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tructur lTechnology andMaterials roupThe Structural Technology and M aterials G roup STM G) C om mittee comprises expertsrepresenting companies and organizations such as : British Steel plc, Rover Group Limited,University o f Nottingham, U niversity of Strathclyde, N AFEM S Limited, University of W ales,Ford Motor Co Limited, Swansea University and, the EPSRC.The STMG Committee serves the membership by organizing relevant seminars andconferences, as well as representing the UK on national and international committees andorganizations.The Terms of Reference of the Structural Technology and Materials Group are: to promote the use of improved methods of designing and assessing the strength ofcomponents and of predicting their life in order to achieve minimum cost without

com promising integrity; to provide designers with information on established materials such as steels, aluminiumalloys, and fibre-reinforced plastics, and on newer materials such as metal matrixcomp osites and ceram ics; to encourage theoretical andexperimental studieson themechanics of materials formingprocesses such as rolling, pressing, and extrusion, and the effect that these have onsubsequent performance of the component; to encourage the development of tools for the estimation of stresses strains, anddeformations in structures. Including finite element and boundary element methods,simplifiedmethods, and experimental methods; to develop computing technology in so far as it isrelevant to materials andm echanics ofsolids; to investigate the criteria covering the failure of components and life cycle analysis, e.g.excessive deformation, fatigue, fracture, creep rupture, combined creep and fatigue,environmental degradation,andstress corrosion; to ascertain th eprop erties ofmaterials n eeded fo rengineering design, includingth e effectof manufacturing, forming,andjoiningprocesses onthose properties; to promote new ideas and publicize new information in a form which practisingmechan ical engineerscanuse.More informationon theworko f thegroupcan beobtainedbywriting to:

Structural Technology and Materials GroupInstitutionofMechanical Engineers1 Birdcage WalkLondonSW1H 9JJ

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Reference Stress Methods nalysing Safety and DesignEditedby

IanGoodall

rofessionalngineeringublishing

Published by Professional Engineering PublishingBuryStEdmundsand London UK.

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First Published 2 3

This publ icat ion is copyr ight under the Berne Convent ionand the I n t erna t ional Copyright Conv ent ion .A ll r ights reserved. Ap ar t f rom any fai r deal ing for the purpose of pr ivate study research cr i t ic ism orreview as permit ted under the Copyr ight Designs and Patents Act 1988 no par t may be r eproducedstored in a retr ieval system or t r ansmi t t ed in any form or by any means elect ronic elect r icalchem ical mec hanical pho toco pyin g recording or otherwise wi tho ut the pr ior permission of thecopyr ight owners. nlicensed multiple copying of the contents of thispublication is illegal Inquir iesshould be addressed to: The Pu bl ish ing Edi tor Professional Engineer ing Publ is hing LimitedNor thgate Avenue Bury St Edmunds Suffolk IP32 6BW UK. Fax: +44 0) 1284 705271.

2003 The Institution of M ech anic al Engineers unle ss otherwise stated.

ISBN1 86058 362 8

A CIP catalogue record fo r thisbook is available from theBritish Library.Printed by TheCromwellPress, Trowbridge, Wiltshire,UK

Th e Publishers are not responsible for any statement made in this publication. Data discussion an d conclusionsdeveloped by authors are for information only and are not intended for usewithout independent substantiatinginvestigation on thepart of potential users. Opinions expressed are thoseof the Authorsand are notnecessarilythoseof the Institutionof Mechanical Engineersor its Publishers.

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Related Titles of nterestTitleCreep o f Materials and StructuresDesigning High-performance Stiffened StructuresEngineering M easurements MethodsandIntrinsicE rro rsEvaluating Materials Pro perties by DynamicTestingFlaw Assessme nt in Pressure Equipm ent andWelded StructuresH a ndb ook o f Mechanical Works InspectionIMechE Engineers Data Boo k Second EditionPractical Guide to Engineering Fa ilureInvestigationRecent Advances in Welding SimulationStress Analysis o f Cracks H a n d b o o k

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ont nts boutthe Editor xIntroduction xDeterminingth basic parametersChapter1 Reference stress requirementsfor structural assessmentR A AinsworthChapter 2 Com putational methods for limit states and shakedownA R S PonterandMJEngelhartChapter 3 Limit loads for cracked piping componentsDGMoffat 33Extendingth approachtoweldmentsChapter4 Some aspects of theapplicationof the reference stress methodinthecreep a nalysisofweldsTHHyde and WSun 57Chapter5 High temperature creep ruptureof lowalloy ferritic steelbutt welded pipes subjectedto combined internal pressureandend loadingsF Vakili-Tahami D R H ayhurst and M T Wong 75 pplicationsChapter6 Codeapplication belowthecreeprangeA R Dowling 3Chapter7 Code application withinthecreep range

G Webster 127Chapter8 Fracture assessment ofreeled pipelinesCArbuthnotandT Hodgson 45Chapter9 The use ofreferencestressesinbuckling calculationsT Hodgson 55Index 7

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bouttheEditorDr Ian Goodall is a mem ber of the Institution s Structural Technology and M aterials G roupwhich promoted the Seminar in November 2000 that forms the basis of this volume. He isalsoa Fellow of the Institution.AfterUniversity hisexperiencewasprincipallyin the nuclear industrywhere hespent over30 years working on structural integrity issues and deve loping strategic research programm eson other engineering matters. He was responsible, with his colleagues, for bringing togetherthe knowledge base required to produce a document which is now called R5 and entitled AnAssessmentProcedurefor the High Temperature Response of Structures This procedure usessimplified methods of assessment wherever they are justified. It was required for ap plicationto components in both the fast reactor and the advanced gas-cooled reactor where the effectsof creep, fatigue, and fracture are impo rtant. It is now used throughout the nuclear industry fo rcomponents operating at elevated temperature. Since leaving the industry he has beenworking as a Consultant in the structural integrity field working on creep, fracture, and fatigueissues in collaboration with various universities.

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ntrodu tionReference stress methods and other simplified m ethods of assessment offer many attractionsin the design process and they continue to have considerable value in spite of the rapidimprovements in finite element analysis. A significant feature of simplified methods is thatthey enable rationalization of informa tion both from analysis and from experiment. They areoften more intuitive than methods based solely on finite element approaches an d allow thesensitivity to input parameters to be assessed rapidly. Consequently they enable design andrisk assessments of structural components to be performed very efficiently. The basicprinciples are set down in Chapters 1, 2, and 3, which define the underlying theory and giveadvice on the determ ination of the required param eters.To put the book into context it is worth setting down its main objectives. At its most general,the principal objective is to collate expert opinion on this topic; this was done by invitingnational experts to present papers at a seminar and subsequently to prepare chapters fo r thisvolume.At a detailed technical level there are two objectives which are identified in Chapters 1 and 2an d may be summ arized as follows. Firstly to simplify theanalysis process, w herever possible, bybasing structural assessment

on the following: elastic solutions - with andw ithout defects; plasticity or limit load solutions - with andwithout defec ts; shakedown solutions for cyclic loading. Secondly to reduce the influence of detailed variations in material properties by suitablenormalization.

It transpires that a particularly useful quantity in assessing structures, both with and withoutdefects, is a reference stress which is based on the limit load. This quantity appearsfrequently in this volume and is defined by the relationship

where a re f is the reference stress, F is the applied load, an d F L is the rigid-plastic limit loadfor the structure with a yield stress ery Chapter 3 gives details of how the limit load may bedetermined fo r complex structures such as cracked piping com ponents.in the world of high-speed computing power, most stress analyses can be performed using

finite element techn iques for both linear an d non-linear analysis. The re is aneed, however, forthe provision of underlyin g theory that enables the analyst to validate his num erical analysisan d also to interpret experimental findings. This is a two-way process as detailed results offinite element analysis may be also used to refine estimates of a reference stress. A goodexample o f this is given in Chapters 4 and 5 where the ambition is to extend these conc epts tothe treatment of the complex situation that exists in weldments.

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The real test of any of these approaches is whether they are used by the design engineereither directly or in developing design codes. In fact the application of such approaches iswidespread and Chapters 6 7 8 and 9 in this volume address the application of thetechniquesto: code developm ents both below ndwithinthecreep range; pipelines; buckling.Finally I wou ld like to thank all the authors for their patience with my comm ents and theirefforts in producing this volume on simplified methods. It is in my view a verycomprehensive introd uction to the topic which will be of value to both design engineers andacademics alike.

Ia n W oodallovem er 2 2

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Reference Stress Requirements ForStructural ssessmentR insworth

bstr ctThe reference stress method is a powerful approximate method fo r describing the inelasticresponse of structures. The method has been developed to enable simplified assessmentprocedures to be produced fo r both defect-free and de fective components. In this Chapter, th ebackground to the reference stress method is briefly described and the accuracy andlimitations of the method are discussed. Then specific uses of the technique and theirincorporation into structural assessment methodologies to guard against component failure bya number of mechanisms are described.

