi) (PCi) (psi/pci)psi/pci)

graphite 3,000 0. 07 43 x 10

aluminum oxide 1,800 0. 13 14 3. 3

iron 1 , 9 0 0 0. 284 6. 7 2. 9

silicon 5 50 0. 087 6. 3 2. 1

613.4 x 10 6

A lthough Table 7 l i sts the best whi sker properti es currently achi eved, the

average properti es of a batch of whi skers wi l l be f ar below these values. F or our

purposes here, i t wi l l be assumed that wi thin a batch of whi skers there i s a normal

distribution of tensi l e strengths between the highest values given i n T able 7 and the

lowest strength whi ch s taken as zero. H ence, the batch tensil e strength would be

1 2 of the T able 7 values.

F or purposes of comparison wi th f i l amentary composites, it i s assumed thatan epoxy matrix may be sui table for whi sker composi tes, Si nce the packi ng density

of the whi skers in the composite wi l l probably not be as high as that achieved for

f i l amentary composi tes, it i s assumed that 50 vol. O/owhiskers and 50 vol. 7 epoxymay be representative. U sing this compositi on and the batch tensil e strength, the

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uniaxi al composite strength/weight rati os (S/p) given n Table 7 were computed.

Degradation effects due to whisker cross-over and improper whisker alignment are

not accounted for.

A compari son of the S / p values wi th the uniaxial composite strength of a full

scale S-994 glass composite (T able 4 ) of S / p = 3. 18 x 10 (psi /pci ) ndicates that

i ron and si l i con whi sker composi tes are not competi tive on this basis. On the other

hand, graphite and aluminum oxi de whisker composites are attractive.

6

In a biaxi al stress fi eld, the whi sker must be ori ented accordi ng to Eq. (18)

so as to car ry both pri ncipal stress components n an optimum manner. In this

respect whi sker ( and f i l amentary) composites are l ess ef f i ci ent than monol i thic

material s. F or all whi sker composites of optimum design, the confi guration

eff iciency coeff icient] C = 3.

In the preceding evaluation] the whi skers were assumed to have the optimum

orientation associated with filamentary membranes in a biaxial stress fi eld.T his may be unreal i stic i n a practi cal sense and, therefore] i t may be important

to consi der randomly ori ented whi sker composites. n this case the composite i s

to be consi dered as a monoli thic rather than f i l amentary membrane. By some

control of the randomness of whi sker ori entati on in the composi te i t should be

possibl e to obtain ani sotropi c monol i thic membranes. T he analyses of monol i thic

membrane structures presented previ ousl y herein would apply to the randomly

ori ented whi sker composite. n parti cular, Eq. ( 4 4 ) and ts representati on nFig. 12 i ndicates that this composi te must have signif i cant compressive strength

in the thickness di rection to be an eff i ci ent monoli thic materi al f or pressurevessel appli cati ons.

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6. Overall Pressure V essel Efficiencies

I n previ ous sections, the confi guration ef fi ciencies of vari ous pressure vessel

shapes were i nvestigated and the eff i cienci es of vari ous materi al s were studi ed in

some detail. Now, we combine the two eff ici ency f actors to determine the overallprimary structural weight eff i ciency of membrane type pressure vessels. For this

purpose we return to the design synthesis relationship in the fol lowi ng form:

w l p v = CpIS (59)

I n Eq. (SS), the desi gn conditi ons are represented by the pressure (p) and

volume (V). T he atter can usual ly be speci f ied n a completely straightforward

manner. T he desi gn pressure, on the other hand, i s usuall y taken as the maxi mum

operati ng pressure mul ti pl i ed by a suitable saf ety f actor.F or a structural reli abi l i ty standpoint, the maximum operati ng pressure i s

stati stical n nature as i s the structural strength (S). C onsequently , for a prescri bed

value of structural rel i abi l i ty, whi ch can al so be taken as a specified design condition,

the specif i c statisti cal vari ations of the structural strength should be charged to the

materi al when comparing the efficiencies of a vari ety of materi al s. In thi s manner

we would be compari ng pressure vessels designed for the same structural reli abi l i ty.

U nfor tunately, there are insuff i ci ent data avai l abl e to permit i ncorporati on of strength

distributions in the present investigation.

Based upon the C and S / p values obtained herein, Fig. 20 has been prepared

to evaluate the overal l eff i cienci es of monoli thi c and f i l amentary materials. T he

cross-hatched regions in F ig. 20 represent materi als that have been util i zed in f ul l

scale aerospace producti on components. t i s to be noted, however, that the aero-

space envi ronment encompasses temperatures other than room temperature upon

which Fig. 20 i s based.

I t can be observed that inf l atable structures as a cl ass (based on data of R ef. 4 )

are nherently much ess eff i ci ent than metall i c and glass-epoxy composites. B ased

on room temperature properti es, sotropi c metall i cs are not as efficient as gl ass-epoxy composi tes. Under the best circumstances for each, a weight saving potential

of approximately 113 can be attained wi th the glass-epoxy composi te.

