6' TECHNICAL REPORT

C)L"u

5 STABILITY OF THIN TORISPHERICAL SHELLS UNDER

UNIFORM INTERNAL PRESSURE

m by

JOHN MESCALL

METALS AND CERAMICS RESEARCH LABORATORIES

U. S. ARMY MATERIALS RESEARCH AGENCYJUNE 1963 WATERTOWN, 72, Mi

The findings in this report are not to be construedas an official Department of the Army position.

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Qualified requesters may obtain copies of this report from Director,Armed Services Technical Information Agency, Arlington Hall Station, Arlington 13, Virginia

DISPOSITION INSTRUCTIONS

Destroy; do not return

Shells, buckling

Stresse s and strains

STABILITY OF THIN TORISPHERICAL SHELLS UNDER UNIFORMINTERNAL PRESSURE

Technical Report AMRA TB 63-06

by

John Mescall

June 1963

AMS Code 5011.11.838

Basic Research in Physical SciencesD/A Project 1-A-0-1050-B-010

METALS AND CERAMICS RESEARCH LABORATORIESU.S. ARMY MATERIALS RESEARCH AGENCI

WATERTOWN 72, MASS.

Shells, buckling

Stresses and strains

STABILITY OF THIN TORISPHERICAL SHELLS UNDER UNIFORMTNTERNAL PRESSURE

Technical Report AMBA TB 63-06

by

John Meacall

June 1963

AMS Code 5011.11.838Basic Research in Physical Sciences

D/A.Project 1-A-O-10501-B-010

METALS AND CERAMICS RESEARCH LABORATORIESU.S. ARMY MATERIALS RESEARCH AGENCI

WATERTOWN 72, MASS.

U.S. ARMY MATERIALS RESEARCH AGENCY

TITLE

STABILITY OF THIN TORISPHERICAL SHELLS UNDER UNIFORMINTERNAL PRESSURE

ABSTRACT

The stability of the toroidal portion of a torispherical shell underinternal pressure is considered from the point of view of the linearbuckling theory. A detailed stress analysis of the prebuckled ahell ismade employing asymptotic integration. The change in potential energy ofthe shell is then minimized using a Raylei-h-Ritz procedure for actualcomputation of the critical pressure. Numerical results reveal thatelastic buckling may occur for very thin shells whose material has arelatively high value of the ratio of yield stress to elastic modulus.

P ther,,ati cian

APPROVED:

E. N. HEGGEActing DirectorMetals and Ceramics Researchl Laoratories

CONT FTS

Page

ABSTRACT

SYMBOLS ............. ............................. .iii

INTRODUCTION ............ ........................... 3

ANALYSIS OF THE ELASTIC STABILITY OF THIN TORISPHERICAL SHELLSUNDER UNIFO44 INTERNAL PRESSURE

Theoretical Background ......... ................... 4

Toroidal Geometry and Strain-Displacement Relations . ... 5

Stability Criterion .......... .................... 7

Prebuckled State of' Stress ........ ................. 10

Simplified Formula for Critical Pressure .. .......... ... 12

Numerical Results - Comparison with Experiment - Discussion 14

ILLUSTRATIONS .......... .......................... ... 16

APPENDIX ........... ............................ ... 21

REFERENCES ............. ........................... 23

SYMBOLS

alp coordinates of middle surface of shell

z coordinate normal to middle surface of shell

A,B Lame" coefficients associated with a,

RR 3 principal radii of curvature

u,v,w displacements in the a, 3, z directions

uIvI-,wI additional displacements in the meridional, circumferential,and normal directions after buckling

ea,¢peap strains at any point in the shell

araap ,aa stresses at any poinl in the shell

U strain energy of thin shell

2,X middle surface strains and curvature changes in a thin shell

Watw angle of rotation of normal to middle surface about tangentto lines P = constant, a = constant

tP, e coordinates in meridional, circumnferential direction ontoroidal middle surface

b radius of cross section of torus

a distance from center line to cenLer of toroidal cross section

r r = a + b sin (r, horizontal distance from center line topoint on toroidal miadle surface

