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OFFSHORE TECBNOLOGY CO:NFERENCE6200 North Central ExpresswayDallas, Texas 75206THIS IS A PREPRINT - - - SUBJECT TO C O R R E C ~ I O N

PAPERNUMBER OTC 1355

The Effect of Axial Tension on Moment CarryingCapacity of Line Pipe stressed

Beyond the Elastic LimitBy

J . C. Wilhoit, Jr. and J . E. Merwin, Rice U. Copyright 1971

Offshore Technology Conference on behalf of American Institute of Mining, Metallurgical, andPetrolaum EngineerB, Inc., The American Association of Petroleum Geologists, American Institute ofChemical Engineers, American Societyof Civil Engineers, The American Society of MechanicalEngineers, The Institute of Electrical and Electronic Engineers, Inc., Marine Technology Society,Society Exploration Geophysicists, and Society of Naval Architects &Marine Engineers.This paper was p r e p a r ~ d for presentation at th e Third Annual Offshore Technology Conferenceto be held in Houston, Tex., April 19-21, 1971. Permission to copy is restricted to an abstractof no t more than 300 w o r c r s ~ ~ ~ I l l u s t r a t i o n s may not be copied. Such use of an abstract shouldcontain conspicuous acknowledgment of where and by whom th e paper is presented.

INTRODUCTIONThe question of pipe ovalization and i tseffect on m o m e n t ~ c u r v a t u r e relationships in theelast ic case have been treated by Brazier1 andothers. 2- 4 The effect ofovalization in th enonlinear (strain hardening) case has beentreated numerically by Ades. 5 Ades' analysisemploys the principle of least W9rk for thecylindrical tube under pure bending when partof th e tube i s beyond th e e las ti c l imi t. Thetube "is allowed to assume that ell ipt icalcross-section which minimizes th e strain energy.Ades' analysis has been successfUlly employedto predict resUlts which were obtained "experimentally in pipe bending tests at Rice U.Since tension is generally employed inoffshore pipeline laying operations, theimportance of this . effect on Ades' analysis isworth investigating. The relationship betweenth e reduction inJTIaximum moment carryingcapac ity as the axial tension is increased is

desired. The pipe is allowed to deform beyondthe e la st ic l im it and th e effect of pipeovalization is included. -Review of Ades' Analysis

Consider th e cylindrical pipe loaded asshown in Fig. 1. Ahollow tube subjected topure bending is known to flatten and take on anovalled shape. The ovalled shape is assumed tobe el l ipt ical (Fig. 2 ), and i t i s assumed thatReferences and i ll us tr at ion s a t end of paper.

the circumferent ia l deformation is inextentionalinsofar as relating the el l ipt ical cross-sectionto the in i t ia l circUlar cross-section. Otherassumptions made by Ades are that the tube i ssubjected to pure bending, i t is long enollghthat end effects may be ignored, and thematerial is isotropic with th e same stressstrain curve in tension and compression.Referring to Fig. 2, the work per unit length inthe elast ic range is1 1dW s [ 2 crL EL + 2 crT ET ] d ~ d S ,

where 01, SL and O"T' E.r are longi tudinal andtransverse stress and strain. Written in termsof strains,dW = 1 - ~ 2 [Ei2+ ET2 + ELET] d ~ d S

This may be written in terms of stress andstrain intensit ies O"i and ei as

where

and O"i = Eei for th e elastic range. Adesshowsthat, also in the plastic range, the work perunit volume is s t i l l g iven by the area under th es t ress intens i ty-s tra in intens ity curve and,hence, that the theory is continuous in the

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1-294 THE EFFECT OF AXIAL TENSION ON MCMENT CARRYING CAPACITY OF... .. LINE PIPE STRESSED BEYOND 'IHE ELASTIC LIMIT OTC 135elastic and plastic 'ranges. I t should be notedthat th e conventional stress-strain curve in auniaxial tension tes t is th e s t ress in tensi ty strain intensity curve for that case. Thus,the standard stress-strain curve from a simpletension test is used in both elastic andplast ic ranges.

Ades implies th e use of the principle ofleast work, bu t th e procedure he follows is tof irs t bend th e tube and then le t i t oval tothat elliptical cross-sect ion for which th estrain energy is a minimum. Since th eexternally applied moment does no work duringth e ovalization, this procedure actually findsthat value of a /r (a being th e semimajor axisof the ellipse and r th e original averageradius of th e undeformed tUbe) which minimizesth e strain energy.

The appropriate expressions needed for th ecalculation of th e strain e n e r ~ , W, for agiven ovalization parameter, alb, ar e repeatedfrom Ades' paper,E: = l . = (y + 1; cos a.)L p m

12 ,y 2 Y 2 --coso. { l+ ( ~ 2 ) [ ( 1 - i 7 ) / ( F ) ) } 21 1 ' 1 1 Ym

E:T 1; [ ( - ) - -] -1lE: =1;[( - ) - - ] - \ l ( -p )PT r LM 'PT r3....L = 2 2 a 2 2 2'ab [b - Ym + ( 1)2") Ym ]PT

Poisson's ratio" l{, is taken a function of eiand is given bY'" .1

\l { l- [(1+0) / (1+e. )l2:} /e i

and J{ = .

