热效应下的屈曲分析
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IN-SERVICE BUCKLING OF HEATED PIPELINES By Roger E. Hobbs 1 (Reviewed by the Pipeline Division) ABSTRACT: Compressive forces may be induced in pipelines by the restraint of axial extensions due to temperature changes or other causes. These forces may cause vertical or lateral buckling of the pipeline. These two buckling modes, which both involve an overall column-type response without gross distortion of the pipeline cross-section, are analyzed on the basis of related work on rail- road track. For normal coefficients of friction, the lateral mode occurs at a lower axial load than the vertical mode and is dominant in pipelines unless the line is trenched or buried. The theoretical solutions are illustrated by numerical re- sults for a typical pipeline and some design implications reviewed. INTRODUCTION The economic importance of submarine pipelines has increased greatly in recent years with the development of offshore oil and gas fields in many parts of the world. The cost in terms of lost production of a failure in such a line is so high that considerable interest has been focused on the stresses caused during the laying (12,16) and modification or repair (5) of submarine pipelines. Similar arguments on a somewhat lower scale of expense apply to pipelines on dry land. However, comparatively little attention has been paid to problems occurring in lines during routine service. This paper addresses one such problem, the assessment of com- pressive axial forces in the pipeline and their consequences. The magnitude of the axial load in the pipe depends on many factors. As well as the mechanical properties of the line and its weight coat, the axial force is a function of the initial tension at the seabed just after lay- ing, the pressure difference across the pipe wall and temperature vari- ations due to hot oil passing through the line. These factors interact with seabed geometry and frictional or trench backfill effects, or both, as well as the influence of end restraints in shorter lines: Akten (1) has treated some effects of partial end restraint at a platform tie-in and Yen et al. (17) have described some analogous phenomena in land based pipelines. Finally, the previous loading history and time dependent changes due to scour, currents and tides are also relevant. Thus it is extremely difficult to say with any certainty what axial force exists at any point in a given pipeline at a given time. Nonetheless, two major causes of compressive forces can be identified, arising from the restraint of the strains associated with thermal and internal pressure loadings. With oil and gas temperatures potentially up to 180° F (100° Q above water temperature and operating pressures over 1,450 lb/in. 2 (10 N/mm 2 ) these effects can produce very significant forces indeed in a long line where the necessary frictional force can develop between pipe 'Sr. Lecturer, Dept. of Civ. Engrg., Imperial College, London, U.K. Note.—Discussion open until August 1, 1984. To extend the closing date one month, a written request must be filed with the ASCE Manager of Technical and Professional Publications. The manuscript for this paper was submitted for re- view and possible publication on January 31, 1983. This paper is part of the Jour- nal of Transportation Engineering, Vol. 110, No. 2, March, 1984. ©ASCE, ISSN 0733-947X/84/0002-0175/$0L00. Paper No. 18691. 175 nloaded 10 Jan 2010 to 118.203.42.237. Redistribution subject to ASCE license or copyright; see http://pubs.asce.org/co
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热效应下的屈曲分析
Transcript of 热效应下的屈曲分析