X X b/a

2Tn11 - 2cpo

h shell thickness

E Young's modulus of elasticity

Poisson's ratio

I Eh3D flexural stiffness of orus: 12(1 - 1)2)

i i i

p applied pressure

Pcr critical pressure

n wave number defining number of circumferential buckles intorus

NIo,Neo middle surface stress resultants prior to buckling

No,Meo middle surface stress couples prior to buckling

iv

INTRODUCTION

Torispherical shells are frequently employed as end closures forcylindrical shells both in missile design and in a wide variety ofindustrial-type pressure vessels. Such shells generally consist ofa shallow spherical cap joined to a toroidal segment, joined in turnto a cylindrical shell. This combined shell is then subjected to in-ternal pressure. It was known' that large, compreesive hoop stresseswere developed in the torus, and that for very thin shells, elasticbuckling was a distinct possibility. For shells whose thickness isheavy enough to avoid buckling, but which may still be rather thincompared to the toroidal radius, plastic deformation may occur. Thishas been examined in detail by Drucker and Shield. ' However, theelastic stability of torispherical shells was not considered a matterfor concern until, recently, such a shell was actually observed tobuckle under internal pressure. (See Figure 1.) The problem is aninteresting one, since the prebuckling state of stress in such a shellis not a simple one. The membrane state, often employed to estimateprebuckling stresses, would actually predict a state of tension in thetorus. Consequently a thorough stress analysis of the shell prior tothe onset of instability must be made. This stress analysis is alsouseful (and necessary) in ascertaining the dividing line between thoseconfigurations which are likely to buckle and those which are likelyto undergo plastic deformation.

In this report, prebuckling stresses in a shell of the type shownin Figure 2, subjected to internal pressure, were determined by asymp-totic integration techniques, with special attention being devoted tothe particular solution. The vertical support on the cylinder was as-sumed to be sufficiently far removed from the junction of the torusthat its specific form had little influence on the toroidal stresses.These results were then incorporated into the stability equations forthe toroidal segment, and numerical results were obtained by applying*a Rayleigh-Ritz approach to an appropriate potential energy expression.Since no prior theoretical results appear to be available, an experi-mental investigation of the same problem was undertaken simultaneously.The results of the two investigations compare favorably. Finally, asomewhat simplified stability criterion is proposed which provides re-sults of essentially the same degree of accuracy, at a considerablesaving in effort.

-3.

ANALYSIS OF THE ELASTIC STABI! ,Y OF THIN TORISPHERICAL SHELLSUNDER UNIFORM INTERNAL PRESSURE

Theoretical Background

Equations governing both the equilibrium and the stability of a thinelastic shell may be obtained in the following manner.4'5 Beginning withan appropriate set of strain-displacement relations and a stress-strainlaw, one forms thelstrain energy integral associated with the thin shellunder consideration. For a homogeneous isotropic thin shell thit may beexpressed as

U e[a€ + a + e + a B]AB da dp dz (1)

Next, the principle of virtual work and the principle of stationary poten-tial energy are applied. In this connection, consider a shell in a stateof equilibrium characterized by displacements o, v w This state issaid to be one of "neutral" equilibrium if there exists an adjacent equi-librium state differing from the first by an infinitesimal variation inthe quantities characterizing that state. We therefore consider an alter-nate equilibrium state characterized by displacements

u " Uo + rui v vo + nv w w Wo + w (2)

where 71 is an infinitesimal, independent of a, 3.