Notation

C * steady-state creep characterizing param eterC( t) transient creep characterizing para m eterE Young s modulusE E inplane stress; E /(I - v2) inplane strainF loadF normalizing value of FFL limit load value of FG elastic strain energy release rateJ characterizing param eter fo r elastic plastic fractureK elastic stress intensity factorK p value of K for primary loadsK s value of K for secondary loads normalizinglengthn creep stress exp one nt

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S t time-depen dent strengtht t imetC D time for failure fo rcont inuum damaget r rupture timeu el elastic displacem entlif steady-state creep displacement rateV factordescribingtheeffect ofsecondarystressE r e f strain atreference stressec creep strain ratee s steady-state creep strain ratev Poisson sratiocr re t referen ce stress< j ^ f reference stressusedtoestimateJcj^ f referen ce stress used to estimate creep ruptu rea y yieldo r 0.2 percent proof stressC TU ultima te stressC T flow stressC Tcl,max ma ximu m valu e of equivalent stress ca lculated elasticallya ss max max imum value of equivalent stress in steady-state creep stress concentration factor

1 1 BackgroundThe essence of the reference stress technique is that the inelastic behaviour of a componentunder a given loading is related to inelastic materials data at a reference stress defined for thegiven loa ding. The method and its backgrou nd have been described in the book by Penny andMarriott 1) in the context of creep problems. Consider, for example, a component operatingunder steady load, F, in the creep range for which an estimate of the steady-state creepdisplacementrate, ii*s, isrequired.Thereference stress estimateis

where 8^(0^) is the secondary creep rate o f the material at a reference stress level,trr c f , andl has dimensions of length.Clearly values for and a ref are required to use the estimate of equation 1.1). If creepanalysisof the com ponent and material o f interest w ere required to d erive these values, thenthere would not be a major advantage in the technique. However, numerical analyses andexperimental data suggest that the reference stress can often be estimated with sufficientaccuracy from aknow ledgeof thevalueof the loadFco rrespondingtoplastic collapse 2).

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Then

where F L is the plastic collapse load defined for a rigid plastic material with yield strengthcr y . Since F L O y ) is directly proportional to 0 y, the reference stress of equation (1.2) isindependent of c r y ; it is proportional to F and depends on geometry through th e term[FL oy)/ay].There then remains a requirem ent to estimate l in equation (1.1). Suppose an inelasticsolution is available for one material (A). Then

ensures that equation (1.1) is exact for m aterial A and allows estimates to be made for othermaterials. If results of inelastic analysis are not available, then it is common to make use ofelastic solutions. The estimate

where E is Young s modulus an d iiei is the elastic displacement under load F ensures, forexample, that equation (1.1) is accurate for a power-law creeping material in which th e stressexponent is unity. Clearly, however, the accuracy will be reduced for non-linear materials andthis is discussed in Section 1.2 below.Further examples of reference stress estimates are described in Sections 1.2 and 1.3, below. Inessence,the approacheshavethefollowing properties: direct use of inelastic materials data in any convenient form w ithout the need for such datato b e described by specific equations such a s power-law creep or plasticity; use of limit load solutions (o r shakedown solutions for cyclic loading) which are widely

available without the need for inelastic analysis; canmake use of theresults ofelastic stress analysis; can be refined to improve accuracy using detailed analysis results where available, orsimplified to provide order-of-magn itude estimates w here rapid results are needed; robust fo r engineering use as they are not sensitive to detailed descriptions of con stitutiveequationso r highly refined analysis.

1 2 Accuracy and limitationsThedevelopment of designm ethods based on reference stresstechniques has beendiscussedby Goodall et al (3) in the context of creep design of non-defective structures. An importantaspect in this is the accuracy of the techniques and any l im itations. Energy theorems may beused to dem onstrate that, quite generally, the reference stress of eq uation (1.2) overestimates,on average, th e creep energy dissipation rate within a structure (4). However, this does notensure that any particular estimate of displacemen t o r strain within a com ponent will be safely

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4 ReferenceStress Methods Analysing afetyandDesign

estimated using approximations such as equations ( 1 . 1 1 . 4 ) . Indeed, at stress concentrationsthe local stress levels would be expected to be higher than the reference stress of equation(1.2) wh ich essentially describes average component behaviour. How ever, the energytheorems do give confidence that reliable estimates will be obtained fo r displacementsconjugateto the applied loads.In this section, some specific examples are used to examine the accuracy and limitations ofreference stress techniques. Consider, first, creep rupture of structures subjected to steady load,for which the m ultiaxial creep rupture criterion is similar to the yield criterion used to define thelimit load in equation (1.1). In this case, it is possible to show (5) that estimating the time for astructure to fail by the spread of creep rupture damage, ICD, is less than the time-to-ruptureobtained from uniax ial stress rupture data at the reference stress of equation (1 .2 ), i.e.

Numericalan d experimental data show that the difference between ten and t r < jr e f ) depends onthe magnitude of the stress concentration factor in the component. D efining a stressconcentration factor,x, as

where (7elm x is the maxim um value of the elastically ca lculated equivalent stress in thecomponent, then th e peak stress in steady-state creep, a ss max is approximately

wheren is the creep stress exponent in a power creep law (6) .For creep brittle materials, overall creep rupture of a component may be assumed to occurwhenlocal rupture at the stress concentration occurs, i.e.

However, for creep ductilematerials there can be a significant time taken fo r damage tospread throughout a component after this local damage initiation (3). A pragmatic approachbased on numerical and experimental data is to define a rupture reference stress, < j* f ,intermediate between aref and ssm ax This takesthevalue

For n > 7, this estimate is over conservative an d equation (1 .7) is used even for ductilematerials withhigh n . The rupture reference stress a* f , which m ay take either th e valuedefined by equation (1.7) or equation (1.9) depending on n and whether or not the material iscreep ductile, leads to improved accuracy compared with the simple use of the limit loadreference stress. This is i llustrated in Fig. 1 .1 .

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As asecond example, consider elastic-plastic fracture where an estimate is required for theparameter J which describes the stress and strain fields at the crack tip. In this case, areferencestress estimateof J hasbeen developedas (7,8

where

is the elastic value of J which is related to the stress intensity factorK and E = E inplanestress and E/(l-v 2) where v isPoisson s ratio in plane strain.The strain e re fis the total(elastic +plastic) strainat the reference stress level .Equation 1.10)isclearly accuratein the elastic regime where sref =o r e f /E.It hasalso beenshown to be accurate in the fully plastic regime by comparison with numerical solutions forfully plastic materials 7). However, it loses accuracy in the small-scale yielding regimewhere J exceeds G but the reference stress may be less than th e limit of proportionality. Aconvenientcorrection to improve accuracyin this regimeis

Thisprovides a correction at small loads which is phasedout as the fully plastic term [the firstterm on the right-hand side of equation 1.12)] become large. This is a convenientapproximate estimateof J in the elastic, small-scale yielding,and fully plastic regimes.The estimate of equation 1.12) requires only a knowledge of the stress intensity factor, todefine G, and the limit load, to define o ref. Thesehavebeen collected in compendia for alarge number of defective engineering components. This approachh as been incorporated intothedevelopment ofpractical flaw assessment procedures suchas R6where theJ-estimateisconverted into a failure assessment diagram (9) of the type shown in Fig. 1.2.If inelastic analysis of the defective component is available, th e accuracy of equation 1.12)may beimprovedby modifying thereference stress definitionas follows. WhenCT rer= ay,equation (1.12)shows that

where

when ay is defined as the 0.2 percent proof stress. Fromthe inelastic analysis,the load, Fsay, at which J/G equals th e value given by equation 1.13) can be identified and then amodified r eference stress is givenby

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Use of this reference stress ensures that equation (1.12) is exact at F = F , is exact at lowloads where behaviour is elastic, and has the correct dependence on material response underfully plastic conditions. This leads to improved accuracy compared to use of the limit loadreference stress an d often to reduced conservatism in assessments since F > F L in manycases.

1 3 Specific usesIn this section, some specific uses of reference stress techniques are listed in terms of theirincorporation into structural assessment methodologies to guard against component failure bya number of mechanisms.1 3 1 Plastic collapseIn design codes, plastic collapse may be avoided by satisfying limits on so-called stressintensities or stress res ultants. A m ore accurate approach is to use a plastic collapse solution ifavailable. This is equivalent from equation (1.2) to the limit

where, additionally, some design margin may be imposed. This approach is used in R5 (10)andhas the advantage that the reference stress can be mo dified subsequently to address creeprupture, deformation limits, an d creep fracture. R5 contains procedures fo r assessing theintegrity of components operating at high temperatures an d addresses the following failuremodes. Excessiveplasticdeformation due tosingleapplicationof aloadingsystem Increme ntal collapsedue to aloading seque nce. Excessive creep deform ation orcreep rup ture. Initiation of cracks in initially undefected material by creep and creep-fatigue m echanisms. Thegrowthofflaws bycreepandcreep-fatigue mechanismsReference stress technique s are used to provide simplified assessment m ethodologies to guardagainst al l these failure modes and some of thesetechniques are described in Sections 1 3 2 1 3 5For low tempe rature fracture assessments, a plastic collapse limit is included in R6 (9) whichcorresponds to

where a is a flow stress to allow for strain hardening bey ond yield and oftena =X ( C T y + < 3 u )whereCTUs theultimate tensile stress R6 addresses fracture of componentscontainingdefects by both b rittle and ductile mechanisms.

