External Pressure - Tube

7/27/2019 External Pressure - Tube

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O A K R I D G E N A T I O N A L L AB OR AT OR Yoperated b y

UNION CARBIDE CORPORATIONfor the

U.S. ATOMIC ENERGY COMMl SSl ONORNL- TM- 166/g

'I

COLLAPSE OF TUBES BY EXTERNAL PRESSUREC. R. KENNEDY1. T. VENARD

NOTICETh is document contains information of a prel imina ry nature and was preparedprimari ly for intern al use at the Oak Ridge Nat ional Laboratory. I t i s s ub i ec tto rev is ion or correct ion and therefore does not represent a f in al report. Theinformation is no t to be abstracted, repr inted or otherwise given pub l ic dis-seminat ion without the approval of the ORNL patent branch, Leg al and Infor-mation Control Department.


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L E G A L N O T IC E

Thi s repor t wos prepared os on occount of Government sponsored work. Neithe r the United States,nor the Commission, nor ony person oct ing on behalf of the Commission:A. Mokes any warronty or representotion, expressed or implied, w ith resp ect lo the occurocy,

completeness, or usefulness of the informot ion contained in t h is repor t, o r t ho t t he use o fony informot ion, apparatus, method, or process disclosed in this report may not inf r ingeprivately owned r ights; or

6. Assumes any l iab i l i t ies w ith respect to the use of, or for damoges result ing f rom the use ofany informat ion, apparatus, method, or process disclosed in this report .

As used in the above, oct ing on behalf of the Commission" includ es any employee orcontractor of the Commission, or employee of such contractor, t o the extent that such employeeor contra ctor of the Commission, or emplo yee of such contra ctor prepares, dissemin ates, orprovides occess to, ony informot ion pursuont to his employment or contract wi th the Commission,or h is employment w i th such contractor.


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Co nt ra ct No. W-7405-eng-26

Metals and Ceramics Division

COLLAPSE OF TUBES BY EXTERNAL PRESSURE

C. R . Kennedy and J. T. Venard

DATE ISSUED

'APR 1 ? 1962

OAK RIDGE NATIONAL LABORATORYOak Ridge, Tennessee

opera ted byUN ION CARBI D 3 CORPORATIONf o r t h e

U. S. ATOMIC EJVERGY COMMISSION


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COLLAPSE OF TUBES BY EXTERNAL PRESSURE

C. R. Kennedy and J. T. VenardABSTRACT

The problem of tube co lla pse by ex te rn al pressure has been in ve st i-gated experimentally. A graph ica l so lu t ion developed t o s im pl i fyin el as t i c co l lap se des ign problems was shown t o agree with t he t e s tr e su l t s. The von Karman reduced modulus was used i n t h e gr ap hi ca ls o lu t i o n t o co r r ec t f o r t h e s t r e s s r ed i s t r i b u t i o n cau sed b y y i e ld in g .The ef fe ct s of th e geometric imperfect ions of ov al i ty and wal l - th ickn essv a r i a t i o n s on co l l ap s e p r e s su r e w ere shown t o b e r e l a t ed t o t h e s t r e s s -s t r a i n behavior of th e mate r ia l . The concept of a " c r i t i c a l t ime" wasdiscussed i n reg ard t o creep-buckling phenomenon.

INTRODUCTIONA s a r e s u l t of t h e i n t e r e s t i n t u b e co l lap s e b y ex t e r n a l pr e s su r e ,

d a t i n g from th e f i r s t exp er im en ts o f ~ a i r b a i r n ' i n 1 858 , s ev e r a l th eo -r e t i c a l and emp irica l so lu tio ns have been obtained. The code methodsder ived from the se so lu t io ns i n genera l con ta in sa f e ty f ac t o r s of un-kncwn magnitude which allo w fo r geometric impe rfec tion s, ma te ri alimperfections, and a con fide nce fa ct or . The us e of such code methodsi s over ly re st r ic ti v e under co ndit ions where optimum desig n i s r equ i redf o r e f f i c i e n t o pe ra ti on , p a r t i c u l a r l y i n r e a c t o r d es ig n a t e l ev a t edtem pera ture s, where neut ron economy d i c t a t e s a minimum m a s s i n t h e fluxf i e l d .