The strain energy associated with this second state becomes

U = U0 + YU I + rpU . (3)

The principle of stationary potential energy implies that for the secondstate to be one of equilibrium,

u + 6V = 0, (4)

where 5V is the negative work done by external loads due to a variation indisplacements. Considering only variations in u,, v,, w,, we have:

6iJ + 6V 1 + i9(W 3a + 6;) = 0. (5)

But, since the first state was assumed to be an equilibrium state, theprinciple of virtual displacements implies that

1J + V =0 . (6)

Hence the condition that the first equilibrium state be unstable is that

W + 6V2 = 0. (7)

To summarize, equation 6 yields equations governing the equilib-rium of a state described by displacements uo, Vo, wo, while equation 7yields equations governing the stability of this state. This general

-4-

statement of the problem is valid for both large and small deflections.Thus, depending upon the generality of' tne strain-displacement rei.ationsand the stress-strain laws assumed, a variety of theories is possible.Kempner5 delineates several of these in considerable detail and withattendant clarity.

Toroidal Geometry and Strain-Displacement Relations

We consider the coordinate system shown in Figure 2 with ,e themiddle surface coordinates in the meridional and circumferential direc-tions and z normal to both, positive when directed inward. The princi-pal radii of curvature R , Re are given by R a b and R sin cp = a + b

sin p r. The so-callea Lame coefficients lor this system are A, N band A R sin cp = r. With this information it is a simple matter toobtain from Reference 4 or 5 (with obvious changes in notation) thefollowing strain-displacement relations, valid for small strains butlarge rotations:

C= *+ i (W02)(P CP 2(w)

1Ce = + 1(W.2)

qeg '%e - RPwe (8a)

where

1q 9 1 z. ZY1 le -toe - ,.xe

Pe e- zx (8b)

and

(u , - w) 0=;{v,o + u cos cp - w sinp)

e : v,) + Ku, o - v cos CP) (8c)

CPO b w -

1 1 ( ) ' C O S C t -eWO ) -(w),P IKe r r

Ye = E - (we)'e (8d)

- (w,e + v sin p) We = -(w,P + u) (8e)

In obtaining these relations, in addition to the underlying assumptionsof thinness of the shell and validity of Kirchhoff's hypothesis, wehave made the further assumptions that components of strain and rota-tion are small compared to unity, and in addition, strains are smallcompared to rotations, i.e., 9 = 0(1) = 0 (w*) << 1. Further, we haveassumed that W, the component of rotation in the plane of the middlesurface of the shell is much smaller than the other two components,and we . Finally, it is possible to effect a considerable saving in

ebra with a minimal expense of accuracy by neglecting in N and wethe contribution of the u,v terms. This may be justified on an orderof magnitude estimate of the terms involved, or by examination of theeffect of this approximation on numerical results. The latter rein-forces the former. This approximation results in what is sometimes re-ferred to as a Donnell-type theory. In what follows, then, we take,

Iw,e) We = - W, (8f)

With a stress-strain law assumed in the form

a --- + Vue) Ce M#l- e+ U-+ )

Ea Po . 2(.l+) 69e (9)

the strain energy of the shell may be written out in detail, as well asthe quantities U. and U., sometimes referred to as the first and secondvariation of the strain energy. Specifically,

U, 2 .JTP (acoo 1 + 7e0 oI + 0eo 0 8e1 )brdmdBdz (10)

--

usr (a2e I f a ' el ',@I 8 , w lcpe

+ 2 eI + e0 eel + oteoae 1 ])brdtodedz (11)

where

2 0 A' + (~)2 ey C~6 PO ~C (P+ NO' 8 eo80 0 0 0- N 0

6(p, Cpl + W06 W " 'e = -e4 + loe' evp Ae, = Ne - 010

Cez i °ez 1 80

S we )2 =((~ 1 =-(pw 8 (12)

and where it is to be understood that the subscripted strain or rotationis to be evaluated in terms of the displacement of the same subscript.Finally, ap, c e and ace are related to the appropriately subscriptedstrains according to equation 9.