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It may be noted that th e reference stress approach may be extended to cyclic loadings usingshakedown concepts rather than plastic collapse concepts. This is addressed elsewhere (11)and is notdiscussed further here.1.3.2 Creep ruptureAsalreadydiscussedin Section 1.2,componentcreeprupturemay be avoided bylimitingtherupture reference stress of equation (1.7) or equation (1.9) to the time-dependent strengthbasedonun iaxial creep rupture data. Thisisincorporatedin R5 for defect-free components.Fordefective components or components with sharp stress concentration features,th e stressconcentration factorx ishighan d equations (1.7) and(1.9)areover-conservative. Inthis case,it is acceptable to base creep rupture on the limit load reference stress of equation (1.2)provided a separate assessment is made of the potential fo r crack extension by creepmechanisms(see below).Thisis the approachadoptedin R5 andalsoin theBritish StandardsdocumentB S 7910 (12).1.3.3 Creep deformationAverage strains in a component during it s service life, t, may be limited by imposing th erestriction

where S t(m ,t) is the stress from isochronous stress-straindata at the tem perature of interestfor th e service life t, for the strain level of m which istypicallyone per cent. Peak strainsmay be similarly limitedby imposing a restriction on the steady-state creep stress of equation(1.7). Typically, peak strains are limited to five pe r cent. In R5 such limits are used.Additionally, creep strains at a reference stress level defined from a shakedown analysis areused to calculated creep damagevia a ductility exhaustion approach. The acceptable levelofcreepdamage dependson the level ofassociatedfatiguedamageand themarginsrequired inth e assessment.1.3.4 Low temperature fractureThereference stress J-estimate of equation (1.12) has been converted into afailure assessmentdiagram approach and is extensively used for defect assessment through R6 (9) and othermethods world-wide 12,13).Although equation (1.2) iswritten for asingle primary load, th ere ference stress m ethod ca nhandle m ultiple primary loadings by suitable definition of FI . It h as also been extended tocombined primary and secondary loadings. Recently (14), it has been shown that equation(1.12) ma y st ill be used prov ided the d efinition of G in equation (1.11) is replaced by

where Kp, Ksare thevalueof K for the primary and secondarystresses,respectively, and V isa factor. Reference stress methods have been used to evaluate th e factor V. For elasticbehaviour, V = 1; at low loads plasticity tendsto lead to values V > 1wh ereas athigh loads(a re f > ay) plastic relax atio n of secondary stresses leads to V < 1 (14).

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1 3 5 Creep fractureIn R5 4) creep crack ini t iat ion an dgrowth are assessed usingthecreep equivalent,C*, of thelow-temperaturefracture parameterJ. A short-term parameter C t)i salso used fo rtimes priorto steady-state conditions. By analogy wit h equation 1.10),C*m ay be est imated from

Modifications similar to those in equation 1.12) fo r small-scale yielding are used to assesssmall-scale creep using C t) 8). Then C* or C t) are used to assess crack ini t iat ion an dgrowth(8,10).

1 4 Closing remarksThis Chapter has briefly described th e background to the reference stress techniques. Theaccuracy of the method and how this may be improved in specificcaseshas been discussed.Finally, thepowerof thetechnique hasbeen illustrated bysumm arizing anumber ofpracticalcases where reference stress methods have been introduced into codes and stand ards.

AcknowledgementThisC hapter is published with perm ission of British Energy Gen eration.

References(1) Penny,R. K. andMarriott,D. L. DesignforCreep, Second edition, C hapman Hall,London 1995). 2) Sim,R. G. Reference stress conceptsin theanalysisof structures during creep, Int.J.Mech.Sci.12,561-573 1970)(3) Goodall, I. W., Leckie, F. A., Ponter, A. R. S., and Townley, C. H. A. Thedevelopment of high temperature design methods based on references stresses andboundingtheorems, ASM EJ.Engng M ater. Techno l. 101,349-355 1979). 4) Leckie,F. A. andMartin, J. B. Deformation bounds forbodies in a state of creep,

ASMEJ.Appl.Mech. 34,411-417(1967).(5) Goodall,I. W. andC ockroft, R. D. H. Onboundingthe lifeof structures subjectedtosteady load and operating within the creep range, Int. J. Mech. Sci. 15,251-263 1973). 6) Calladine,C. R. Arapid methodfo restimatingth e greatest stress in a structure subjectto creep, Proc. IMechE Vol. 178, Part 3L,198-206 1964). 7) Ainsworth,R. A. The assessment ofdefectsin structureso f strain h arden ing materials,Engng Fract. Mech.19,233-247 1984). 8) Webster,G. A. andAinsworth,R. A.High Temperature Com ponentLifeA ssessment,Chapman Hall,London(1994). 9) R6, Assessment of the Integrityof Structures Containing Defects, Revision 4, BritishEnergy Generation 2001).10) R5, Assessment Procedure for the High Temperature Response of Structures, Issue 2,British EnergyGeneration 1999).

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11) Ponter, A. R. S. Computational methods for limit states and shakedown, ReferenceStress Methods - Analysing Safety and Design, Professional Engineering PublishingLimited 2002). 12) BS 7910: 1999, Guid e on meth ods for assessing of the acceptability of flaws in metallicstructures, BSi, London 2000).13) Zerbst, U., Ainswo rth R. A., and Schw albe, K.-H. Basic principles of analytical flawassessment methods, Int.J.Pres. Ves. Piping71,855-867 2000). 14) Ainsworth, R. A., Sharples, J. K., andSmith, S. D. Effects of residual stresses onfracture behaviour - experimentalresultsandassessment methods, J. Strain Analysis

35,307-316 2000).

Fig. 1.1 Estimate of creep rupture time

Fig. 1.2 The R6 failure assessment diagram

R insworthBritish Energy Generation Limited Barnwood UK

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omputational Methodsfor LimitStates nd Shakedown R SPonterand M JEngelhart

bstractThis Chapter discusses a com putation al technique the linear match ing method for the directevaluation of parameters that determine strength characteristics of a structure subjected tocomplex histories of loading. Here we discuss limit loads shakedown limits and an extendedshakedown limit associated with creep rupture. he method consists of the solution of asequence of linear problems for a constant residual stress field. The solutions provide amonotonically reducing limit load or shakedown limit upper bound. On convergence themethod provides the least upper bound associated with the class of displacement rate fieldsdefined by a finite element mesh. The method may be implemented within a conventionalfinite element code and the solutions discussed here were all generated using the commercialcode ABAQUS. The efficiency and accuracy of the method is illustrated through a sequenceofsolutionsof typicallife assessment problems.

otation vonMises effective stresss vonMises effective strain ratey yield stress? y inelastic strain ratesAf? A J incrementso finelastic strain over aloading cycleA scalar loading param eterP i applied load temperaturea - linear elastic stress history

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p ij time-co nstant residual stress field /7 shear m odulidefined in thelinear ma tching method

2 1 IntroductionLife assessment methods anddesign codes were originally developed with theunderstandingthat, at most, only linear elastic solutions were available. These days com plete sim ulation ofcomponent performanceis possible although there remainstheproblem of theavailabilityofsufficient material data forcon stitutive equationsand the fact that full analysis isbest doneatthe later stages of a design process. Between these tw o extremes there exist a number ofanalysis methodsof asimplified nature that provide sufficient information for design or lifeassessment decisions, based upon less demanding calculations. Among such methods, thosebased on limit analysis and shakedown analysis provide relevant examples. Suchcomputationalmethods havethe attractive featureo f inve rting analysis, in the sense that theyprovide load ranges forwhich certain typesof structural performance occurs, although basedupon simple material models.This Cha pter discusses the linear matching m ethod, a recent advance in com putation almethods for shakedownand related problems based upon aparticularlyuseful methodology.The procedure originates from the elastic com pensation and related metho ds 1, 2) where asequence of linear problem s is solved w ith spatially varying linear modu li. In refe renc e 3) itwas demonstrated thatthe method may be interpreted as a non linear programming methodwhere the local gradient of an upper bound functionaland the potential energy of the linearproblem arem atcheda t acurrent strain rate orduring astrain rate history. This interpretationmay be used to formulate a very general method for evaluating minimum upper boundsolutions. Provided certain convexity conditions are satisfied, it is possible to define asequence of linear problems where the upper bound monotonically reduces. The sequencethen converges to thesolution that corresponds to the absolute minimum of the upper boundfunctional, subject to constraints imposed by the class of strain rate histories underconsideration. Thetheoretical bases for the method andconvergence proofsare discussed inreferences 4, 5). A full discussion of the current range of application of such methods issummarized in reference 6). As a result of these theoretical considerations it is possible togenerate limit load and shakedown limits that are the absolute minimum of allupper boundvalues given by the kinematicso f a finite element mesh. Such values are the most accuratethat may be obtained within the formal structure of the stiffness finite element method. Forthe solutions described in this Chap ter the general code AB AQ US was used .The Chapter consists of three main parts. Section 2.2 contains a summary of the method,based upon the theoretical structure of 4, 5) but specialized to a von Mises yield condition.Section 2.3 isconcerned with the implementation of themethod withina finite element codefor limit analysis. This is followed, in Section 2.4, by the solution of two shakedownproblems involving variable load and variable temperature. Finally, in Section 2.5, th esolution of an unconventional shakedown problem is discussed. The history of load isprescribed and the shakedown limitisrequiredintermso f aminimum creep rupture stressf oramaximumcreep rupture life.Thisproblem occurs in the life assessment method of BritishEnergy,R5 7),an ddemonstratestheflex ibilityof themethod.