This inv es t ig a t ion app l ies prev ious so lu t ions fo r ins tan taneousco l l ap s e of p e r f ec t t u b es t o exp e rimental r e s u l t s o b t a in ed a t e l ev a t edtemperatures. The e ff e ct of geometric imperfec tions in reducing th ec r i t i c a l p res sure f o r c o l lapse was determined so th a t more appropr ia tesa fe ty fa ct or s could be used. The conf idence fa ct or i s not d iscussed

'w. Fa ir ba irn , Phi l. Trans. Roy. Soc. London 1 4 8 ( ~ ) , 8-13(1858).


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s ince it must be determined independently f o r each desig n by balan cingopera t ing eff ic iency and i n i t i a l cos t aga ins t fa i l ur e and rep lacementcosts .

Time-dependent or c reep co ll ap se experiments have not yet been pe r-formed, but th e problem i s considered i n view of i t s importance i n designa t th e elevated temperatures.

SpecimensTube specimens were fabricated from seamless type 304 stainless

s t ee l pipe and tub ing t o g ive rad ius t o wa ll th ickness r a t io s rang ingf r m 10 t o 25. These ra ti os were obtained by honing th e ins ide diameterand grinding the ou tside diameter t o clos ely held tolerances. The re-quired range of r a ti os neces sitate d the use of severa l hea ts of mate rial .

Delib erate wall- thickness v aria tion s of up t o ?lo$ were producedby of f- ce nt er gr in ding in a number of specimens. Specimens were al s odeformed i n a press t o obtain tu bes of an oval shape. A l l specimenswere annealed a t 1900F ?or 1 hr i n hydrogen a f t e r machining and fo rni ngoperat ions.

The fin is he d and annealed specimens were provided with s l i p - f i tend plugs welded with an edge fu si on weld, with a r es u l ti n g unsupportedlen gth of 11.0 in. i n most cases. End de fe ct s requi red t h a t two0.025-in. -w all specimens be sh orten ed t o a 9.0-in. free length and two0.015-in. -wall specimens t o an 8.0-in. fr e e length. A schematic drawingof a t y pi ca l specimen showing end plugs and l/4 -in . vent t ube i s shownin Fig. 1.

Tensile data were obtained by pul ling 3.0-in. sec tio ns of ty pi ca lspecimens f i t t e d with spe cia l end plugs f o r gripping.

CapsuleThe experimental capsule cons is te d of a len gt h of 2-in. sched-40

Inconel pip e with welded pipe-cap end clos ure s; a cutaway view i s showni n Fig. 2. The top cap was d r i l l e d t o recei ve t he specimen vent tube,


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UNCLASSIFIEDORNL-LR-DWG 64882

. VENT TUBE

12 in.

HFig . 1. Tube Collapse Specimen


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UNCLPSSIFIEDPHOTO 56339R

THERMOCOUPLES

SPECIMEN

VENT

I ir/i VENT TUBE

% A I 2o3 I N S U LA T I O NON PROBE

I PREHEATER LOOPi PROBE

>a1203 INSULATIONI : ON PROBE

2 - ~ n .SCHED-40INCONEL PIPE

Fig. 2. Cutaway View of Tube Collapse Capsule.


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th re e swaged thermocouples, and a pr es su re li n e . The to p cap assemblywas wel@d togethe r and ins er te d in to th e capsule , and then the c losureweld was made.

An i n l e t gas p rehea te r loop a round the ou ts ide of the capsu le p re -vented th e specimen from cooli ng during pr es su riz at io n. Test temperatures,as dete rmined by th e top., middle, and bottom thermocouples, were main-tai ned withi n '9OF over th e specimen len gth duri ng th e te s t .