Stability Criterion

Employing linearized strain-displacement relations we may obtainfrom equation 6 the equilibrium equations governing the (rotationallysymmetric) prebuckling state of stress in the shell. We prefer tosolve these in a somewhat different formulation however, and omit theirpresentation here. Proceeding directly to the stability relations weobserve first that the prebuckled state is one of small deflections,governed by the classical (linear) theory. As Novozhilov' pointsout, then, we are justified in simplifying the expression for Us byomitting the wo terms in e,,, e and epe,, Upon integrating throughthe thickness, h, we may write

u 2(1-e') J + 2 e ' + 2 V le + e . brdcpd8

+ Eh ' 2 + X 2 + 2"X, 0 + 2(l _)Ne 2brdtpde2 4 (1-v2) 1 + 1 1 1 1

+ J N (We )2 + NO( wtp )2 - 2Neo (U V we) brdcpde (13)

2 Dk p o 1 0 1 - 7 - -

where Nco, Neo are the stress resultants of the prebuckled shell, and0a 0.due to rotational symmetry prior to buckling, N&Pe O

The potential energy of the external force system, V2 . is equal tothe product of the external load and the increase in volume enclosed bythe shell. An approximate expression for V. for the toroidal shellsegment under uniform internal pressure p is 8

"~ f (u,( - w) + -"v,, + u costp - w sin cp)

vsinP _~ -U v---- brdPdO. (114)r b uw, (P - rd~e•(4

ftability is governed by the relation

6(U2 + V2 ) = 0. (15)

It is clear that solution of the differential equations governing thestability of the toroidal shell segment would be a formidable task evenif the prebuckled state were a simple one. For this reason we employa Rayleigh-Ritz approach and obtain an upper bound for the criticalpressure. In this connection, guided somewhat by experimental observa-tion of the buckled pattern in the toroidal segment, we choose the fol-lowing set of displacements

U, a r2 cos (cp'p) cos (nO)

v = B r2 sin (w' ) sin (n8)

W, = 6 rs sin ("'- 0 ) cos (nO) (16)

where n is the number of lobes in the circumferential direction and1, 1, C are undetermined coefficients. The variations in u1, v,, w. are

reduced to variations in A, B, U. Inserting equation 16 into equation15 and requiring that the resulting equation be satisfied for arbitraryvalues of fa, 6B, 8), we obtain a system of three linear homogeneous

algebraic equations in Y, B, C, whose determinant set equal to zeroyields an estimate of the critical pressure. Omitting details of inter-mediate algebra we may write

- 8-

vhere

4a1 1',a 3 - a, 1 .a*, - a,22 a33 + a1*a1 3a~ - ,a3

D2a aqb + a 28 bl + a 2b3 + 2a1 3 agb 1 + i

-a12(a.3b13 + a.3 b 3) - a33 (a. 2 bil + a,1b,,)

where:

2(2 + 72 )6 + 1-v)n2 (u*\a,, =b 4 (5 (12 b (2+v)l

a12 n(1 + 5x') 2 vnX 2(1 )- nT(i-)(21)

aa - 2(2+u * + 27 126 - 2(1 + 2u)(18)+ 2 2(*)

am.~ N ++ ((b) 3\

\b 3

(26*) + 2li 22* + 11 8(i+v)+433 ~ /sr3 ( 12bib k7)

+ e(17+80)ii +1* - )A 8(26*\($ + 20 Z+"

-4 72(1+v)(t 40 - LK(4+v)(118) + 4214A+ n

+ i~2(.) 23(~)(2 t) + 2 X~n(c)N - 2( +3(*

+ 2n25Z(1-6)(8*)+ 2n2(l-T111A

k'bT -9 -b

2 \b /

bs 27_ + -2*~_(5\

2 2 ;2-2b

where the quantities (1*), etc.,are definite integrals of the form

(l*) = / r sin2 7(cp-o)dyp,

and are listed in the Appendix. All these integrals may readily beevaluated explicitly in terms of trigonometric functions. Finally,

TO/2Ke 40'Noe0r 3 sing 3(tp- dc

KI - Nr 2r r,V sin X(rp-eo) + K r2 cos K(cp-po40 0

(18)

Prebuckled State of Stress

In U. we encounter the integrals

1 f NT ) brd(Pd6, Ne brdcPdO (19)

which assess the contribution of the prebuckled state of stress to thestrain energy of the buckled state. Since the shells under considerationare very thin, the asymptotic method of solution of Reissner's equationsfor symmetric deformation of shells of revolution is most appropriate.