IN-SERVICEBUCKLINGOFHEATEDPIPELINES ByRogerE.Hobbs1 (Reviewedbyt hePi pel i neDivision) ABSTRACT:Compressiveforcesmaybe inducedin pipelines bytherestraintof axial extensionsdueto temperaturechangesor othercauses.Theseforcesmay causeverticalorlateralbucklingofthepipeline.Thesetwobucklingmodes, whichbothinvolveanoverallcolumn-typeresponsewithoutgrossdistortion ofthepipelinecross-section,are analyzedonthebasisofrelatedworkonrail-roadtrack. For normal coefficientsof friction,the lateral modeoccurs at a lower axialloadthantheverticalmodeandisdominantinpipelinesunlesstheline istrenchedor buried.Thetheoreticalsolutionsareillustratedby numericalre-sultsforatypicalpipelineandsomedesignimplicationsreviewed. INTRODUCTION The economic importanceof submarinepipelines has increasedgreatly inrecentyearswiththedevelopmentofoffshoreoil andgasfieldsin many partsof the world.The cost in terms of lost productionof afailure insucha lineis so highthatconsiderableinteresthas beenfocusedon thestressescausedduringthe laying(12,16)and modificationorrepair (5) of submarinepipelines.Similarargumentson a somewhatlowerscale of expenseapplyto pipelines on dry land.However,comparativelylittle attentionhasbeenpaidtoproblemsoccurringinlinesduringroutine service. This paperaddressesone suchproblem,the assessmentof com-pressiveaxialforcesin the pipelineandtheirconsequences. Themagnitudeof the axial loadin the pipedependson manyfactors. As well as the mechanicalpropertiesof the lineand its weightcoat,the axial forceis a functionof the initialtensionat the seabedjustafterlay-ing,the pressuredifferenceacrossthe pipewallandtemperaturevari-ations due to hot oil passing through the line. These factorsinteractwith seabedgeometryand frictionalor trenchbackfilleffects,or both,as well asthe influenceof end restraintsinshorterlines:Akten(1) hastreated someeffectsofpartialendrestraintataplatformtie-inandYen etal. (17) have describedsome analogous phenomenain land basedpipelines. Finally,the previousloadinghistoryandtimedependentchangesdue toscour,currentsandtidesarealsorelevant. Thus it is extremelydifficultto say withany certaintywhataxialforce exists at any point in a givenpipeline at a giventime.Nonetheless,two majorcausesofcompressiveforcescan beidentified,arisingfromthe restraintofthestrainsassociatedwiththermalandinternalpressure loadings.With oil and gas temperaturespotentiallyup to 180 F (100Q abovewatertemperatureand operatingpressuresover1,450 lb/in.2 (10 N/mm2)theseeffectscanproduceverysignificantforcesindeedina longlinewherethe necessaryfrictionalforcecan developbetweenpipe 'Sr. Lecturer,Dept. of Civ. Engrg., Imperial College, London, U.K. Note.Discussionopenuntil August1, 1984.To extendthe closing date one month, a written request must be filed with the ASCE Manager of Technical and ProfessionalPublications.The manuscriptforthispaperwas submittedfor re-view and possible publication on January 31, 1983. This paper is part of the Jour-nal of TransportationEngineering,Vol. 110, No. 2, March,1984. ASCE, ISSN 0733-947X/84/0002-0175/$0L00.Paper No. 18691. 175 Downloaded 10 Jan 2010 to 118.203.42.237. Redistribution subject to ASCE license or copyright; see http://pubs.asce.org/copyrightandseabed,orinshorterlineswitheffectiveendrestraints.Denoting thecross-sectionalareaofthepipebyA,Young'sModulusbyE,the coefficientoflinearthermalexpansionbyaandthetemperaturechange byT, theforceP0 createdbyfullrestraintofthermalexpansionis,simply, P0 =EAaT(1) Thefreeaxialstrain,e,duetoapositivepressuredifferencepbetween theoilandtheseaisgivenintermsofthewell-knownthinwallaxial andhoopstressesinthepipeby e = i(^-vHT)(2) E \2tt /W in whichv is Poisson'sratio,andt andr arethepipewallthicknessand radiusrespectively.Then,ifeiscompletelyrestrained,theaxialcom-pressiveforcegeneratedandavailabletoparticipateinbucklingis EAe Apr J- ( 0.5- v ) . (3) This paperaddressestwopossibleresponsestothecompressiveforce generated.Both involvesignificantbendingoftheinitiallystraightpipe-line,similartothebendingdeformationoccurringintheelastic(Euler) bucklingofanaxiallyloadedcolumnasitscriticalloadisapproached. As ina longcolumnfabricatedfromacircularholloi^section,thecross sectionofthepiperemainscircular,at least intheinitialstagesofbuck-ling,althoughtheresponsesdiscussedheremaywell leadto pipeover-stressandlocalfailurebyyieldingandovalization.Theinitialabsence ofgrosscross-sectionaldistortionmaybecontrastedwiththesituation inapropagatingbuckletriggeredbyexcessivebendingandexternal pressurizationduring laying operations(13). Thefirstbuckling mode(Fig. 1(a)) addressedhereinvolvespartofthelineliftingitselfverticallyfrom theseabed,whilethesecond(Fig.1(b))resultsin varioussnakinglateral movementsinthehorizontalplaneagainstfrictionalresistance.Bothof thesemodescanbedemonstratedinaverysimplemodelemploying, -77777. El evati on l