The capsule was placed i n a furnace and brought t o 1200F, th e t e s ttemperature. When equ ilib riu m was reached, th e capsul e was pr es su ri ze da t approximately 300 ps ig of argon per minute u n t i l c ol lap se occurred.The capsule pressu re was recorded on a s t r i p ch ar t recorde r reading theou tpu t of a s t r a in -gage f lu i d -p res sure c e l l . A re la y system which wasact uat ed by th e probe shorting against the specimen wall and whichsimultaneously energi zed an event marker on th e pre ssur e time chart anda l i g h t on t h e c o n t r o l p a ne l was i n i t i a l l y used t o i n d i c a t e f a i l u r e o rspecimen co ll aps e . The prob e system was abandoned, however, when i tbecame evident t h a t the d isc ont inu i ty obtained on the pres sure vs t imerecord ing p lus the de f in i te and qu i t e aud ib le snap o f t he co l lapse weres u f f i c i e n t i n d i ca t i o n s of f a i l u r e .

Time of t e s t , maximum p re ss ur e , and load ing ra te fo r each te s t wereobtained by examinat ion of th e pressure vs t ime ch ar t .

The exper imenta l r esu l t s a re t abu la ted i n Table 1, which givesmater ia l ident i f icat ion, information on the specimen dimensions ,co l laps ing p res sure , and t ime of t e s t . S t re s s - s t r a i n curves ob ta inedf o r ma te ri al s 23999X and 24555 a re shown i n Fig s. 3 and 4, r e s p ec t i v e ly .I t i s important t o note th e agreement between t he two curves shown i nFig. 3 d es p i t e t h e s t r a i n r a t e s d i f f e r i n g by a f ac t o r of 3 0. F ig u re 5presen ts some ty pi ca l specimens a f t e r te s t . I t was i n t e r e s t i n g t oobserve th a t specimens having wal l - th ickness va r i a t i on s of l e s s than5% exhib i ted a uniform two-lobe col la pse, whi le tho se having 10 t o 13%var ia t ion s f a i l e d i n a tw is ted and uneven manner a long th e i r l en g th .


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Table 1. instantane ous Tube Colla pse Data f o r Type 304 S t a i n l es s S t e e l a t 1200F( A l l specimens annealed a t 1900F f o r 1 h r i n h y drog en )

Mean In i t i a lAverage Average Radius W a l l - Def lec t io nW a l l Outside Average Thickness Average Specimen Coll apse TestTes t Heat Thickn ess Diameter W a l l V a r i a t i o n W a l l Length Pr es su re Time

No. No. ( i n . ) ( in . ) Thickness (k ) Thickness ( in. ) (p s i ) (min )23999X23999X23999X23999XMcJunkin

McJu nk in23999X23999X245552455524555McJunkin245552455524555245552455524555245552455524555245552455524555


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0 0.1 0.2 0.3 0.4 0.5STRAIN (% )

Fig. 3. Ten si le Curves for Type 304 S ta in le ss St ee l Specimensat 1200 OF - Heat 23999X.


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UNCLASSIFIEDORNL- LR-DWG 64884

0 0.1 0.2 0.3 0.4 0.5 0.6STRAIN (7%)

0.025 in./in./min0.2 % YIELD STRENGTH =44,900 p s i

I

Fig . 4 . Tensi le Curve f o r Type 304 S ta i nl es s St ee l Specimena t 1200OF - Heat 24555 .

I

I


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DISCUSSION AND CONCLUSIONSThe at te nt io n given t o the problem of th e col lapse of long, th in ,

c i r cu la r , cy l in d r i ca l sh e l l s by ex te rna l p r essu re dur ing the pas t hundredyea rs has re su lte d i n a number of t he or et ic al and empir ica l so l u t i o n swhich form the b as is f o r the design of such vess els .