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IO-^

pc ips i

-

10

Films

Structures

\

Filamtwt WwndTtuture hbrdenedM h l Cy/inders

c- Monolithies-

\,/sotropic Metallics

6 lo 7S /p psl/pci

Figure 20 Overall M embrane E fficiencies of P r essur e V essel s at RoomT emperature

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F or other materi als concepts which have not, as yet, reached the aerospace

production stage, f i l ament wound i sotropi c metal cyl i nders represent an i nherent

improvement over monoli thic sotropi c metal l i cs. However, at room temperature,

the glass-epoxy composites still appear to be at an advantage. On the other hand,

the development of anisotropi c metals can represent a signif icant weight saving

potential as compared to currently used materi als. T hi s potential depends strongl y

upon the degree of anisotropy that can be achieved wi th high strength metals and the

configuration of the pressure vessel . T hi s i s also true for f i l ament wound texture

hardened metal cyl i nders.

An important improvement in overall eff ici ency appears possibl e wi th oriented

whi sker composi tes. However, on the basis of the anal ysi s used herein the potential

of such composi tes appears to be far l ess dramati c than predi cted by Hoffman . Infact, only the low densi ty whiskers such as graphi te and aluminum oxi de appear to

be attractive when used in the form of ori ented whi sker composites.

2

Should it not be possible on a production basis to ori ent properl y the whiskers,

then randomly or i ented whi sker composites may be a practical solution. In thi s case,

the composite woul d end o act as an anisotropic monolithic materi al with a conse-

quent large reduction i n structural strength as compared to the ori ented composite.

In pressure vessel appl i cations thi s reduction i n strength woul d be compensated to

some degree by the i nherently more ef f i ci ent confi guration coeff i ci ent associated

with monolithic materials.

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R eferences

1. Schuerch, H. , "A nalytical Design or Optimum Filamentary Pressure V essel s, I t

AIA A Preprint 2914-63, April 1963.

2. Williams, M. L ., Gerard, G ., and Hoffman, C. A ,, S el ected A reas of

Structural Research n R ocket V ehicl es, X I I nternational A stronautical

Congress, Stockholm, Vol. 1, pp. 146-166, 1960.

3. Pi pkin, A . C. , and Rivlin, R. S . , "M inimum-W eight Design or P r essu reV essels Reinforced wi th I nextensibl e F i bers, I J ournal of A ppl ied M echanics,

Vol. 30, No. 1, pp. 103-108, M arch 1963.

4. Brewer, W. N. and eppeson, N. L . , "M ethods of E valuation of I nfl atableStructures for Space A ppli cations, I t AIA A F i fth A nnual Structures and

M ateri al s Conference, pp. 344-360, April 1964.

5. J ohnston, G . s. , "Weight of Ellipsoid of Revolution with Optimum T aperedT hickness," ournal of the A erospace Sci ences, VOl. 29, No. 10, pp- 1269-

1270, Oct. 1962.

6. Hoffman, G. A. , "M inimum-W eight Proportions of P ressure-V essel Heads,"

J ournal of A pplied M echanics, Vol . 29, No. 4, pp. 662-668, Dec, 1962.

7. Bert, C. W . , "Ellipsoidal C losures or M inimum Weight Pressure V essels,"

Canadian A eronautics and Space ournal, Vol . 9, pp. 133-136, M ay 1963.

8. Gerard, G. , 'I Structural Significance of Ductil ity n A erospace PressureV essel s," ARS J ournal, V ol. 32, No. 8, pp. 1216-1221, Aug. 1962.

9. G erard, G. and Papirno, R. , "Ductility Ratio of A ged Beta Titanium Alloy, I t

T ransactions of the A SM, V O ~. 5, NO. 3, pp. 373-388, Sept. 1962.

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10. M i l l s, E. J . , A tterbury, T . J . , Cassidy, L. M. , Eiber, R. J . , Duffy, A. R. ,I mgram, A . E ., and M asubuchi, K . , "Design, Performance, Fabrication

and M ateri al Considerations or High-pressure V essel s, U. S. A rmy M issi l e

Command, Redstone Scientific nformation Center, RSI C-173, M arch 1964.

11. M orri s, E . E. , "Glass Cases or the Biggest Solids, I 1 A stronauti cs and A ero-nautics, V ol. 2, No. 7, pp. 28-38, uly 1964.

12. B ert, C. W. , and Hyler, W.S. , "Design Considerations n Selecting M ateri al sfor L arge Sol id-P ropel l ant Rocket-M otor Cases, I f Defense M etals nformation

Center Report 180, Dec. 1962.

13. Backofen, W. A . , Hosford, W. F. , and Burke, J . J . , "T exture Hardening,"

T ransacti ons of the ASM, Vol. 55, No. 1 , pp. 264-267, M arch 1962.

14. Hill R., T he M athemati cal T heory of P l astici ty, Oxford University Press,

L ondon, 1950, pp. 318-320.

15. Sliney, J . L. , Corrigan, D. A. , and Schmid, F. , "P rel i mi nary R eport on theB iaxi al T ensi l e B ehavi or of A nisotropi c Sheet M ateri al s, U. S. A rmy M ateri al

Research A gency, T ech. Rept. AM RA TR 63-11, Aug. 1963.

16. M arti n, W . J . , M atsuda, T. , and K aluza, E. F. , "Study of T i tanium AlloyT ankage at C ryogenic Temperatures, I ' Douglas A ircraft Co. Report No.

SM -43116, M ay 1964. ( U. S. Dept. Commerce, OTS, AD604 053).