b)FIG.1.Vertical and LateralBucklingModes 176 Downloaded 10 Jan 2010 to 118.203.42.237. Redistribution subject to ASCE license or copyright; see http://pubs.asce.org/copyrightforexample,rubberstripsrestingona horizontalmachinedsurfaceand loadedbyasmallscrewjack,andbothhavebeenrecordedinpipelines inpractice. ANALYSISVERTICALMODE Thismodeof bucklinghasattractedagreatdealofattentionoverthe years fromrailway engineersconcernedwithananalogousproblem,the bucklingofcontinuouslyweldedrailroadtrack.Kerr(8)haspresented areviewpaperwithnearly50 referencestothisproblem,whileMarek andDaniels(10)havedescribedanapparentlyindependentanalysisof the vertical buckling of crane rails. Their work(as corrected indiscussion byGranstrom(3))agreesinallrespectswiththetheoryfirstdeveloped in1936byMartinet(11).Accordingly,itseemsappropriatemerelyto summarizethemethodandresultsfortheverticalmodeatthispoint. The firststepis tosolvethelineardifferentialequationforthedeflected shapeofthebuckledpartofthepipeline,whichistreatedasabeam column underuniformlateral loadequalto theselfweight.It isassumed thatthebendingmomentatthelift-offpointiszero.Thisassumption hasbeenexaminedelsewhere(4),anditshouldalsobenotedthatthe useofthelinearequationreliesontheusualcolumnbucklingassump-tionofsmallslopes,apointtobeborneinmindlaterwhenassessing theresults.InthenotationofFig.2(a), y"+ n2y+ -(4x2-L2) =0(4) 8 inwhicha primedenotesdifferentiationofthedisplacementywithre-spect tothe longitudinalcoordinatex,theselfweightis w perunitlength, andthesecondmomentofareaofthepipeis I.tn= w/EIandn2=P/ EI, andthebucklelengthisL.Eq.4 hasthesolution mIcos nxn2x2n2L2\ y=ni~~^- v+ir+1(5)\C0ST/ The unknownlengthof buckleL is thendeterminedfromthecondition ^J^^J^^^ttt^^ L ^J .L^ _ men sions I b l Axialforcedistribution FIG.2.--DetailsofVerticalBuckle 177 Downloaded 10 Jan 2010 to 118.203.42.237. Redistribution subject to ASCE license or copyright; see http://pubs.asce.org/copyrightthattheslopeattheendsofthebuckleshouldbezero.Thisyields nLnL... tan = (6) 22 oraslowestroot nL=8.9868(7) ThenextstepistocomparetheaxialloadPinthebucklewiththeaxial loadP wellawayfromthebuckle(Fig.2(b)).PisclearlylessthanP becauseoftheextralengthroundthebucklecomparedtoL itself.The dropin forceis notas largeas wouldbeexpectedat firstglancebecause twoadjacentlengthsofpipeLs slide intowardsthebucklegivingthe axial forcedistributionshownin Fig. 2(b). Thediscontinuitiesinthis dis-tributionateachlift-offpointareassociatedwiththeconcentratedver-ticalreactionsof0.5wLwhichoccurthere. Settingupacompatibilityequation(10),thefollowingresultsare obtained: E7 I5/ P =80.76(8) zvL P0 = P + [1.597 X 10~5 EAfywL5-0.25(cb7)2]1/2(9) in whichis thecoefficientoffrictionbetweenpipeandsubgrade.The maximumamplitudeofthebuckle ,wL4 y =2.408x10- 3-(10) andthemaximumbendingmoment,atx=0,is M= 0.06938 wL2(11) whilethemaximumslopeis wL3 V= 8.657 x10~3(12) "EI Thislastresultisusefulforcheckingthevalidityofthesmallslopeas-sumptioninparticularnumericalcases,i.e.conventionally^s0,1for "small"slopes.Afurtherresultofpracticalinterestisthesizeofthe slippinglengthadjacenttothebuckle, P0-P Ls = 0.5L(13) Thustheminimumtheoreticaldistancebetweenthecentersoftwoad-jacentbutindependentverticalbucklesis L +2LS = ^ ~1^(14) Eq.9isawkwardandmaybecomparedwiththeresultforavery 178 Downloaded 10 Jan 2010 to 118.203.42.237. Redistribution subject to ASCE license or copyright; see http://pubs.asce.org/copyrightlargecoefficientoffriction(i.e.,Ls=0) EI,w2AEL6 P= 80.76 77 +1.597X 1CT5 =~ V(EI) It iseasytoshowthatEq.15 hasaminimumatabucklelength L= 1.6856 x10s(Elf w2AE (15) (16) Eqs.15 and16 areusefulwhensolvingEq.9 forparticularcasesasde-scribedlater. ANALYSISLATERALMODES The lateralmodesobservedinasmall scaleexperiment(e.g.Fig. 1(b)) all resemblesine curves decayingeitherside of a central peakamplitude. Asafirststep,alateralmodeofthesameshapeastheverticalmode wasconsidered,mode1 inFig.3.Withtheverticalanalysistohand,it wassimpletodeterminetherelationshipbetweenbucklelengthLand axialforceP0foralateralcoefficientoffrictionassumedequaltothe frictioncoefficientforaxialmovementsoftheslippinglengthsL$.Un-fortunately,forequilibriumthismoderequiresconcentratedlateralforces ofwL/2attheendsofthebuckleanalogoustotheconcentratedver-tical