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OTC 1355 J. C. WIIlIOIT, JR. and J. E. MERWIN 1-2955. The entire process is repeated fordifferent p 'S to obtain the moment M as afunction of p for a fixed value of FA. Themaximum value of M as a function of p is usedas a point on th e M-vs-FA curveCalculations were made for four differentcases. Two different stress-strain diagramswere used: one with n = 20 corresponding to a

large degree of strain hardening, and theother with n = 200 corresponding to a nearlyelastic-perfectly plastic material (Fig. 4).For both cases, aO.7 was taken to be 55,000psi. These stress-strain relationships areconsistent with stress-strain curves obtafneafor steel pipe used in th e pipe bending testsat Rice U.Pipes with two different diameter-tothickness ratios were used: a 10-3/4 in. x0.233 in. (D/t = 45) and 16 in. x 0.255 in.(D/t = 62).The maximum bending moment was determinedfor various axial loads. The results of thesecomputations are plotted in a nondimensionalform (Fig. 5). Mo is th e maximum moment thatcan be applied with no axial force. Whileordinari ly for strain-hardening material therewould be no theoretical maximum moment, therewill be a maximum moment when ovalling of thecross-section is considered. Fo was arbitrarily taken to be the force which would cause anaxial stress of (JO.7. For an ideal elasticperi'ectly plastic material ( i .e . , n approachingini ' inity in th e R8mberg-osgood -relat:Lon) . F ~ - - -would be the maximum axial force that could beapplied. An exact . ~ a l y s i s for the .elastic

perfectly plastic casewith no ovaIHng is .included in th e Appendix and gives the resultMIMo = cos [( ~ 2 ) ( F / F O ) ] . This curve is alsoincluded in Fig. 5.

ResultsAn examination of th e moment-forceinteraction diagrams indicates that there isl i t t le dependence on th e diameter-to-thicknessratios considered. For a rather f lat stressstrain diagram, th e results do no t differsignificantly from th e ideal elastic-perfectlyplastic relationship which did no t includeovalling. However, for a material with a largedegree of s train-hardening, the ideal momentforce relationship does not adequately describeth e behavior.Although axial tension is sometimesemployed to decrease th e external moment appliedto a pipe during laying operations, this sameaxial force also decreases th e moment carryingcapability of the pipe.

REFERENCES1. Brazier, L. G. : "On th e Flexure of ThinCylindrical Shells and Other Thin S e c t i o n s ~ ,Proc., Royal Society of London (1927) 116,Series A, 104-114. -2. Chwalla, E.: "Reine Biegung schlanker,dllimwandiger Rohre mit gerader Achse", ZAMM(1933) il, 4S-53. -3. Wood, J . D.: "The Flexure of a UniformlyPressurized, Circular, Cylindrical Shell",J. Appl. Mech., ~ , Trans., ASME (195S) SO,453-458.4. Reissner, E.: "On Finite Bending ofPressurized Tubes", J . Appl. Mech., 26,Trans., ASME (1959) 81, Series'E, 3S6=392.5. Ades, Clifford S.: "Bending Strength of 'Tubing in th e Plastic Range", J. Aeronautical Sciences (Aug., 1957) 605-610.6. Stang, A. H., Greenspan, M. and Newman,S. B.: "Poisson's Ratio of Some structuralAlloys for Large Strains", J . Res.,National Bureau of Standards, (Oct., 1946)21, 211-221.7. Ramberg, W. and Osgood, W. R.: "Description of Stress-strain Curves by ThreeParameters., NACA TN 902 (July, 1943).APPENDIX

Elastic-Perfectly Plastic Bending of aTube Under Axial Force With No OvallingConsider th e tube shown in Fig. 6' in th e.fully plas ti c s ta te , so that the maximum moment,M, is being transmitted with an axial i'orce, F.Defining Mo as th e maximum moment transmittedwith no axial force and Fa th e maximum axial

force transmitted with no bending moment,M 4r2 tcr0 yF 2rt'ITcr0 y

and -a .!.ry.. 2f t rde + 2 trda'IT y-2" ' -a4 cr t r ay

-aM= 2 i-atr r sin ada-.!. y2

'IT

+ 2 j \ t r r sin a d a-a

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THE EWCT OF AxIAL _W~SION ON MOMENT CARRYING CAPACIZY OFI-296 LINE PIPE SmSED BEYOND TdE ELASTIC LIMIT OTC 13. I=4crytrLcosa.Therefore,

F w. mFTo m c=~FFo

and hence,M ITF10 =Cosa=cos( ).2F o

I

(1 ---t)__________

1. 0

s\o.5z

o. (

Fig. 1

Y

I

60

40x

.:b20

Fig. 30-

\\

\\ \.

~=COS(~F / h ) \.\

\\ \ 4.

0!5F/F. I!0

Y

Fig. 2

J.x___

t 1 1 1 I 1 r 1 I 1 1 , ,.000 I.005 I.010 .015

n=20n=200

v =0.3E=30,000,000psin=20 n=200

a7s55,000 psi 55,000ps55249 54,7556PL=0.0015 0.0018

kA r I =2(3 ed/t=45+ I- I.20 d/t=62 Fig. 4 - Idealized work-hardening0 n.20c) d f=gs stress-strain diagrams.x n=200 dlf=62

I r-- -aI t-B-JFig. 5 - Moment force interaction diagram. Fig.

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