The solu tfon fo r el a s t i c collap se of pe rf ec t tubes can be accomplishedby using th e von ~ i s e s ~e l a t i o n sh i p :

whereP = t h e c r i t i c a l pr es s ur e f o r c o l la p se ( p s i ) ,c ra = th e mean radius of t he tube ( i n . ) ,

h = average or nominal wall thic kne ss of tube ( i n. ),0E = reduced modulus of e l a s t i c i t y ( ps i) ,r7 = P o i s so n ' s r a t i o 0 .3 ,a = tube length between supports ( in .) ,n = number of lobes i n coll aps ed tube.

The re su lt s of ca lcula t ion s using Eq . (1)a re shown i n Fig. 6,where a /a i s p lo t te d vs @. t i s seen t h a t a s ingle curve for a givenvalue of a/h i s formed by combining port ions of curves fo r i n te gr a l0values of n. The c r i t i c a l pressure f o r any given tube may be obtainedthrough Fig. 6; however, th e tube geometry necessary t o re s i s t a givenpressure cannot be obtained without th e use of unwieldy tr i al -a nd -e rr orprocedures. As a means of elim ina tin g th e ne ce ss it y f o r such proceduresth e fol lowing grap hica l sol ut i on was developed.

2 ~ .on Mises, Z . Ver. deut. Ingr. (VDI 2.) -8, 750 (1914).-


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UNCLASSIFIEDORNL-LR-DWG 58722R1 - 1 - 1 T I I 1 x 1 1- I --I---

[ 9 - v 2 ) ~ 0 (9 - "2 )EQUATION = =E ho-7

Fig. 6. Master Curve f o r C ri t i ca l Externa l Pressure To CollapseCyl in dr ica l Vesse l s.


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Equation (1)may be modified t o

where ( f ) = t h e r a t i o of @ fo r a g iven .l/a and @ f o r an i n f i n i t e - l en g t htube , obtained from Fig . 6. I t i s observed i n Eq. (1) and more c le ar lyshown i n Eq. ( 2) t h a t t h e c r i t i c a l p r es s ur e i s a l in ea r func t ion of t hereduced modulus f o r fi xe d valu es of .l/a and a/h .0

The s t r es s d is t r ib ut ion across the tube wal l var ie s depending onwhether the deformation i s e l a s t i c o r p l a s t i c . The ave rage s t r e s s ,however, i s given by

where u = s t r e s s i n t a n g e nt i a l d i r e ct i o n ( p s i ) and P = pressure ( ~ s i ) .acombining Eqs. (2 ) and (3 )

where

8y sub s t i tu t i ng va lues o f ( f ) and a/ho i n Eq. (4) , a s e r i e s ofs t ra ig ht par al le l l i ne s may be generated on a logar i thmic plo t of s t r es svs reduced modulus. Sinc e th e des ign er i s primarily concerned with thetube geometry necessary t o re s is t a pa r t i cu la r pressure , Eq. (3) i s usedt o const ruct s t r a i gh t pa ra l l e l i so bars superimposed on the previouslyobtained plot . Each isobar in t ers ec ts the constant ( f ) and a/h l i n e s0a t a s t r e s s s a t i s f y i n g Eq. ( 3 ). A plot as descr ibed above i s shown i nFig . 7 fo r i n f in i t e - l eng th tubes [ (f = 1 1 .The addi t ion of a mat er i a l l i ne t o Fig . 7 w i l l now y i e ld cond ition sof s t ab i l i t y and i n s t ab i l i t y f o r t ub es u nd er ex t e r n a l p r e ssu r e. T hi smat eri al l in e should id ea ll y be obtained from compression te s ts ; however,t h e u se of t e n s i l e d a t a t o t h e s t r a i n s of i n t e r e s t w i l l not introduces i g n i fi c a n t e r r o r .


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The reduce d modulus, de fi ne d by von arma an, c o n s i d e r s t h e s t r e s sr e d i s t r i b u t i o n due t o y i e l d in g :

:


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5 lo7MODULUS ( p s i )

Fig. 9. Logarithmic Plot of Stress vs Reduced Modulus for Type 304Stainless Steel Specimen at 1200'~- Heat 24555.