forcesatthelift-offpointsintheverticalmode.Onarigidseabed concentratedverticalforcesarepossible(andpracticalcasescloselyap-proachtherigidideal(4))butitisnotpossibletogenerateatrulycon-centratedlateralforcebyfriction.Refs.8,9 and11 whichcametohand ratherlaterdealwiththispointatlength,butintheinvestigationde-scribedherethenextstepwastorepeattheanalysisusingadifferent assumptionwhichayoidsthedifficulty. Mode1Mode2 FIG.3.LateralBuckling Modes 179 Downloaded 10 Jan 2010 to 118.203.42.237. Redistribution subject to ASCE license or copyright; see http://pubs.asce.org/copyrightLookingattheexperimentallyobservedlateralmodesoncemore,it wasassumedthatthefamilyofdecayingtrigonometriccurveswereall initial imperfectiongeneratedvariantsofonefundamentalconstantam-plitudeperiodiccurve.Inotherwords,itwasassumedthataninitially perfectpipe wouldbuckle into an indefiniteseries of halfwavesasshown inFig.3,mode.Thisassumptionhasthecomputationaladvantages thatthenodesofthehalfwavepatterndonotslideparalleltotheaxis ofthepipeand,moreimportantly,canbemadetosatisfylateral equilibrium. Lookingattheconsequencesofthisnewassumptionindetail,the lineardifferentialequationgoverningthedeflectedshapeisunchanged, (Eq.4),exceptthatmisnow$w/EI,i.e.,itisassumedthatthelateral frictionalforceisfullymobilizedeverywhere.Thedisplacementbound-aryconditionsareunchangedbutforthezeroslopeconditionatx= L/2whichisreplacedbyashearforceconditionatthesamelocation. ThusEq.5 stillholdswiththenewvalueofm,whileaftercarefulcon-siderationofsignstheshearconditionyieldsthelengthofthebuckle from nL t a n y= 0(17) orasthelowestnon-trivialroot nL=2TT/(18) Compatibilitythenrequiresthat f t/ 2 (19) AEfL/2l2 thatis:thereductioninaxialforceinthebuckleequalstheproductof theaxialstiffnessandtheextensionroundthecurve.Thisleadstothe followingresultsfortheinfinitelateralbucklingmode ,EZ P = ^2J J (2) P0 = P + 4.7050xIQ-5AE(JL6(21) Forcomputationaluse,itisnotedthatthisequationhasaminimumat --[2-7969X105(E7)3]0125 1 ~[(4 >W)2A E J(22) Themaximumamplitudeofthebuckle y = 4.4495x10~3 L4(23) yEIK' andthemaximumbendingmoment,atx=0,is til= 0.05066 = 0.7.Thepipeconsideredhasan outsidediameterof25.6in.(650mm),andawallthicknessof0.59in. (15 mm)givinga cross-sectionalareaof46.38 in.2(299.2 cm2)andasec-ondmomentofarea3,625 in.4(150,900cm4).Itssubmergedweight(in-cludingconcretecoating)hasbeentakenas260lb/ft(3.8kN/ m).The resultshavebeenpresentedintermsofthetemperaturerisenecessary togeneratetheaxialforceinequilibriumwiththebucklelengthsand amplitudesshown,andthecoefficientoflinearthermalexpansion,a, hasbeentakenas11x10~6/Cforthispurpose. Theresultswereobtainedfromasmallcomputerprogramusingthe followingprocedurewhichrecognizesthatitismucheasiertofindthe Temperaturf rise T l*C) 200 240 reZOO 160 120 80 40 " -Pipeline5utmergedweight 650 15 3.8 urn O o \ mmW.T. kN/m 1 i * large////'/ i, sss 1 * 0 4 ,0-3 " -/ CM" -" - 0 0 1 -fi=0 020406080100120140160 Buck l e l engthLI m) 00050102051251020 Buckle amplitude y(m) FIG.4.Results forVerticalBuckling 40606010012014000.2040.6QB Buckle LengthL( m)Buckle Amplitude y( m) FIG.5.ResultsforLateralBuckling Mode 182 Downloaded 10 Jan 2010 to 118.203.42.237. Redistribution subject to ASCE license or copyright; see http://pubs.asce.org/copyright32006080100120140 *0. 20.61.0U1.82.22.6 Buckle lengthI(m)Buckle Amplitude y(m) FIG.6.Comparisonof theLateralBucklingModes force(ortemperaturechange)correspondingtoagivenbucklelength L thanviceversa. 1.Verticalmodehighfrictioncoefficient (a)ComputeL usingEq.16. (b)Forarangeof20valuesofL between0.1L and3L,compute P using Eq.15, T using Eq.1, andthe buckle amplitude yusing Eq.10. 2.Verticalmoderealfrictioncoefficients Forarangeoffrictioncoefficients, and20valuesofLbetween 0.1 L and3 1, computeP0 usingEq.9,T usingEq.1 andthebuckle amplitudeyusingEq.10. 3.Lateralmodesinfinitemode (a)Foreachofarangeoffrictioncoefficients, computeusing Eq.22.j (b)Foreachfrictioncoefficient, andarangeof20valuesofL between/0.5Land1.5L,computeP0 usingEq.21,TusingEq. 1andthebuckleamplitudeusingEq.23. 