- 10 -

Clark7,8 has presented asymptotic representations for both the homoge-neous and particular solutions for the toroidal shell. These were

employed in the present investigation.

In this connection, it may be well to digress for a moment to dis-cuss the particular integral for toroidal shells. Galletley' has givenample warning that the frequently employed practice of usin, the membranesolution as a particular solution to the nonhomogeneous thin shell equa-tions is not valid in the case of the torus. His results of a numericalintegration of the complete differential equations for just such a tori-

spherical-cylindrical shell show a markedly different stress patternwhen compared to a solution using the membrane state as a particularsolution. However, when the first two terms of an asymptotic particularsolution developed by Clark6 are used in conjunction with the homogeneousasymptotic solution, the results agree remarkably well with Galletley's.

(See Figure 3 .) This agreement is obtained at a value of the asymptoticparameter (. ' 15) which is considerably smaller than the values assumedby the shells whose stability we wish to consider, and thus assures

validity of application of the asymptotic method.

Very briefly, the prebuckling stresses may be written in terms ofthe two basic functions 0 and Y according to the relations:

N =r- cos V + 0 sin cp)yo0 rm

N Eh2 , E- rbpH) (20)o mb ("P Eh

where

S(+XsinP)'/2Q (Aohir - Bohii + Coh ar - Doh, i + p.2/3(X/D)1/2

* [FoTr - GoTi) + (VQI)-1(X/D)1/2[GoQ-3 - G(cp)]]

¥ (l+Xsin)-j/2 QfBohir + A0hi + Doh 2 r + i(-/(/D)/

* [FoTi + GoTr)_ (p.)-i (X/D)l/ 2[F Q-3 - F(cp)]) (21)

and

- 11 -

X b/a, i ah' 2m y = l/3 ( )/3,

"" ) - o o' 1/ad

-M rV - - -- j rbpdp, F(qp) hlsin) (rV)cos ,

1/2g

G(cp) _k)irP -co bsincp) (rV) -2b2 p sin cp coo t

Mhr r

- bpr coD I' F0 = F(90 ) Go = G(w) (21a)

PH and PV are the horizontal and vertical components of applied pressure.In the above, hr, h1i, har, h2i are the real and imaginary componentsof the modified Hankel functions of order one-third, of the first andsecond kind respectively. Their argument is understood to be (iy).Tr, Ti are the real and imaginary parts of a special function introducedby Clark, and satisfying

T' - iy T(y) = 1. (22)

Al, Bo, C9, D. are constants to be determined from the boundary condi-tions of %he specific problem considered. In the present analysis, ashallow spherical segment was assumed joined to the torus at = to,

and a cylinder at t - - The vertical support on the cylinder was2"

assumed to be sufficiently far removed from the junction of the torusthat its sp6cific form had little influence on the form of the toroidalstresses. Nco , Neo, Mcpo and Mo were evaluated along the length of the

torus. A typical set o1 results is shown in Figure 4. The integrationindicated in equation 18 was carried out numerically. When this is donefor a specific choice of geometric parameters, Pcr/E remains a function

of n. Tne minimum value of critical pressure was found by calculating

Pcr/E over a range of n, subject to the condition that n be an integer.

Simplified Formula for Critical Pressure

The procedure outlined above was carried out initially usingequation 17. It soon became apparent that certain terms could be

- 12 -

omitted without significantly affecting the numerical results, and atthe same time affording considerable simplification of the relation forcritical pressure. The validity of the approximation rests on the factthat the value of n found to minimize Pcr/E was always rather a large

number compared to unity. This result is consistent with experimentalobservations of the buckling pattern. (See Figure 1.)