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UNCLASSIF IEDORNL-LR-DWG 64889

"

i 4 2 14 i 18 2 0 2 2 2 4 2 6CI- mean rad iush, (average wal l th ickness

\-\.\ - ATA MA TE RIA L IDENTIFICATION

Fig. 12. Instantaneous Collapse Pressure vs a/ho Curvescalculated from tensile data.

\\ \ \

0 2 3 9 9 9 X2 4 5 5 5A McJUNKlN>'\ \

\

I


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p r o p o r t i o n a l i t y of ( f ) and c r i t i c a l p r e ss u r e i s no lo n g er v a l i d f o rin e l as t i c col lapse s ince the reduced modulus var ies wi th s t r es s . Thef a c t o r ( f ) f o r a g iv en ,4/a r a t i o a l s o v a r i e s s l i g h t l y wi th t h e a /h ora t i o , and t he re fore t he same ( f ) cannot be used t o sh i f t a l l t he a /hol in es i n F igs. 10 and 11. A method which may be used t o d eterm ine t h eef fe c t of reducing the ,4/a ra t i o i s t o make an a p p r o pr i at e s h i f t of t h em a t e r i al l i n e t o t h e r i g h t f o r a g iv en ,4/a r a t i o .

For design purposes the minimurn expected m at er ia l pro pe rti es oftype 304 s t a i nl es s s t e e l a t 1200F should be used. The graphic solut ionbased on these minimum values i s shown i n Fig. 13 and in clud es t he e f f e c tof lowering th e ,4/a ratio. This was done by s hi f t in g the mate r ia l l i n et o the r ig ht through the use of F ig. 6.

This so lut ion does not imply th a t tubes under a pressure l e ss th anthe c r i t i c a l p re s sure fo r p ro longed pe r iods o f t ime w i l l not col lapse ,s ince c reep i s important a t 1200F. In f ac t , t ime-dependent l i ehaviormay be important even under ra pi d increas es i n pres sure f o r ce rt ai nmate r ia l s and tempera tures . For th i s pa r t ic ul ar case of type 304 s t a i n-l e s s s t e e l a t liOClF, t he t en s i l e d a t a were una f fec ted by s t ra in - r a t eva r i a t i ons i n t he t e s t i n g range, and time dependency i s t he re fore f e l tt o be n e g l i g ib l e i n t he s e r e s u l t s .

Se l ec t i on of a su i t ab l e sa fe ty fa c to r i s dependent upon sev e ra lc r i te r ia , f o r example , a nt i c ip a te d perfe c t io n of the tube , minimumexpected s t ren gth of the mater ia l , and cos t of fa br ica t io n and opera t ionvs cost of f a i l u r e and replacement . The major imperfect ion s i n tubin g,ova l i t y , and wa l l- t h ickness va r i a t i o n w i l l now be evaluated.

The most severe red uct io n in co l laps e pressu re r es u l t s from tubeov s l i t y o r ou t -of - roundness. ~ i m os he nk o~as shown th a t t h e maximumst re ss ac t in g on the tube wal l of an ova l spec imen i s given by

W- Pa 6Pa ou - - + -max h P 90 h 2 l - -0 PCr's. Timoshenko, Theory of E l a s t i c S t a b i l i t y , McGraw-Hill,

New York, 1936.