4.Lateralmodesmodes1-4 Repeatsteps3aandbusingEqs.27 and28 forP0 andy.Toestab-lishtheminimumformode1 valuesofL upto2.0 L mayhaveto beused. NUMERICALEXAMPLE If,asanillustration,thecoefficientoffrictionistakenas0.5, Fig.4 shows thatthe largest"safe"temperaturechangeto avoidverticalbuck-linginthisparticularpipelineisabout61 C.Forlateralbuckling(Fig. 6) the"safe"temperaturechangeisratherlower,about44 C.Then,if (as anexample)a temperaturechange betweenthesetwofiguresofsay, 45 C is expected,theconsequencesof later bucklingshouldbechecked, whileverticalbucklingisunlikelytobeaproblem. FromFig.6,bucklesinmodes2,3and4arepossibilitiesat45 C. FromaninterpolationofthedatausedtoplotFig.6,thehalfwave-lengthsofthebucklesinthesemodesareshowninTable2.Giventhe wavelength,theamplitude,bending momentandaxial force(andhence 183 Downloaded 10 Jan 2010 to 118.203.42.237. Redistribution subject to ASCE license or copyright; see http://pubs.asce.org/copyrightTABLE2.NumericalExampleLateralBuckling Buckling mode (D 2 3 4 Buckle lengthL ft(m) (2) 268.6 (81.86) 260.3 (79.35) 237.4 (72.36) Amplitude y ft(m) 0)4.96 (1.51) 8.17 (2.49) 5.73 (1.75) MomentM 103Ib-ft (kN-m) (4) 1,022 (1,386) 1,266 (1,716) 1,088 (1,475) Axialforce P 103lb(kN) (5) 413.7 (1,840) 379.7 (1,689) 378.1 (1,682) Bendingplus axial stress 103lb/in.2 (N/mm2) (6) 52.1 (359) 61.6 (425) 54.1 (373) %ofyield ifCT,= 65,000lb/in.2 (448N/mm2) (7) 80 95 83 thecorrespondingbending,axial,andtotalstresses)canbecalculated usingEqs.28,29and26respectively.Theresultsofthesecalculations areincludedinTable2,whichcomparesthetotalstresswiththeyield pointofanAPIX65gradeline.Mode3isclearlythemostcriticalat some 95% of yield.It is concludedthatlarge amplitudebucklesarelikely to occur,andthatlarge bendingstresseswill begenerated.Thusitwould be prudentto reduce the design temperaturechange or to preventlateral bucklingbyraisingtheeffectivefrictioncoefficient,perhapsbytrench-ingtheline. ANALYSIS!j ItisapparentfromFigs.4-6thatthelateralmodesbecomepossible atasmallertemperaturechangethantheverticalmodeforrealisticfric-tioncoefficients.Thusunlesslateralrestraintisprovided,bytrenching thelineforexample,thelateralmodeswill bedominant.Thisfeatureis confirmedbysmallscaleexperimentsandpracticalexperience.Anun-buriedlinewillsnakelaterally,whilea buriedlinemayburstoutofthe seabed(ordesert!).Oncea buriedline haslifted,aninteractionwiththe lateralmodemayoccur,asthebuckleitselfnowhasnolateralrestraint ( =0).Alternativelythebucklemayrollortwistlaterally. Figs.4 and5 areotherwiseverysimilaranditisusefultodiscussthe commonfeaturesintermsof idealizedloadortemperatureagainstlength of buckleandamplitudecurves(Fig.7). Thefirstpointisthatintheory anideal,perfectlystraightpipelinewouldnotbuckle.Thepipeisin equilibriumforallvaluesofaxialloadinanundeformedconfiguration and while the curve forthe deformedshape(Eqs. 9,15 or 21)approaches theverticalaxisasymptoticallyathightemperaturesandsmallwave-lengths,itneverquitemeetstheaxis.AsKerr(9)hasremarked,thisis a consequenceof theassumptionof fullymobilizedfrictioneven forvan-ishinglysmalldisplacements,andthesituationmaybecontrastedwith theconventionalpin-endedEulercolumnwherethepost-bucklingequi-libriumpathactuallycrossestheinitialequilibriumpathontheaxis. Supposing,next,somethingmarginallylessthaninitialperfectionin thepipeline,it isclearthatatsometemperatureT theimperfectionwill beenoughtobridgethegapbetweenthetwoequilibriumpaths,bring-ingittotheequilibriumpositionA.Thepartofthesecondaryequilib-rium pathfromAtoB is clearlyunstable,anda dynamicsnapwilloccur 184 Downloaded 10 Jan 2010 to 118.203.42.237. Redistribution subject to ASCE license or copyright; see http://pubs.asce.org/copyrightBucklelengthLBuckleamplitudey FIG.7.EquilibriumPaths and imperfectionSensitivity-Schematic fromAtoC (chaindottedline)ata constantloadP0 inthepipelinewell away fromthe buckle,althoughthe buckle itselflengthensandpartially unloads.Because,onthisargument,nosnapcanoccurbelowthetem-peratureT atB,fhas beencalledthe"safe"temperatureintherailroad literature. Theseideascanbepresentedwithslightlymorerigorbyconsidering thetemperature/bucklelengthcurvesforinitiallyimperfectsystems sketchedinFig.7(dashedlines).Forverysmallimperfectionsalarge snapisindeedseenexperimentally.Astheinitialout-of-straightnessis increasedthesnapoccursatlowerforcesandlessdramatically.Even-tually,forlargeenoughimperfectionsthesnapiseliminated,tobere-placed by a single-valuedmagnificationof the