The simplified relation for critical pressure may be written:

Pcr h N-c h- N w (n >> 1) (23a)

where

4 !, 1 2 a33 - 1 33a-.

D- 3 3 ( 2 -4 ill 2) (23b)

aV) -n 2"

n 2v- + (1-0a (211))a22 W n2 (:)

- ~ b3J ~/ag =-2n Il i + 1, 12 0*

b3 b6 / b' 2b\b/

's bs3 = + 222A + K2 ( . + + Kb

Ka n/2 N r3 sin3 7(cp-qo)dp

K% S - I/2 Ner(2rrcp sin (cP.) + 3 r2(coS(CP-(po) ) dcp.IPO (23c)

- 13 -

Equation 23 is to be preferred over equation 1 for computational pur-poses when n is large. In all cases considered in this investigation,the value of n minimizing Pcr was greater than 40 and in most cases wasgreater than 60.

Numerical Results - Comparison With Experiment - Discussion

Computations of the critical pressure have been carried out thusfar for a limited number of parameters. A typical set of numericalresults is shown in Figure 5. The parameters in these curves correspondto those involved in an experimental program concerned with the sameproblem. In each numerical evaluation of the critical pressure, astress analysis of the unbuckled shell was made, the parameters K1 andKa determined, and then the stability criterion evaluated.

In the experimental program, scale models representative of thoseused in missile applications were tested. These models were made ofpoly-vinyl chloride, a material chosen among other reasons for its rel-atively high ratio of yield strength to elastic modulus. Some of theresults of these tests are also shown in Figure 5. In the case of thethickest shells tested, (h/b > .007) the disagreement between theoryand experiment increased significantly. The prebuckling stress analysisfor these shells revealed that at pressure levels below the predictedbuckling pressure, the difference in principal stresses in the shellnear the junction of torus and spherical cap had exceeded the yieldstress of the material. Since our analysis has assumed elastic behaviorthroughout, it would not be expected to apply to the experimental mate-rial in this range of the parameters.

The following experimental result may also be of some interest. Analuminum torispherical bulkhead was tested under internal pressure. Theparameters involved were

TO = 350 a = 34.43" b = 18.07" haverage = .081" E = I0 U P = .3.

Dimples began to appear in the toroidal section at about p = 25 psi.The pressure level was increased to 40 psi, and when the load was re-leased the dimples remained. The theoretical buckling pressure for ashell of this material(see Figure 5)is b4 psi. Hfowever, the prebucklingstate of stress reveals that in the vicinity of e = 45, the value of

on the inside and outside surfaces reaches a value of 1300 and

- 14 -

1015, respectively. Thus, for aluminum with a yleld stress of 35,000psi (as was the case in the test described) a value of p = 25 psi wouldproduce stresses very close to yield.

The plastic models of this configuration, however, appeared tobuckle elastically at 3.2 psi. (Theory predicts 2.9 psi.) This corre-sponds to an elastic buckling pressure of 70 psi for aluminum withE = l0 7 . The stress level induced near 45* in the torus in the plasticmodels for p = 2.9 psi is in the vicinity of 3700 psi, which is belowthe yield stress of the plastic material used.

It is clear, then, that the phenomenon of elastic buckling oftorispherical shells under internal pressure occurs only for very thinshells whose material has a relatively high value of the ratio of yieldstress to elastic modulus. With the increasing role being played bysuch materials in space structures, it is believed that the analysisdescribed in this report will assist the designer toward his objectiveof increased efficiency. Finally, it should be pointed out that theanalysis may be applied to toriconical shells simply by replacing thespherical cap by a cone in determining the prebuckled state of stress.

- 15-

IIMN

Figure I. EXPERIMENTAL SETUP

-16-

-hSM

0 =

-J

-____ ___ C

(D

C.'

0h.2as

LL

-17-

CL .

C * 0

U L

InIus

C84

ana

43L4

*0go

wIf

~~2IL

an .