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The ease of implementation, efficiency, and reliability of the method indicate that it hasconsiderable potential fo r application in design and life assessment methods where efficientmethods are required for generating indicators of structural performance of structures.

2 2 Shakedown limitfor a von Mises yield conditionConsider a body composed of an isotropic elastic-perfectly plastic solid that satisfies the vonMises yield condition

where a = - f ~ crj } denotes the von Mises effective stress, o- . = cr -\Sijakk th edeviatoric stress, and ay is auniaxia l yield stress. Theplastic strain rate, s? , isgivenby theassociated flow rulein the formo f thePrandtl-Reuss relationship

where(2.3)

denotesthe vonMises effective strain rate.Consider the following problem. A body of volume Va nd surface S is subjected to a cyclichistory of loadAP^x^t over ST , part ofS, andtemperature W(Xj,t) withinV,where /I isascalar load pa rameter. On theremainder ofS,namely Su, th e displacement rate ui = 0. Thelinear elastic so lution to the problem is denoted by /lov . In the following we assume that theelastic solutions arechosen sothat A > 0. Theobjectiveo f shakedown analysis is to define avalue of A= / ls , the shakedown value, so that for any < , shakedown will always occur.The l inear matching method is an upper bound approach wh ere, through an iterative processwecalculate Asor aleast upperbound to As. his relies upon tw o separatetheoreticalresults,the upper bound shakedown theorem and the convergence criteria for the l inear matchingmethod, both discussed below.The upper bound shakedown theorem may be expressed in the following form. W e define aclass of incompressible kinematically admissible strain rate histories, ~, with acorresponding displacement increment fields, A ,c , and associated com patible strainincrement

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Thestrain ratehistory, e^, which neednot becom patible,satisfies th econdition that

In terms of such ahistory of strain rate anupper bound on the shakedown limit is given by8-11 ,

where X^ > A, s, with /ls the exact shakedown limit. In the follow ing we describe aconvergent method w here a sequence o f kinematically adm issible strain increment fields, withassociated strain rate histories, corresponds to a reducing sequence of upper bounds. Thesequence converges to the shakedown limit A s, or the least upper bound associated with theclassof displacement fields and strain rate histories chosen.The linear matching method relies upon the generation of a sequence of linear problemswhere the moduli are found by amatchingprocess For the vonMisesyieldcondition theappropriate classofstrain rates chosen areincom pressibleso thelinear problemisdefined bya single shear modulus n which varies both spatiallyandduring th ecycle. Corresponding toan initial estimate of thestrain rate history e~, ahistory o f a shear modulus fj(x t.,t) may bedefined by amatching condition

i.e. th e effective stress defined by the linear of the material is the sameas the yield stress forth e e -. A corresponding linear problem for a new kinem atically adm issible strain rate history,s~ , and atime constant residual stress field, ~pf, may now bedefined by

where A. = X m ,th eupper bound equation 2.6) correspondingto e^ = e -.On integrating equation 2.8) over a cycle we obtain

and

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where p f denotest hedeviatoric components of p f, etc. No te that equation 2.9) definesalinear problem for compatible Asj and equilibrium p f. Th e convergence proof, given byPonterandEngelhardt 5), then concludes that

where equality occurs if, and only if, l tj = sf an d A . { B is the upper bound corresponding to y = e~ The repeated application of theprocedure will result in amonotonically reducingsequence of upper bounds that converge to a min imum when th e difference betweensuccessive strain rate histories has a neg ligible effectupon the upper bound.The residual stress field pf from the solution to equation 2.9) also provides a lower bou ndshakedown limit, /l{fl, as the largest load parameter fo rwhich the yield conditionissatisfiedby the history of stress A^o^.+p~. If the solutions were carried out exactly such lowerbounds would themselves be exact, but if a Galerkin definition of equilibrium is used then itis possible to show that the lower bound converges to the least upper bound 3, 12) andprovides no additional information, other than an independent check on the accuracy of afinite element implementation. Hence such lower bounds are referred to as pseudo lowerbounds. The accuracy of implementation and the role of the pseudo lower bound is discussedbelow.The choice of the linear problem of equations 2.7)- 2.10) has a simple physicalinterpretation. For the initial strain rate history, e~, the shear modulus is chosen so that therate of energy dissipation in the linear material is matched to that of the perfectly plasticmaterial for the same strain rate history. At the same time the load parameter is adjusted sothat the value corresponds to a global balance in energy dissipation through equality ofequation 2.6).Ino ther words,the linear problemis adjusted sothatit satisfiesasmanyof theconditions of the plasticity problem as is possible. Th e fact that the result ing solution, whenequilibrium is reasserted, is closer to the shakedown limit solution and produces a reducedupper bound sh ould be no surprise. H ow ever, we need not rely upon such intu itive argum entsas aformalproof ofconvergence exists 3-5,12).A u l l description of the procedure as a general non-linear programming method withapplicationsto creep problems has been given by Ponteret al. 6).

2.3 Implementationof themethod limitanalysisThe method has been implemented in the commercial finite element code ABAQUS. Thenormalmodeofoperationof the code fo rmaterial non-line ar analysis involvesthe solutionofa sequenceo f linearized problems fo rincrem ental changes in stress, strain, an ddisplacement

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in time intervals corresponding to a predefined history of loading. A t each increment, userroutines allowadynamic prescriptionof the Jacobian whichdefines th erelationship betweenincrements of stress and strain. The implementation involves carrying through a standard loadhistorycalculation for thebody, but setting up the calculation sequence and Jacobian valuesso that each incremental solution provides the data for an iteration in the iterative process.Volume integral options evaluate the upper bound to the shakedown limit which is thenprovided to the user routines for the evaluation of the next iteration. In this way an exactimplem entation of the process m ay be achieved. The on ly source of error arises from the factthat ABAQUS uses Gaussian integration which is exact for a constant Jacobian within eachelement. The condition in equation 2.7) is applied at each Gauss point an d results invariations of the shear modulus ft t) within each element. There is , therefore, acorresponding integration errorbut theeffect ofthison theupper boundissmall.Theprimaryadvantages of this approach to implementation are practical. An implementation can beachieved whichi s: easily transferabletoother usersof thecode; requires fairly minor additions to the basic routines of the code so that a reliableimplementationcan be achieved; can beintroduced for awide rangeo felemen t an dproblem types.For the case of constant loads the formulation in the previous section reduces to the solutionof equation 2.8) or, equiv alen tly, equation 2.9) for a shear modulus distribution defined byequation 2.7). In the upper boun d equation 2.6) the time integration is not required. Thisformulation differs from theformulation givenbyPonter and Carter 3)where eachsolutionin the iterative process involves a stress field in equilibrium with an applied boundary loadwhereas in equation 2.7) the external loads are introduced through the linear elastic solution/ 1 < T ; The two formulations are entirely equivalent fo r linear elastic solutions which aresolved by the same Galerkin finite element method as used in the iterative procedure.However, the sequence of calculations is not identical and, as was mentioned earlier, theGaussian integration is not exact. In practice we find differences between th e results of thetw o formu lation s which are negligible compared w ith the approxim ation errors of the finiteelementmesh.Figure 2.1 shows the finiteelement mesh for the symmetric halfof aplane strain inden tationproblem where a line load P is applied over a strip of wi dth D. It may be recognized as one ofthe standard examples in the ABAQUS examples manual. The convergence of the upperbound is shown as lineA in Fig. 2.2 for the mesh shown in Fig. 2.1 and a refined mesh, asline B, where each element has been subdivided into sixteen identical elements. The analyticsolution the Pran dtl solution) for the half space is also show n. The observed behavio urillustrates two points, common to all the following solutions. As the method is a strict upperbound th e solution converges to the analytic solution from above. The accuracy of theconverged solution depends entirelyon the abilityof the class ofdisplacement fields definedby the mesh geometry to represent the displacement field in the exact solution. Generallythereis theneed for a sufficient densityofelements in theregions wherethe deformationfieldvaries most rapidly. In this case th e ref ining of the mesh geometry has a very significanteffect on theaccuracyof the convergedsolution.