initial bow.Foranyinitial imperfection,the (behaviorisultimatelyasymptotictothestablecurve BC fortheperfectsystem. The behaviorindicatedinFig.7 issupportedby recentworkbyTver-gaardandNeeetleman(15)onthelateralbucklingofinitiallyimperfect railroadtracks.Inafiniteelementanalysisinwhichthebucklewave-lengthisconstrainedtobethesameasthewavelengthoftheinitial imperfection,anda limitedlengthoftrack isconsideredsothatunload-ingintothebuckleisrestricted,curvessimilartoFig.7arepresented forvariousinitialimperfections,andthemostcriticalinitialwavelength identified.Itis,however,questionablewhetherconstrainingthewave-lengthtomatchthatoftheinitialimperfectioniscompletelylegitimate, andKerr(9) hasshownthattherearesignificanttracklengtheffects.In smallscalemodelssubstantialwavelengthchangesdooccurduring buckling,andthewriterfeelsthatmoreworkontheeffectsofinitial imperfectionisneeded.Itisclearthatsuchworkwillbebasedonnu-mericalratherthananalyticalstudies.Whetherfiniteelementorfinite difference(2)studiesareused,thebuckledwavelengthshouldbefree todevelopas it will,andparametricstudiesusedtodeterminetheworst-caseinitialimperfectionpattern.Suchastudycouldusefullybecom-plementedbysmallscaleexperimentalwork,usingforexamplealong oil filled0.5 in.(12.7 mm)copperpipewithaninternalheatingelement. In practical terms, it may even bedesirableto deliberatelycreateinitial 185 Downloaded 10 Jan 2010 to 118.203.42.237. Redistribution subject to ASCE license or copyright; see http://pubs.asce.org/copyrightout-of-straightness,toeliminatethepossibilityofasnap.Thisissimple enoughforlandbasedlines whichoftenincludedog legsandexpansion loops, but it is notso easy forsubmarinelines. Working froma lay barge, the pipelengthsare weldedtogetherwitha1:480straightnesstolerance andsentover thestinger.Deliberatekinkscouldoverstressthe linedur-ing laying.As the weatherworsensinvoluntarylargeradiuscurvesmay beinducedattheseabedbylaybargemotionbutitisnotpracticalto laythelinetospecifiedradii.Norisitworthtrying;Martinet(11)and smallscaleexperimentsbothshowthatalargeradiusisineffectivein eliminatingthesnaporinchangingitswavelength. Onunloadingthepipelinedifferencesappearbetweenthevertical andhorizontalmodes.Averticalbucklewilltracethesecondarypath backdowntoB,and,iftheloadisfurtherreduced,dropbacktothe straightconfiguration,althoughasmallresidualcompressionwillarise at the site of the buckle becauseof frictionaleffects.A chain ofhorizontal buckleswill initiallytrytotracethepathdowntoB, butthechangesin wavelengthcoupledwiththereversalofthelateralfrictionforcewill leaveitatzerotemperaturewithsomeresidualout-of-straightnessand a small tensionin the line. On reloading the pipeline theseimperfections mustpredisposethelinetobucklelaterallyatalowerloadthanbefore bymagnificationoftheimperfections.Intheverticalmode,withsmall residualaxialforcesthelineislikelytobuckleatthesamelocationas beforebutnotnecessarilyatasignificantlyreducedtemperatureasthe straightnessofthepipeisunimpaired. Asmighthavebeenpredicted,itisclearfromFigs.4and5 thatthe valueofthefrictioncoefficientonlybecomesparticularlyrelevantto theresultsinthepost-bucklingregime.However,theseresultsdonot showanyextremesensitivitytothevalueof.Sincethevalueoffor agivenstretchofseabedisonlyknownapproximatelyandisnotnec-essarilyindependentofthemagnitudeofthedisplacementthisisjust aswell. Turningnowtoacomparisonofthevariouslateralmodes,Fig.6 pre-sents resultsforthefivepossibilitiesconsideredherefora singlefriction value,=0.5.Itisclear,asKerr(5) hasshownforrailtracks,thatthe anti-symmetricmodes2and4occuratverysimilartemperature/initial imperfectioncombinations,whicharerathersmallerthanthoseneeded totriggerthesymmetricmode3.However,asfoundinthenumerical examplepresentedearlier,mode3seemstobeassociatedwithrather larger amplitudesand bendingstresses.Whichofthesethree modesde-velopsinaparticularcaseisthoughttobedeterminedbytheinitial imperfectionsina given pipeline,andwithlinesover100 miles(160 km) long now commonplace it is clear that examplesof all three modesmight be foundina given pipeline.Mode1,occurringat highertemperatures, seems unlikelyto beof practicalconcern.Mode is ofacademicinterest becauseit can be involvedin theinitiationoftheothermodes.Referring to Fig. 6, atsmallamplitudesmode