-I

an

0

00 0

i~. c

u~n n

xr 3

+ +A

30 -I.;o

.20-

APPENDIX

The following definite integrals occur in the expression for criti-cal pressure (equation 17 or 23). In each instance the limits of in-tegration are from go to v/2. The variable r is given by r n a + b sin (.AU integrals may be evaluated explicitly in terms of trigonometricfunctions.

1* - r sin2 3:(c-9o )dc

2* = r sin c cos2 p sin2 K(ip- 0 )dt

3* w r cos2 tp sin2 K(tp-To)dc

4* - r cos c sin2 3(:-(Cp )dCp

5* " J'r cost c sin2 3(c-qO)dc

6* - T r 2 sin cp sin2 3(Cpop)dCp

7* = r 2 sin cos 2 cp sin2 3(Cpqo)dq

8* M r 2 cos t sin 3(p-Vo )cos K(p- )dc

9* - f r 2 coOs c sin (p-v 0 )cos !(- o)dcp

10* - r 3 sin2 (--o )dC

11* - r3 cos 2 K(tP-go)df

12* - r3 cos 2 9 cos 2 (cp-(p.)dcp

13* = r3 cos2 cp sing 3(c-p.)dp

14* - T r 3 sin2 V sin K(v-too)d(P

-21-

f5 r 3 cos2 W sin2 7(P-9%)dcP

16,* - r 3 cos (P sin8 7(cp-cp0)dtp

17* = r cos tp sin X(Y-P) cos :C-P)(

18*" = r 3 sin CP cos cP sin 3:((P-Vp) cos CP9)(

19* = r i 3 sin tp sin" 3(P-Cp)dcp

20* = r4 sin8 5:(P-cp,)dcp

21* - r4 cos2 7(tp-tpo)dtp

22* = fr4 sin cp sin3 :c-co)c

23* - r 4 sin cp cos2 7(p-po)dcp

=4 - r4 COS2 Cp COS2 Tp-ep0)dcp

25* = fr 4 cos ep sin T(cpop) cos Tc-p)c

26* - r 5 sin2 7(p-cp0 )dcp

27*, = fr 5 COS2 K(cp-,, 0)dcp

-22-

REFERENCES

1. GALLETLEY, G. D. Torispherical Shells - a Caution to Designers.Journal of Engineering for Industry, Transactions, American Society ofMechanical Engineers, v. 81, Series B, 1959, p. 51-62.

2. DRUCKER, D. C., and SHIELD, R. T. Limit Analysis of SymmetricallyLoaded Thin Shells of Revolution. Journal of Applied Mechanics,Transactions, American Society of Mechanical Engineers, v. 81, Series E,1959, p. 61-68.

3. SHIELD, R. T., and DRUCKER, D.C. Design of Thin-Walled Torisphericaland Toriconical Pressure-Vessel Heads. Journal of Applied Mechanics,Transactions, American Society of Mechanical Engineers, v. 83, Series E,1961, p. 292-297.

4. NOVOZHILOV, V. V. Foundations of the Nonlinear Theory of Elasticity.Graylock Press, Rochester, N.Y., 1953.

5. KEMPNER, J. Unified Thin Shell Theory; Part I of A Symposium on theMechanics of Plates and Shells for Industry Research Associates.Polytechnic Institute of Brooklyn, March 9-11, 1960.

6. ALFUTOV, N. A. (Moscow). On the Analysis of Shell Stability by theEnergy Method. Translated by M. 1. Yarymovych, Columbia University.Avco B & D, Translation, May 21, 1958.

7. CLARK, B. A. On the Theory of Thin Elastic Toroidal Shells. Journalof Mathematics and Physics, v. 29, 1950, p. 146-178.

8. CLARK, R. A. Asymptotic Solutions of Toroidal Shell Problems. Quarterlyof Applied Mathematics, v. 16, 1958, 47-60.

-23-

U.S. ARMY MATERIALS RESEARCH AGENCY.