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From experience the only source of computational error of any significance that can beidentified arises from Gaussian integration of the stiffness matrix of the linear solutions asdiscussed above. However, there is an indirect check on the magnitude of this error. In theabsence of this error; the upper and lower bound load parameters converge to a commonvalue, the least upper bound associated with the class of displacement fields defined by theelement structure. Hence differences between the converged bounds give an indirectindication of the significances of this error.This phenomenon is shown in the example shown in Fig. 2.3, where a plate withsymmetrically placed cracks is subjected to uni-axial tension. The elements are four nodedquadrilateral elements and this mesh would be expected to give a poor solution. This is solvedas a limit load problem for a von Mises yield condition. The convergence of the upper an dlower bound load parameters is shown in Fig. 2.4. The upper bound converges to the optimalupperboundfo rthis m esh, avalue that lies abov etheknown analytic solution,asshown.Thedifference between the optimal upper bound and the analytic solution is primarily a functionof themesh geom etry.Thelower bound conv erges, more slowly, to avalue closeto butbelowthe optimal upper bound and above the analytic solution. The difference between the twobounds is also a function of the mesh geometry but arises from the error in Gaussianintegration. The effect is exaggerated here as the mesh is coarse and the element degrees offreedom areinsufficient to capture the rapid changes ofstrainin the vicinity of the crack tip.This example is included as a demonstration of the modes of behaviour of the method.Generally the mesh is refined until the converged upper bound does not significantly reducefor increasing mesh density. In Fig. 2.5 the relative error in the upper bou nd,defined as

Relative error= optimal upper bound analytic value)/ analytic value)is shown foruniform meshes for a range of crack problems. The characteristic dim ension of atypical element,h is defined as the d iameter of the smallest circle that surrounds an element.The distance, a is the size of the uncracked length, the crack ligament. It can be seen thatconvergence is near linear w ith h, implying that convergence is primarily concerned w ith theconvergence of equilibrium w ithin the element rather than between elements). Solutions witherrorsofless than 1 percentmay beachievedwithout difficulty. This andotherempiricallyderived information allows the generation of meshes that are likely to yield upper boundswith errors of less than 0.5 per cent. With increasing mesh density the upper and lowerbounds more closely approach at convergence and a difference of less than 1 per cen t isachieved. The lower bound does not, however, always converge m ono tonic ally and it can takeaconsiderable number ofcyclesfor the lower bound to converge. F or a convergence criterionthat there should be no change in the fifth significant figure of the upper bound loadparam eter in five iterations the 5/5 criterion), problems involving the von M ises yieldconditionconverge in about 50 iterations. The method has been used to solve a large numberof problems involving structural components with cracks. Accurate limit load solutions forsuch problems are required for the application of life assessment methods in the powerindustry 2).It is worth commenting on the sensitivity of solutions to the assumptions within the method.In the examples considered above, near incompressibility in solving equation 2.9) for theresidual stress field p~ w as achieved by using hybrid elements with a Poissons ratio of0.49999. In the convergence proof discussed in references 4, 5) a sufficient condition fo r

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convergence is provided by the requirement that th e com plemen tary energy surface for thelinear material defined by the shear m odulu s // wh ich touches the yield surface at the'matching' point, giving rise to equation (2.7), otherwise must either coincide with th e yieldsurfaceor lie outside it. For the von Mises yield condition the complementary energy surfacecoincides with the yield su rface when Poissons ratiov =0.5 , but for v

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Theimplementation of themethod involvesthe following sequence ofcalculations:aninitialsolution assumes that plastic strains may occur at all r possible instants in the cycle. Initial,arbitrary, values of the moduli fj .m =1 are chosen. A s a result of this initial solution, theiterative method described in equations 2.13)- 2.15) is applied. The plastic straincomponents at instants where there is no strain in the converged solution then decline inrelative magnitude until they m akeno contributionto theupper bound.In the following solutions the iterative method was continued until there was no changed inth e fifth significant figure in the computed upper bound for five consecutive iterations.Thenum ber of iterations required depend ed upon the nature of the optim al me cha nism . Forreverse plasticitymechanisms thenumber of iterations requiredcould bequite high, inexcessof 100, whereas fo r mechanisms where all the plastic strains occurred at a single instantateach point in the body although not necessarily the same instant) the number of iterationsrequired was significantly less and 50 iterations was a typical number. For a less exactingconvergencecriteria asignificantly smaller number ofiterations are required andvariationofthe upper bound with iteration numbers shown in Fig. 2.2 is typical of both limit load andshakedown solutions.Figure 2.7 shows the symmetric section of a finite element discretization for a plane stressplate, with a circular hole, subjected to biaxial tension. The shakedown limit has beenevaluated for the two histories of Pl t ) , P2 t)) showninFig. 2.8.The interaction diagramofthe shakedown limit evaluated by the method are shown in Fig. 2.9 together with the limitload fo rmonotonic increase in P1,P2 ) . The elastic limit is also shown, i.e. the highest loadlevels fo rwhich theelastic solutions jus t liewithinth eelastic domainfor the prescribed yieldstress and also for a yield stress of 2 ay . It may be observed that in allcases the shakedownlimit is given eitherby the limit load for the loads atsome point in the cycleor at the elasticlimit fo r 2 cry .A s the initial loading po int is zero load, this later condition corresponds to thevariation of the elastic stresses lying within the yield surface if superimposed upon anarbitrary residual stress. This is a well known result and arises from the fact that themechanism at theshakedown limit corresponds to areverse plasticity conditionat thepointofstress concentration in the elastic solution, on either the major or minor axis of the holesurface.Figure 2.10 shows the classic Bree problem where either a plate or a tube wall thickness issubjected to axial stress and a fluctuation temperature difference, A0, across the plate orthrough thewallthickness. Theproblemhas beensolvedbothas planestressplate problem,where curvature of the plate due to thermal expansion is restrained, and as an axisymmetriccylinder.The two solutionsfor atemperature independent yield stress areboth shown inFig.2.11 in terms of a, , th e maximum principal thermoelastic stress due to A 0. T heplatesolution coincides with the classic B ree solution for a Tresca yield condition the problem isessentially one dimensional) whereas the solution for the axisymmetric problem lies outsidethe classic Bree solution to a maximum^extent of 15 per cent, the maximum differencebetween the Tresca and von Mises yield condition. The reverse plasticity solution, whichcorresponds to a , = 2 ay inboth cases, isoverestimatedb yboth computed solutions. Thisisdu e to the way reverse plasticity limits are evaluated. The optimizing strain rate historyconsists of incrementsof strain which result in azero accumulationof strain over thecycle.Thec ontribution of asingle Gauss p oint or inth is problema row ofGauss points) dom inates

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overthecontributionof allother Gauss pointsand the l im i t isgovernedby thevariationof theelastic stress at that point. In Fig. 2.11 we adopt for and Def thecontributions to the total dissipation Dp givenby those volumes and those times whereth elow temperature andcreep stress operate

Weassumethefollowingform forac tf, 9)

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then we can derive, from equations 2.17) and 2.18), th e following relationships betweensmall changes in A U B an d R for aparticular mechanism of deformation

where

This relationship forms th e basis for an iterative process which converges to the value of Ran dhencetherupture time tf corresponding to theshakedown for /I = 1.W ebegin by choosing an initial value of R = R 0 and tf so that the shakedown limit in theconvergedsolution isexpected to be /I < 1. For fixed R 0 theiterative process is allowed toconverge until the k-th iteration yields the first upper bound value of the load parameterwhich satisfies J ^ U B

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In Fig. 2.12 th e solutions for the Bree problem are shown with 6 > 7 . = 4 0 0 C and othermaterial constan ts appropriate to a ferritic steel.In Fig. 2.12 the shakedown limit is shown for the three cases corresponding to R =0.1, 0 .4,and 1.5. The contours shown were evaluated by converging to the value of the load parametercorresponding to the prescribed yield conditions given by equation (2.16) although it is worthnoting that the dependency of the yield stress on temperature causes a change in the yieldstress ateachiteration.For the caseofR =1.5, ay a throughoutthevolume.In Fig. 2.13we show the inverse problem w here the load param eter A is prescribed and the value of R isevaluated corresponding to a shakedown state. The two solutions shown in Fig. 2.13corresponding to points A and B in Fig. 2.12 for R = 0.1. The two phases of the process canbe seen where th e initial value of R = 0 .05 is ma intained constant for the first fe w iterationsuntil L k U B

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eferen es1) Ponter, A. R. S. andCarter K. F. Limit state solutions, based upon linear elastic

solutions with a statially varying elastic modulus'. Comput . Methods Appl. Mech.Engrg.,Vol. 14 0(1997)pp237-258.2) Ponter, A. R. S. and Carter, K. F. 'Shakedown State simulation techniques based onlinear elastic solutions', Com put. M ethods Appl. M ech. Engrg., Vol.140 (1997) pp259-279.3) Ponter,A. R. S.,Fuschi P., an dEngelhardt,M. LimitA nalysis for a GeneralClassof Yield Conditions', European Journal of Mechanics, A/Solids, Vol.19 (2000), pp401^21.4) Ponter, A. R. S. and Engelhardt, M. 'Shakedown Limits for a General YieldCondition', European JournalofMechanics, A/Solids, Vol. 19(2000),pp401^21.

5) Ponter, A. R. S., Chen, H., Bou lbib ane, M., and H ab ibu llah, M. The LinearMatching M ethod for the Evaluation for Limit Loads, Shakedow n Limits an d RelatedProblems, Keynote Lecture, Proceedings of the Fifth World Congress onComputational Mechanics (WCCMV), July 7-12, 2002, Vienna, Austria, Editors:Mang,H. A.;Ram merstorfer, F. G.;Eberhardsteiner, J. ,Publisher: Vienna Universityo fTechnology, Austria, ISBN 3-9501554-0-6, http://wccm.tuwien.ac.at. (the papermay bedownloaded fromthis site until 2008).6) Koiter, W . T. 'General theorems of elastic-plastic solids', in Progress in SolidMechanics, eds. Sneddon , J. N., Hill, R., Vo l. 1, pp167-221, 1960.7) Gokhfeld, D. A. andC herniavsky, D. F. Limit Analysis of Structures at ThermalCycling , Sijthoff\ Noordhoff. Alphenan DerRijn,The Netherlands, 1980.8) Konig, J. A. Shakedown of Elastic-Plastic Structures , PWN-Polish ScientificPublishers,W arsaw and Elsevier,Amsterdam,1987.9) Polizzotto, C., Borino, G ., Caddem i, S., and Fuschi, P. Shakedown Problems formaterials with internal variables, Eur. J . Mech . A / Solids,Vol. 10, pp621-639, 1991.10) Seshadri, R. and Fernando, C . P. D. 'Limit loads of mechanical components andstructures based on linear elastic solutions using th e GLOSS R-Node method', TransASMEPVP,Vo\ . 210-2, SanDiego,pp125-134,1991.11) Mackenzie, D. andBoyle J. T. 'A simple method ofestimating shakedown loads forcomplexstructures', Proc. ASM EPVP, Denver,1993.