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wherea = maximum st r e ss i n tube wa ll ( p s i ) ,max

w = maximum i n i t i a l de fl ec ti on of tube0= l /4 (ODmax - OD ) ( i n . ) ,minP = c r i t i c a l pr e ss u re f o r c o ll aps e of pe r f e c t tube ( ps i ) .c r

In h i s d i scuss ion of the above f o m ~ l a imoshenko indica ted tha t themaximum st r e s s which cou ld be r e s i s t e d i s t h e y i e l d s t r e s s o f t h emater ia l . With mater ia ls which do not exh ib i t a def ini te y ie ld po int ,some uncer tainty ex is ts a s t o the proper va lue f o r a I t was foundmax'th a t se t t in g amax equal t o the va lue of t he 0.2%-offse t yi e l d s t re ngt hobtained from the te ns il e curve of the mat eri al resu lte d i n a goodcor rel ati on with experimental res ul ts . Equation (6) may be solved f o rthe r a t io p0/pcr a s fol lows:

W 2 amax w0]P c r + 1 + 6 ) [ ( - 1 - )~ cT ho + ) A p - hr o

2 , ( 7 )c rwhere

P = pressure fo r col lapse of the oval tube (p s i ) ,0P = pressure for col lapse of a perfect tube of same a/h ( ~ s i ) ,Cr 0a = 0.2%-of f se t y i e ld s t rength of ma te r ia l (p s i ) ,maxa = c r i t i c a l s t r e s s fo r co l lapse of a pe r fec t tube of sameCr

The co rre lat ion of th e foregoing equation and experimental values i sshown i n Fig. 14. I t should be noted th a t the e f f ec t of ova l i ty va r ie swith wo/ho and with amax/ucr the l a t t e r va ry ing wi th the ma te r ia l .Wall-thickness variation w i l l al so cause a reduct ion i n th e col lapsepressure ; however, t h i s reduct ion i s re la t i ve ly smal l i n comparison withth a t of oval i ty . ~ l l i n ~ t o n ~n considering t h i s problem proposed th e

5 ~ . . El lin gto n, The Cr i ti c a l Pressure of a Tube with an Ecc en tri cBore, DEG-43R) (1960) .


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U N C L A S S I F I E DO RN L - L R - DWG 6 4 8 9 0

u,,, = 44,900 psia,, = !1,640 psiP,,= 827 psi0- - 14.07

Fig. 14. Plot Showing Effect of Ovality on InstantaneousCollapse Pressure.


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fo l lo win g r e l a t io n sh ip :

where v = minimum wal l-th ick nes s va ri a ti on hohmin) and Pv = p r e s s u r efo r co l l ap se of tu b es hav in g wa l l -th i ck ness v a r i a t io n s v . It i s i n t e r -e s t i n g t o n o te t h a t t h i s r e l a t i o n s h i p i s i nd ep en de nt o f a /h and ~ / a .0A co mpa ris on of E l l i n g t o n ' s r e l a t i o n s h i p w it h t h e t e s t r e s u l t s i s g i ve ni n F ig. 1 5. I t can be seen th a t a :LO$ v a r i a t i o n i n w a l l t h ic k n e s s p ro -d uces o n ly a 3 % red u c t io n of th e c r i t i c a l p r e ssu re , an e f f e c t which i sconsidered t o be i ns ig n i f i ca n t s in ce most vendors can rea d i ly producetu b in g wi th in such a to l e r an c e and s in ce th e sc a t t e r i n th e ex p er imen ta lr e su l t s ex ceed s 3% .

The foregoing d isc uss ion serves t o demonst rate how the c r i t i c a lp ressu re f o r ins tan taneo us co l laps e may be ob ta ined and what th e e f f ec tsof tube imperfectio ns ar e. Therefore, wi th the minimum expected st re ng thof th e m at er ia l and the e-xpected imperfe ctions known, i t i s p o s s ib l e t od e si g n w i th a ny d e s i r e d c o nf id en ce f a c t o r t u be s t o r e s i s t c o l l a p s e .

Time-Dependent CollapseAlthough no exp er im en tal work has be en performed on time-dependent

co l l ap se , it i s f e l t t h a t t h i s phenomenon w a r ra n ts d i s c u ss i o n .An exact s o lu t io n of th e c reep-buckl ing problem i s l i m i t e d by t h e

lack of p ro p e r and ex ac t g en e ra l i za t io n s o f th e t en s io n -c r eep s t r e s s -s t r a i n r e l a t i o n s t o m u l t ia x i a l s t r e s s w it h c ha nging pr i n c i p a l s t r e s sd ir ec ti on s. S ev era l t en ta t i ve so lu ti on s have been p r ~ ~ o s e d ~ , ~hicha re approximate because of th e many neces sary sim plif ying assumptionsmade.