occursat lowertemperaturesthan theothermodes,butisclearlyinunstableequilibriumandwillrapidly snapintoalargeamplitudestableformofoneoftheothermodes.This argumentsupportstheintuitiveviewofthelateralmodesbeingimper-fectiongeneratedvariantsofthemode.Indeed,Tvergaardand Needleman(15)arguethatthisprocessisbutoneofmanyexamplesof 186 Downloaded 10 Jan 2010 to 118.203.42.237. Redistribution subject to ASCE license or copyright; see http://pubs.asce.org/copyrightthelocalizationofbucklingphenomena. A cautionary pointshouldbemadeabouttheanalysesof modes3 and 4,inparticular.Itisrecognized(8)thattheycontainanapproximation whichwillleadtoanoverestimateoftheamplitude.Intheanalysesit has beenassumedthattheaxial forceP is constantthroughouttheentire buckle,inspiteoftheinwardmovementfromtheadjacentunbuckled pipeline.Onthecontrary,theaxial forcein the innerhalfwavesofmodes 3and4 mustbesignificantlylessthanintheouter,smalleramplitude, halfwavesbecauseofaxialfrictionandthetrueamplitudeoftheinner waves will be smaller as a result.If thesearguments were to beextended to modes with 5, 6, 7 . . .halfwavesa situation wouldrapidly bereached wherethe inner,mostsignificant,buckles werenot receiving anyenergy atallfromtheinwardslidingoftheunbuckledline,andindeedthe innerwaveswouldapproachtheconditionsassumedintheanalysisof modeoo. This,then,isacogentargumentfornotpursuingthepresent analysisroute(withconstantaxialforce)tomodeshigherthan4half waves,becauseitis unrealistic.Ontheotherhand,a morerealisticcal-culation is most unlikely to produce lower "safe" temperatures thanmodes 2,3 and4 becausemode hasa higher"safe"temperaturethanthese modes.Theanalysesofmodes2,3and4 presentedherecantherefore berecommendedforlowerbounddesignuse. Finally,thefollowinglimitationsoftheanalysespresentedinthispa-pershouldbeborneinmind: 1.Onlyperfectsystemshavebeenrigorouslyexamined.Inthever-ticalmodenoaccounthasbeentakenoftheinitialout-of-straightness, andtheapproachtothelateralmodeneedsfurthernumericalorex-perimentaldevelopmentor both.Theclassical column(Fourier)analysis ofinitiallackofstraightnessifinappropriateherebecausethebuckled lengthchangesprogressivelyastheloadincreases.Itissomecomfort toknowthatthemagnitudeandnatureoftheinitialimperfectionsin pipelinesaslaidarealsounknown,andextremelydifficulttoassess,so thatanimprovedanalysiswouldnotofitselfbeveryhelpfulexceptas alowerbounddesignaid. 2.Perfectelasticity andsmall slopes have beenassumed.Theanalyses presentedassumethatthefullelasticmodulusofthepipeisavailable toresistbending.Onceplasticityoccurstheanalysislosesitsvalidity: yieldingwillobviouslyleaveitsmarkonthepipeonunloadingandact asanimperfectiononsubsequentreloading.Ofcourse,mostlinepipe is competenttodevelopa plastichingewithouteffectonitsserviceability. The analysesalso usethelineardifferentialequationforbendingwhich is strictlytrueonlyforsmallslopes,conventionallylessthan0.1radian. However,moreexactellipticintegralsolutionshavebeenobtainedfor cantilevers(14) andpin-endedcolumns(6). Inthe cantilever,foranaxial load1.5%greaterthanthatpredictedbythelinearequation,theend slopeis0.35radians.Similarly,inthepin-endedcolumn,foranaxial load1.2%greaterthantheEulerload,thegreatestslope is0.31 radians. These resultssuggestthatthe 0.1 radianlimiton"small"slopesisrather conservative;acut-offlinecorrespondingtothislimithasbeenmarked onFig.4asaguide,andtrueloadsbeyondthislimitmayberather higherthanshown.The0.1radianlimitisnotexceededinFig.5. 187 Downloaded 10 Jan 2010 to 118.203.42.237. Redistribution subject to ASCE license or copyright; see http://pubs.asce.org/copyrightCONCLUSIONS Twopotentialbucklingmechanismsinpipelinessubjectedtoaxial compressionhavebeenidentifiedandanalyzed.Itisfoundthathori-zontalsnakingmodesoccurat a loweraxial loadthanthe verticalmode, andahorizontalmodeisthereforedominantunlesslateralrestraintis providedby trenching whenaninteractive bucklingmodebecomespos-sible.Fullaccounthasbeentakenoffrictionbetweenthepipeandthe groundintheverticalmode,relyingontwoindependentbutformally identicalreferencesonthebucklingofrailtracksandcranerails.An analysisofalateralmodeinvolvinganinfinitesequenceofhalfwaves withfullydevelopedlateralfrictionhasbeenpresentedandcompared with earlier workon isolated buckles