12) G oodall, I. W ., G oodman, A . M., Chell, G. C., Ainsworth, R. A., and W illiams , J. A.'R5: An Assessment Procedure for the High Temperature Response of Structures',Nuclear Electric Limited, R eport, Barnw ood, Gloucester, 1991.

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Fig 2.1 Finiteelement m eshfor plane strain modelofindentationofhalf space eightnoded quadrilateral elements were used

Fig 2 2 Convergenceo f theupperboundto thelim it load for the indentation p roblemof Fig 2 9 The lower curve corresponds to a mesh wh ere each elemen t show n in Fig 2 9hasbeendivided into16elements.TheanalyticPrandtlsolution isshown forcomparison note that theyield stress ishere d enotedby

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Fig.2.3 Doub l e edge cracked plate su bjected to u niaxial tension.The elementsa re f ou rnoded qu dril ter lplanestresselements.The limit load solution expressedas anapplied pressure isindependent of thewidth of theplate

Fig. 2.4 Conve rgence of up pe r and lower bou nds for the proble m shown inFig.2.3 for ayieldstresso f TO = 200 MPa

5

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Fig. 2.5 Va riation of the relativeerrorof a range of crack pro blems with differingelement types withtheelem ent size compared withthecrack ligament sizea

Fig. 2.6 Ahistoryofloadwhichdescribes astraight linepath in load space a)producesa historyofelastic stresses which describes straight linesinstress space, b) - as aresultiti sknownapriori th at plastic strains only occurat the rvertices d urin gth ecycle:r = 4inthe Figure

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Fig 2 7 Finite element mesh for planestr ssproblem of Fig 2 8

Fig 2 8 Loading historiesfor theshakedown solutions shown in Fig 2 9

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Fig 2 9 Limit load and shak edow n limits for the geom etry and loa ding histories show ninFig 2.8 andmesh showninFig 2.7 notethat theshakedown limitisidenticalto theleastof thelimit loador the reverse plasticity limit denotestheplatethickness

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Fig 2 10 Bree problem aplateoraxisymmetrictubeissubjectedto fluctuatingtemperature differences andaxial stress mesh geometriesofeight n odedquadrilateral elements

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Fig 2 11 Sh ak edo wn limits for theBree problem ofFig 2 10 modelled asbothaplanestress plateand an axisymmetric thin walled tube- thesolid lineis Bree s solution for aTresca yield condition

Fig 2 12 Sha ked ow n limitsfor the problem discussed in Section2 5 for prescribedvalues ofR Thediam ond s correspond to the Bree solution which coincides withtheco mp uted solution forR = 1 5 Po ints A and B refer to thesolutionsinFig 2 13forR 0.1

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Fig 2 13 Convergence of R to R = 0 1 for theextended shakedown method discussed inSection 2 5 Curves labelled and Bcorrespondto theconvergenceto thecorresponding pointsinFig 2 12 Theslow convergenceof caseB is due to thedominanceof areverse plasticity mechanism

R S onterDepartmentofEngineering UniversityofLeicester UKM J ngelhartAirworthinessandStructural Integrity Group QinetiQ Farnborough UK

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Limit Loads for racked Piping omponentsD GMoffat

bstractThis Chapter summarizes recent work on the influence of cracks on the limit loads of pipingelbows and branch junctions subjected to internal pressure and mom ent loading.Notation cr ckdepthA constantB constantd branch pipe mean diameterD elbow or branch run pipe mean diameterM L limitmomentM 'L limit mom ent/plain pipe limit mom entM L limit mom entofcracked comp onent/limit momentofuncracked componentM LO single load case limit mom entM P twice-elastic-slope plastic loadPL limit pressure limit pressure/plain pipe limit pressureP L limit pressure of cracked compon ent/limit pressure of uncracked compon entPL O single load case limit pressu reR elbow radiusr pipemean radiusT ru npipe thicknessforbranch junc tionst branch pipe thicknessforbranchjunctionso relbow thicknessY uniaxial yield stress

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P elbow circumferentialdefecthalf angle0 elbow axialdefectsubtended anglea branch junction crack half angle bend characteristic tR/r2orbranch junction characteristic d/D)(D/T)05

3 ntroductionSince the late 1970s, structural assessment procedures forcracked components have evolvedan d the introduction of the two-criteria approach (1) hasmatured into the R6 method (2)nowused internationally. Thelatter requires th edefected structureto beassessed by a failureassessment diagram FAD), whichis atwo-dimensional plotof a fracture mechanics measureagainst aplastic collapse measure. Theformer aspectof theproblem is notconsidered here.For theevaluation of thelatter measure, knowledgeofboth applied load and thelimit loadofth edefected structure are required. Limit load solutions for avariety of cracked structuresalreadyexist, especiallyforsimpler geometries such asplatesandcylinders(3, 4).These wereobtainedby analytical stress analyses procedures usingtheLower Bound Theorem [e.g. 5)].However, formore complex components such aspiping elbows andbranch junctions tees),such data are sparse. Over the past few years, work has been underway on piping systemcomponents at The University ofLiverpool with the aim of providing data on elbows andbranch junctions that will helpt o fillthis gap.

An FE parametric studywasconducted on short-radius elbow radius/pipe radius, R/r = 2)piping elbows with internal axial cracksat thecrownorcircumferential cracks at theintrados,the loading being internal pressure, opening-bending, or a combination of the two 6). Theterminology used in the elbow study is shown inFig. 3.1. Inorder to provide confidenceintheelbow FEmodelling procedures, anexperimental investigation was conducted on aseriesof 13 short-radius elbows with cracks an d subjected to opening-bending, th e latter beingdefined as shown in Fig. 3.1. The global results arepresented in (7) and the local results in8).Theterms global and local willbedefined below.During th e elbow study, some difficulties were experienced in modelling components withlarge, deep cracks a/t=0.75).Tothrow some lightonthis,a studywas conductedonaxiallyloaded,plainpipeswith fullycircumferential, internal, part-penetrating cracks. The FEresultsarepresentedin (9) where it wasshown that a focusedmesh was the mostaccurate method ofevaluating th e limit load with a biased, standard mesh agood economic alternative. Someexperimental results from good quality components 10) confirmed th e accuracy of the fullnon-linear FEanalyses asdistinct from limit load analysis). Thetests also indicated that,forthese cylindrical, ful lycircumferentially, cracked components, therewas afactorofabout1.6between th emeasured, ultimate load-carrying capacity and thecalculatedFElimit loadfo r allcrackdepths tested.

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The next stage of the work was concerned with assessing limit loads fo r cracked, weldedpiping branch junctions. R eference 11)presents th e results from an FE parametric study ond/D = 0.5 junctio ns subjected to internal pressure, branch out-of -plane bending, or acom bination of the two. Th e FE results have been extended to d/D = 0.95 and experimentshave been conducted on two uncracked and three cracked tees. These latter results arepresented in 12)and will be published in due course.In the FE work, the cracks have been simulated using the node release method and in theexperimental work the crac ks have been produced using the electric discharge ma chining(EDM ) process.The remainder of this Chapter will focus on work at Liverpool University on elbows andwelded branch junctions and will present sample data from these studies. First, however, anattemp t willb e made to clarify th e procedures used to define limit loads an d plastic loads.

3 2 Definitions of limit loads and plastic loadsIn the ASME III design code 13) and the CEN TC54 unfired pressure vessel draft standardprEN13445, Part 3 on Design 14), there are design-by-analysis (DBA ) sections whichprovide alternative procedures to the design -by-fo rmu la (DBF) routes for designing pressurevessel components. In each case the DBA procedures rely on the concept of limit analysistechniques to assess limit loads fo r components derived from load versus displacement( displacem ent here meaning any relevant measure of change of shape) plots using num ericalanalysis procedures. In both design codes the term limit load is used to deno te the calculatedmaximum load found (b y analytical or numerical procedures) assuming elastic/perfectlyplastic material properties and using small displacement analysis.In ASME III the twice-elastic-slope (TES) method is used to determine what is referred to asa collapse load from a load versus displacement plot determined using full non-linearanalysis (i.e. m aterial and geometry) or exp erimental data. Gerdeen 15) has appealed againstthe use of the term collapse load since, for pressure vessel structures, the true collapse loadis often m uch higher than the TE S load. Gerdeen recomm ended that the term plastic load beused to define loads obtained using the TES procedure (o r equivalent) and this term will beused herein. Gerdeen also recommended that the area under the load versus displacement plotshould represent energy (e.g. moment versus rotation or pressure versus volume change) butrecognized that this would not always be possible since for example, volume change isdifficult to assess, especially fo r cracked components. ASME III warns that particu lar careshould be given to ensure that the strains or deflections that are used are indicative of the loadcarrying capacity of the structure, but does not specify the criteria upon which this decisionshould be based.