6 ~ . . Hoff, W. E . Jahsman, and W. Nachbar, A Study of Creep Collapseof a Long Cylin der Under Uniform Ex te rn al Pr es su re , LSMD-2360(March 28, 1 95 8) .

7 ~ . . El lin gto n, Creep Collapse of Tubes Under Ext ern al Pre ssu re,DEG-162 ( R ) (1960) .


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An approximate so lu ti on which may be more e a s i l y handled tha n th os ere fe rre d t o above can be obtained by using an ana lys is of th e "deformationt h ~ o r y " ype. Such an approach t o in el as t i c buckl ing assumes th at t o t a ls t r a i n a t a g i ve n tim e i s descr ibed by t he equat ion resu l t in g from pr i n-c ip a l s t r a i ns be ing a lways p ropor tiona l t o p r inc i pa l s t r e s ses wi th noro ta t io n of th e p r inc i pa l ax i s occurr ing . When the l a t e r a l de f l e c t ionof t he tube wall , o r buckling, begins t o occur because of inc rea sin gst re sse s , the magnitude of t h i s d ef l ec t ion i s computed as though th enew increased s t re s s had been appli ed throughout th e en ti re loadingper iod . This method should the n provide a lower l i m i t t o t h e i n e l a s t i cbuckling loa d of members with lim it ed geometric impe rfect ions . Appli-c a t i o n of t h e i n e l a s t i c - b uc k l i n g so l u t i o n t o p i n- j oi n t ed c o l m s ha sdemonstrated the conservatism of such a predi ct i on of t ime t o fa i l ur e.

Inelas t ic-buckl ing solu t ions fo r tube col lapse are obta ined byrepla cing th e reduced modulus i n th e previously developed equationswith an ef fe ct iv e modulus. Two such ef fe ct iv e moduli ($ and E ) havetbeen proposed f o r s ol ut io n of column buck ling^:

which was proposed by ~ob otn ov* nd defin ed as t h e tang ent slo pe of t heconstan t s t r a in - r a t e s t r e ss vs s t r a i n curve ; and

proposed by shanley g and defin ed as t he t ange nt slop e of th e isochronouss t r e s s - s t r a i n c u r v e s .

Expressions f o r b ot h moduli may be obtained from the creep equation

where E = c r e e p s t r a i n ( i n . / i n . ) , t; = t ime ( hr ) , and A,a,m = m a t e r i a lC'G. N. Robotnov and S. A. Shester ikov, J . Mech. and Phys. S o l ids 6,

27 (1957). -9 ~ . . Shanley, "P rin ci pl es of Creep Buckling, " Chap. 19 i n Weight-

St re ng th Ana lys is of A i rc r a f t St ru ct ur es , McGraw-Hill, New York, 1952.


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constan ts , and from t h e t o t a l s t r a i n e qu a ti o n

where E = t o t a l s t r a i n . The r e s u l t i n g e x pr e s si o ns a r e t h e ntEe =[A $1-1

and

It sh o uld b e n o ted th a t i f m were t o become equa l t o 1, & wouldgo t o z e r o f o r a l l va lu es of s t r e s s r e g a r d le s s of t h e m a t e ri a l s t r e n g t h .Since i t i s not uncommon t h a t m = 1 f o r many ma ter ia l s , i t i s f e l t t h a tth e Robotnov ef f e c ti v e modulus i s n ot s u i t a b l e f o r u se i n t u be -b u ck l in gequat ions . The s t r e s s - s t r a i n r e l a t i o n s h i p s t o b e c o ns id er ed i n tu b ec o l l a p s e a r e t h o s e o c c ur r in g i n t h e t a n g e n t i a l d i r e c t i o n , a nd it hasbeen shown1' t h a t f o r capped end tu be s