WATEiBTOWN 72, MASSACHUSETTS

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Report No.: AMRA TR 03-06 Title: ;tability of Thin TorisphericalJune 1963 Shells under Uniform Internal

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Commanding General, Headquarters, U. S. Army Tank-Automotive Center,Detroit Arsenal, Center Line, Michigan

1 ATTN: SMOTA-RCM.1

Commanding General, U. S. Army Test and Evaluation Command,Aberdeen Proving Ground, Maryland

1 ATTN: AMSTE-LM, Technical Library

Commanding General, U. S. Army Transportation Research Command,Fort Eustis, Virginia

1 ATTN: Physical Science Division, Dr. d. D. Sands6

Commanding General, U. S. Army Weapons Command,Rock Island, Illinois

1 ATTN: qhief Scientist

Commanding Officer, U. S. Army Ballistics Research Laboratories,Aberdeen Proving Ground, Maryland

1 ATTN: Dr. Coy Glass

Commanding Officer, U. S. Army Chemical Corps NuclearDefense Laboratories, Army Chemical Center, Maryland

1 ATTN: Nuclear Physics Division

1 Commanding Officer, U. S. Army Engineer Research andDevelopment Laboratories, Fort Belvoir, Virginia

Commanding Officer, U. S. Army Quartermaster Research andEngineering Laboratories, Natick, Massachusetts

1 ATTN: Pioneering Research Division, Dr. S. D. Bailey

Commanding Officer, Harry Diamond Laboratories, Washington 25, D.C.1 ATTN: AMXDO-TIB

Commanding Officer, Frankford Arsenal, Bridge and Tacony Streets,Philadelphia 37, Pennsylvania

1 ATTN: Pitman-Dunn Laboratories1 Research Institute

Commanding Officer, Picatinny Arsenal, Dover, New Jersey1 ATTN: Feltman Research Laboratories1 Technical Library

Commanding Officer, Rock Island Arsenal, Rock Island, Illinois1 ATTN: 9320, Research and Development Division

No. ofCopies TO

Commanding Officer, Springfield Armory, Springfield 1, Massachusetts1 ATTN: SWESP-TX, Research and Development Division

Commanding Officer, Watervliet Arsenal, Watervliet, New York1 ATTN: Research Branch

I Commander, Office of Naval Research, Department of the Navy,Washington 25, D. C.

I Director, Navy Research Laboratories, Anacostia Station,Washington 25, D. C.

Commanding General, Air Force Cambridge Research Laboratories,Hanscom Field, Bedford, Massachusetts

1 ATTN: Electronic Research Directorate

Commanding General, Air Force Materials Central,Wright-Patterson Air Force Base, Ohio

1 ATTN: Thysics Laboratory1 Aeronautical Research Laboratories

1 Commander, Office of Scientific Research, Air R&D Command,Temporary Building T, Washington 25, D. C.

U. S. Atomic Energy Commission, Washington 25, D. C.

1 ATTN: Office of Technical Information

1 U. S. Atomic Energy Commission, Office of TechnicalInformation Extension, P.O. Box 62, Oak Ridge, Tennessee

!Director, George C. Marshall Space Flight Center,

Huntsville, Alabama1 ATTN- M-S&M-M, Dr. W. Lucas

Director, Jet Propulsion Laboratory, California Institute ofTechnology, Pasadena 3, California

1 ATTNt Dr. L. Jaffe

1 Director, Lewis Research Center, Cleveland Airport,

Cleveland. Ohio

1 Director, National Bureau of Standards, Washington 25, D. C.

1 Director, Research Analysis Corporation, 6935 Arlington Road,

Bethesda, Maryland

No. ofCopies TO

Commanding O2ficer, Ij. S. Army Materials Research Agency,Watertown 72, Massachusetts

5 ATTN: AMXMR-LXM, Technical Information SectionI AMXMR-OPT1 AMXMR, Dr. R. Beeuwkes, Jr.1 Author

63 -- TOTAL COPIES DISTRIBUTED

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