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In the CE N TC54 d raf t standard (14) th e tang ent intersection (TI) method is recommen ded fordetermining limit loads, a s def ined above, f rom load versus displacement plots. In an a t temptto remove the subjectivity involved in the TI method, the d raf t standard states that, If there isno m axim um (in the load versus d isplacement plot) in the region of principal strains less than5 per cent, the greatest tan gen t intersection value shall be used with one tangen t through theorigin, th e other through a point wher e th e maximum principal strain does not exceed +5 percent. Limit loads determined using the TI method, can be referred to as TI l imi t loads . Theterm TI plastic load can be used to def ine a plastic load obtained f rom a full non-l inearanalysis load versus d isplacemen t plot and using the TI method.Figure 3.2 il lustrates the constructions used to determine the TES and TI plastic loads, L TESand L T I. In Fig. 3.2(b) th e load L 5 is the load a t which th e maximum principal strain is 5 percent. A tangent is drawn to the load versus displacement curve at load L 5 and the point atwhich this intersects th e extended elastic line is the TI plastic load L T I o r limit load L LTI , a sth e case ma y be.In a recent publication (16), the Liverpool group have attempted to assess the m erits of thesedifferent definit ions of plastic load and have confirmed that a range of limit loads and plasticloads can be obtained from the same basic analysis, d epend ing on the def init ion adopted. Thisuncertainty concurs with th e 1999 EPERC Design by Analysis Manual (17) where, ...thedifficulty of extracting meaningful plastic design loads f rom elastic-plastic finite elementanalysis is em phasized. The conclusions in (16) were that, (a) the A SME III TES proceduregives a reasonable estimate of plastic loads but, unfortunately, does not give a unique valuefo r any one component and, (b) the 5 per cent principal strain TI method in CEN TC54 doesgive a unique value of plastic load but can significantly underestimate limit load plateauvalues.In this Chapter on cracked components, l imit loads are obtained as indicated above and usingth e fifteen-times or the five-times elastic slope criterion to ensure that a representativeplateau value is obtained from th e limit analysis load-displacement plot (see Fig. 3.3 as anexample of this using the fifteen-times elastic slope criterion). Plastic loads are obtained fromthe u l l non-linear FE analysis, or from experimental load-displacement plots, using th eASMEIII TES procedure [Fig. 3.2(a)], notwithstanding the concerns expressed above.In the R6 manual (2), th e terms g loba l limit load and loca l limit load a re used. The fo rmeris obtained from load versus displacement plots where the displacement selected isrepresentative of the overall behaviour of the component - for example, diametral growth ornozzle rotation. The term local l imit load is only relevant to components with part-penetration cracks and is defined in R6 as, the load needed to cause plasticity to spreadacross the remaining liga m en t. . . . In the work on elbows to be presented in the next section,an a t tempt was made to measure experimental local plastic loads using electrical resistancestrain gauges (8). However, the conclusion reached w as that this was no t l ikely to be feasibleand therefore the only viable method of assessing local l imit loads is considered to be FEana lysis where the spread of plasticity through the ligam ent can be assessed. In wh at follows,any referen ce to l imit loads o r plas tic load s will imply the adjective g lobal rather than local.

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Limit Loadsfor racked Piping ompone nts 7

Elbows3 3 1 Defect free elbowsThe Liverpoo l group have recently 18) summ arized the available literature on limit loadassessment of defectfreepiping elbows. Reference is made to the theoretical work by Spenceand Findlay 19), Calladine 20), and Goodall 21) which has led to the development ofclosed form expressions for the determinationof limit loads fo run defected pipe bends underin-plane bending. The combined loading case of internal pressure and in-plane bending hasbeen addressed by Goodall 21). M ore recently, Shalaby and You nan 22, 23) used the FEtechnique to investigate limit loads of pressurized, defect-free piping elbows under bothclosing and opening in-planebending. The limit load was found to increase then decreasewith increasing pressurefor allelbows fo rboth closing an dopening modes ofbending.The work by Chattopadhyay et al 24) involved elastic/strain-hardening FE analyses toevaluatethep lastic mom ents of six elbows A , =0.24 to 0.6) underthe effect ofcombined in-plane closing/opening bending with a varying level of internal pressure. Plastic moment datawas obtained by the TESmethod from moment versus end rotation plots. Curve fitting wasapplied to the results to pro duce twoclosed form equations for the closing and opening plasticmoments. For some of the cases where the elbows were unpressurized, the results were notedto be higher than those predicted from the formulae given in 19, 20, or 21). This can beattributed to the stiffening effect of the connecting straights, material strain hardening, andnon-linear geometriceffects see below).In contrast with these analytical assessments, extensive experimental work by Greenstreet25) was carried out in the US at Oak Ridge National Laboratory. Greenstreet perform edroom temperature testsontwen ty, 6-inch 152mm) nominal diameter, schedule40 7.11 mmthick), and schedule 80 10.97 mm thick) long and short radius commercial piping elbows short radius R/r == 2, long radius R/r == 3). Sixteen were made of carbon steel A STM -A106B)and theremaining four were stainless steel A 312-304L).The limit moment expressions mentioned above from Refs 19-21) share the same form andcan be written as

where n can be taken as 2/3 20, 21) or 0.6 19), and A can be 0.8 19), 0.94 20), or 1.0421). If A istakento be 0.94 and n as2/3,thenit is shown in 18) thatequation 3.1)underpredicts the available experimental data fo r opening bending by Greenstreet 25), Griffiths26), and Yahiaouiet al 7) in the range20-50per cent, depending on the definition used forthe experimental plastic moments.A significant outcome of the above analytical and experimental work on plastic loads asdistinct from limit loads) is that elbows subjected to closing in-plane bending have a lowerplastic mo men t than nom inally identical elbows under opening in-plane bending. Th is iscaused by the non-linear ovalizing of the elbow cross-sections where the ovalizing effectweakens or strengthens the elbow depending on whether the applied moment is closing oropening respectively. While the above is true fo r uncracked elbows, fo r elbows withcircumferential cracksat the intradoso raxial cracksat the crown, closing bending causesthe

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38 Reference S tressMethods- Analysing SafetyandDesign

cracks to close rather than open as has been observed inreferenc e 26). For this reason,thecracked elbows consideredin thenext section have been subjectedtoopening bending.3 3 2 Cracked elbowsThe terminology adopted for the Liverpool cracked elbow work is as shown in Fig. 3.1.ABAQ US 20-noded brick elem ents C3D20R) w ere used in the FE study. Four elementsthrough thethickness were found to beadequatefo rmost casesbut for long, deep a/t=0.75)cracks, additional mesh refinement was needed 6). Figure 3.4 illustrates the defo rm ationmode for an elbow under opening bending with an internal axial crack at the crown 0 = 75, a/t = 0.75).Theresults of the elbow limit loadparametric studyaresummarized inFigs3.5-3.8. Figures3.5 and 3.6 are foropening bending andpressure respectivelyforinternal crown axial cracks.Notethe false zeroonthesetw oplots.)Figures3.7 and 3.8 are for the same loadsfo rinternalintrados circum ferential cracks. The conclusion from this data isthat part-penetrating defectsup to half the thickness deep have little influence on opening bending and pressure limitloads. It is only when the defects are deep or through-wall that their presence significantlyreduces limit load levels.Table 3.1indicates the models used for the experimental elbow w ork under opening bendingloading. The experimental arrangement is show n in Fig. 3.9 and the FE m odel simu lated thisset-up.Forcomparison with the experimental data, the FE models were re-run using full non-linear analy sis non-linear geom etry plus true stress versus strain curve). The true stressversus strain curve is given in Fig 3.10. The comparisons presented below are encouraging,bearing in mind that as-manufactured elbows were used in the tests outside diam eter 88.9mm; thickness 5.49mm bend radius 76.2 mm).