3E = - EC 4 - c ' (15)8where E = t a n g e n t i a l c re ep s t r a i n und er t a n g e n t i a l s t r e s s a an dC8 8F = u n i a x i a l c r ee p s t r a i n u nd er a u n i a x i a l s t r e s s e q ua l t o uc 8 '

Su b s t i tu t in g i n Eq. ( 14 ) y i e l d s

"c. R. Kennedy, W. 0. Harms, and D. A . Douglas, Trans. Am. Soc.Mech. Engrs . J . Basic Engr . 81 ( s e r i e s D ), 599 (1959).-


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The re su lt in g tube collap se rel at i on shi p from Eq. (2) i s th en

P = ( f )c r 4 ( l"t r 2 ) (5).As was prev io usl y poi nte d ou t, Eq. (19) does not yi el d an exact

so lu t ion fo r the c r i t i c a l time of co l lapse bu t y ie lds , ins tead , thec r i t i c a l time f o r t h e l o s s of s t a b i l i t y i n t he c l a s s i c a l s en se . A ft erthe c r i t i c a l time has e lapsed, any small d is turbance appl ied t o the tubeshould re su l t i n def le ct io ns which increase wi th t ime. Collapse may thenbe expected a t some time gre ater than the c r i t i c a l time f o r los s ofs t ab i l i t y calcula ted by Eq. (19) .

A s imple r e la t ionsh ip o r p ropor t ion a l i ty w i l l p o ss ib l y e x i s t f o rc r i t i ca l time vs ac t ua l co l l apse time bu t it must be determined experiCmentally. U nt il such experiments have been performed, the c onserv ativepre di cti on i n Eq. (19) should be used.

An importan t r e su l t of the above a n a 1 y s i s . i ~ ha t the p rev ious lydeveloped gra phic al so lut ion may again be ut i l iz ed . The su bs ti tu t i onof a ma ter ial curve of E vs a corresponding t o the des ign l i fe t im et eof the ve ssel f o r the mat er ia l curve of Er vs a w i l l produce thesol ut ion of Eq. (18 ).

CONCLUSIONSThe following conclusions on tube col lap se a t 1200F f o r type 304

s t ai nl es s s t e e l have been reached:1. The use of th e von &m an reduced modulus i n col la ps e equat ion s

adequately accounts f o r in el as t i c behavior of t he mater ia l under rapid lyappl ied loading.

2. The gr ap hi ca l method developed, which superimposes m at er ia lbehavior on tube geometry, allows the a pp li ca ti on of known sa fe ty fa ct o rsi n design t o r e s i s t ins tan taneous co l l apse .


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3. The e f fe c t o f o va l i ty , a s e r io us imper fec tion which g r ea t lyreduces the pressure f o r ins tantaneous co l lapse , can be predic ted.

4. The reduct ion of col la pse pressure due t o wal l - th icknessva ri at io ns wit hin normal manufacturing t ol er an ce s may be neglected.

5. A c r i t i c a l t im e, a f t e r which co l l ap s e may occur , ex i s t s f o ra t u be u nd er an ex t e r n a l p r es s ur e l e s s t h an t he c r i t i c a l p r es s ur e f o rins tan taneous co l lapse .

The a uth ors wish t o thank F. L. Beeler and C . W. Walker f o rt h e i r a s s i s t a n c e i n the experimental program.


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C . F. k i t t e n , J r .A. L. Lot t sH. G . MacPhersonW. D. ManlyW. R . Mart inH. E . McCoy, J r .P . P a t r i a r c aM . J . SkinnerG. M. SlaughterR . L. StephensonR . W. SwindemanA. TaboadaW. C. ThurberJ . T. VenardJ . R . Weir, J r .J . W. WoodsA. F . Z u l l i g e rDivis ion of Technica l

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