over a shorter length whichinvolve longitudinalaswellaslateralsliding. Theeffectsofimperfections,unloadingandreloadingarediscussed anditisconcludedthatfurthernumericalworkontheeffectsofinitial imperfectionswould be valuable.The theoreticalsolutionsareillustrated bynumericalresultsforatypicalpipeline. ACKNOWLEDGMENT TheauthorisindebtedtoDr.TaylorandMr.G^n(SheffieldPoly-technic)forpointingoutsignificantcorrectionstothenumericalvalues inEqs.21 and22.,f APPENDIXI.REFERENCES 1.Akten,H.T.,"NewDevelopmentsinSubmarine Pipeline In-Place Stability Analysis," Proceedingsof Arctic/Offshore/DeepseaSystemsSymposium,Ameri-can Society of Mechanical Engineers, NewOrleans, 1982. 2.Cassell, A. C,and Hobbs, R. E., "Numerical Stability of Dynamic Relaxation Analysis of Non-linearStructures," International Journal of Numerical Methods in Engineering,Vol.10, 1976, pp.1407-1410. 3.Granstrom, A., discussion of "Behavior of Continuous Crane Rails" by P. S. Marek and J. H. Daniels, Journal of the Structural Division, ASCE, Vol. 98, No. ST1, Jan.,1972, pp.360-361. 4.Hobbs,R.E.,"TheEffectofSoil ModulusonPipelineStresses," Journalof the Transportation Engineering Division,ASCE, Vol. 106, No. TE6, Nov., 1980, pp. 775-786. 5.Hobbs, R.E.,"SolutionsforPipelineTie-inandRepair Problems,"Proceed-ingsSecondSymposiumonOffshore Mechanics and Arctic Engineering,American Society of Mechanical Engineers, Houston,Tex.,1983, pp. 538-552. 6.Home, M. R.,and Merchant, W.,TheStability of Frames,Pergamon,Oxford, 1965, pp. 10-12. 7.Kerr, A.D.,"OntheStability oftheRailroadTrackintheVertical Plane," RailInternational,Vol. 5, Feb.,1974, pp. 132-142. 8.Kerr, A. D., "Analysis of Thermal Track Buckling in the Lateral Plane," Acta Mechanica,Vol. 30, 1978, pp. 17-50. 9.Kerr, A. D., "On Thermal Buckling of Straight Railroad Tracks and theEffect of Track Length on the Track Response," Rail International, Vol. 9, Sept. 1979, pp. 759-768. 10.Marek, P. J., and Daniels, J. H., "Behavior of Continuous Crane Rails," Jour-nalof theStructuralDivision,ASCE, Vol.97, No.ST4, Apr.,1971, pp. 1081-1095. 11.Martinet, A., "Flambement des Voies sans Joints sur Ballast et Rails de Grand 188 Downloaded 10 Jan 2010 to 118.203.42.237. Redistribution subject to ASCE license or copyright; see http://pubs.asce.org/copyrightLongueur"("BucklingofTrackswithoutJointsonBallastandVeryLong Rails;" in French), Revue Generate des Chemins de Fer,Vol. 55/2,1936, pp212-230. 12.Palmer,A.C,Hutchinson,G.,andElls, J. W.,"ConfigurationofSubmarine PipelinesduringLayingOperations,"Transactionsof ASME,Journal of Engi-neering for Industry,Vol.96,1974,pp.1112-1118. 13.Palmer,A.C,andMartin,J.H.,"BucklePropagationinSubmarinePipe-lines,"Nature,Vol.254,Mar.,1975, pp.46-48. 14.Timoshenko, S. P., andGere, J. M.,Theory of Elastic Stability,2nded.,McGraw-Hill,Kogakusha,NewYork,1961, pp.76-81. 15.Tvergaard,V., andNeedleman,A.,"OnLocalizedThermal TrackBuckling," International journal of MechanicalSciences,Vol. 23, No.10,1981, pp.577-587. 16.Wilhoit,J.C.,Jr.,andMerwin,J.E.,"PipeStressesInducedinLayingOff-shore Pipelines," Transactions of ASME,Journal of Engineering for Industry,Vol. 89,1967, pp.37-43. 17.Yen,B. CTsao,C.H.,andHinkle,R.D.,"Soil-PipeInteractionofHeated Oil Pipelines," Journalof the PipelineDivision, ASCE, Vol.107, No.TE1, Jan., 1981,pp.1-14. APPENDIXII.NOTATION The followingsymbolsareusedinthispaper: A E I K1K2K3K4K5 L Ls M m n P Po V r T t w X y a e V O-y Superscripts = = = = = = = = = = = 7 U = = = = = = = = = = cross-sectionalarea; Young'smodulus; secondmomentofareaofcrosssection; constantsseeTable1; lengthofbuckle; lengthofslippingpipeadjacenttobuckle; bendingmoment; w/EI; (P/EI)1'2; iaxialforceinbuckledpipe; prebucklingaxialforce; pressure; radius; temperatureincrement; wallthicknessofpipe; submerged weight of pipeline (including weightcoat) perunitlength; coordintatealongpipeaxis; coordinateperpendiculartopipeaxis; coefficientoflinearthermalexpansion; strain; coefficientoffriction; Poisson'sratio;and yieldstress. peak,maximumvalue; firstderivative,slope; secondderivative,curvature;and valueassociatedwiththe"safe"temperature increment. 189 Downloaded 10 Jan 2010 to 118.203.42.237. Redistribution subject to ASCE license or copyright; see http://pubs.asce.org/copyright