Table3 1 Test elbow detailsTestelbowidentification

E0 note2)E1E2 note3)E2A note 4)E3E7E8E9E4E6E10E12E11

Typenote 1)

-AAAAAAACCCCC

21212121757575

DefectLength degrees)

2/7

4646120120120

a/t

-0.50.50.51.00.50.751.00.51.00.50.751.0

Notes: 1. A =axial crown),C =circumferential intrados)2. Defect free elbow3. Duplicate testof E14. Duplicate testof E1

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Limit Loadsfor racked Piping omponents 9

For reasons that w i l l not be explained here, there were three, nominally identical, openingbending tests w ith a short internal axial crack at the crown 6 = 21 degrees, a/t = 0.5). Figure3.11 shows the inevitable scatter in theexperimental moment versus crosshead displacementdata w ith the FE d ata passing through the middle of the latter. Figure 3.12 gives the m omentversus crosshead displacement data for an intrados crack having 2p = 46 degrees, a/t = 1.0.There is agood agreementup to amomentofabout 10 kNmwhenthetest model crack startedto grow, after which the experimental moment falls off. Figure 3.13 presents th e momentversus displacement plots for anintrados crack w ith 2 3 = 120degrees, a/t =0.75. Inthis case,the agreement isgoodup to amomentofabout8 kNm atw hich stagethe ligament failed andthe load dropped off. The data in Figs 3.11 3.13 are generally representative of thecomparison betw een the test dataand the full, non-linear FEpredictions.The factthat thereisgood agreement up to the point at which the cracks propagated, or the ligaments failed,suggests that the FE modelling used in the limit load parametric study 6) to produce Figs3.5 3.8was acceptable thus giving confidence in this data.InFigs3.11 3.13 moment versus displacement curves the twice-elastic-slope FE plastic loadis indicated b y M Pand the FE l imi t load by MI., the latter being o btained using the yield stressof 328 MPa from Fig. 3.10. In each case the limit moment ML is a conservative estimate ofthe TES plastic load. This w as true of all elbow s tested 7).Figure 3.14 presents a summ ary of the experimental data w hich again emphasizes how toleranttheseshort radius piping elbow sare tocracks, insofarasplastic/limit loadsa reconcerned.

3 4 W elded branch junctions3 4 1 Defect freejun tionsThe main concern in the Liverpool work was to assess limit loads of cracked branchjunctions. However,prior to embarking on the parametric study of cracked components, astudyw as completed on uncracked branch junctions subjected to pressure loading. The rangeofparametersin thestudy was asfollow s, w ithanadditionalset ofmodels not listed here) tostudytheeffect ofbranch/run pipe thickness ratio t/T:

Theresultso fthis w ork w illbepresentedinreference 27) w hich includesaprocedure, basedon the results obtained, fo r calculatingth e limit pressure of anybranch junction in the rangedefined above. In 1968, Cloud and Rodabaugh 28) produced an analytical expression forestimating the l imit pressures PL ofuncracked branch junctions. T his w aspresented using afactor,p*, toadjust th eTresca plain pipel imitpressure equation:

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This compares quite well with th e Liverpool data but is somewhat conservative fo r thick-walled, equal pressure strength (t/T=d/D) junction s.3 4 2 Crac ked branch junctionsThe terminology used in the cracked branch junction work and the location of the cracksinvestigated are shown in Fig. 3.15. Sample results from reference (11) are presented here. Thebranch junction s studied had a branch/run pipe mean diameter ratio d/D = 0.5, a thickness ratiot/T = 1.0, and diam eter/thickness ratios D/T of 10, 20, or 30. One of the FE m odels is shown inFig. .16 where it can be seen that the FE me sh has been biased towards the crack tip. Onlythrough-wall cracks were investigatedand themodelshad three elements throughthethicknessinthe junc tion region. L oading was either internal pressure o r branch out-of-plane bending, thelatterbeing applied in the direction that w ould cause crack opening (Fig. 3.17). The global limitload results w ere o btained frommoment versus nozzle end rotation curves.Limit mo ments were normalized to the branch pipe limit mom ent d2tY and limit pressureswere normalized to the run pipe limit pressure 2TY/D. These normalized values are referredto here as M'L an d P'L,. The normalized limit loads versus crack ang le 2a are presented in Fig.3.18for the three D/T ratios 10, 20, and 30. Alterna tively the limit loads could be norm alizedon the basis of the uncracked branch junction values taken from the ordinate of Fig. 3.18.These are referred to as M Land P L and arepresented inFig. 3.19 where it can be seen thatthe points collapse nicely on to two lines, representing the two load categories. Notsurprisingly, Fig. 3.19 indicates that large, through-wall cracks h ave a significant wea keningeffect on the limit loads of these d/D = 0.5 branch junctions.Finally, the interaction of pressure and moment loads on cracked junctions was investigated byvaryingthe ratio of the loads applied. In Fig. 3.20 the non -dimen sional parameters ML/M LO an dPL/P LO (where MLo and PLO represent the single load case limit moment and pressure) are usedto indicate the interaction behaviour for the D/T = 20 case. It is clear that for theuncrackedmodel20A the interaction diagram m ay be reasonably assumed to be circular. For mo del 20Dwiththe longest crack (2a = 140 degrees), the interaction relationship tends toward s linear,withthe intermediate cracked models (20B, 2a = 49degrees and20C, 2a = 95degrees) distributedevenly between the uncracked case and the 2a = 140 degrees case. Similar results wereobtained for D/T = 10 and 30 and for d/D = 0.95 branche s (12) .

3 5 ConclusionsLeavingasidethepossibility offailure by fast fracture, short radiuspip ingelbows arecapableofsustaining quitelargedefects witho ut dramatic reductionsintheir limit load capa city. Part-penetrating d efectsup tohalfth ethickness deep have littleinfluence onpressureandopeningbending limit loads. It isonly whenth e defects are deepor through-wall that their presencesignificantlyred uces limit load levels.

40 eferenceStress Methods AnalysingSafetyandDesign

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LimitLoadsfor CrackedPiping Components 4

For the d/D = 0.5 piping branch junctions with throug h-wall cracks at the flank, subjected tointernal pressure or out-of-plane branch bending, l imit loads are significantly influenced byth e crack length. If the l imit loads are norm alized on the basis of the uncracked limit loadsan d plotted against crack length, the results converg e to single lines for each of the tw o loads.Limit load interaction diagrams for pressure versus out-of-plane branch bending show that,for th e uncrackedcases, circular interaction is relevant, but that for increasingcrack length,there is a clear trend towards l inear interaction.

3 6 AcknowledgementsThe work presented here w as part sponsored by British Energy on behalf of IMC (BritishEnergy and BNFL) and the UK Engineering and Physical Sciences Research Council. Theirsupport is gratefully acknowledged. M y grateful thanks to colleagues Kadda Yahiaoui, MikeLynch, and Dave Moreton for all their hard work in producing this data - this Chapter isreally a summary of their work.

References(1) Dowling, A. R. and Townley, S. H. A., The effect of defects on structural failure: Atw o criteria approach, Int. J. Press. Vess. Piping 1975, 3,77-107.(2 ) Assessment of the integrity of structures containing defects, Nuclear Electric Ltd

document ref R6, Revision 4, April 2001.(3 ) Miller,A. G,, Review of limit loads of structures containing defects, Int.J. Press. Vess.Piping Vol.32 , 1988,197-327.(4 ) Zahoor, A., Ductile Fracture Handbook Vo l. 1-3, Elec tric Power Research In stitute,EPR INP-630 1-D/N14, Palo Alto, California, USA , 1989, 1990, and 1991.5) Calladine,C. R.,Plasticity for Engineers EllisHorwood, 1985.(6 ) Yahiaoui, K.,M offa t, D. G., andMoreton,D. N., Piping elbows with cracks - Part 1 :A parametric study of the influence of crack size on limit loads due to pressure andopening bending , J. Strain Analysis Vol.35, No 1, 2000, 35 -46.(7 ) Yahiaoui, K.,Moffa t ,D. G., and Moreton, D. N., Piping elbows with cracks - Part2:

Global finite element an d experimental plastic loads u nde r opening b end ing , Journal ofStrainAna lysis Vol. 35, No 1, 2000,47-57.(8 ) Yahiaoui , K ., Moreton, D. N., and Moffa t , D. G., Local finite e lement andexperimental limit loads of flawed piping elbows under opening bending , Strain J. ofBritish Soc.F or S train M easurement Vol.36, No 4, 2000, 175-186.(9) Lynch, M. A., M of fa t, D. G., Moreton,D. N., and A inswor th , R. A., Limit loads forcylinders with fully circumferential cracks in tension: Comparison of analytical andfinite e lement d ata , Procs 9th Int. Conf. on Pressure Vessel Technology Sydney 2000.(10) Lynch, M. A., Moreton, D. N., and Moffa t , D. G., Limit loads for axially loadedcylinders having full circumferential cracks: Experimental, analytical an d numericalstudies , subm itted toSTRAIN BSSM.(11) Lynch, M. A., Mof fa t , D. G., and More ton , D. N., Limit loads for a cracked pipingbranch junction under pressure and b ranch out-of-plane bending , Int.J. Pressure Vesselsan d Piping Vol. 77, No 4, 2000, 185-194.(12) Ly nch, M. A., Lim it loads of piping branch junc tions with cracks , PhD Thesis TheUniversity of Liverpool, 2001

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(13) ASME BoilerandPressure Vessel Code Section III, 1995.(14) Draft European Standard, prEN 13445-3, Unfired pressure vessels - Part 3: Design annexB,section B9, CEN, 1999.(15) Gerdeen J. C. A critical evaluation of plastic behaviour data and a unified definition ofplastic loads fo r pressure componen ts , PVRC Welding Research CouncilB ulletin 254,

section16, ASME, 1979.(16) Moffat D. G. Hsieh M. F. andLynch M., An assessment of ASM E III and CEN TC5 4methodsof determining plastic and lim it loads fo r pressure system co mp onents , J. StrainAnalysis IMechE, 2001 V ol. 36, No 3,301-312.(17) The Design-by Analysis Manua l European Pressure Equipment Research Council,European Com mission Joint Research Centre,EUR 19020 EN, Section 1.3, 1999.(18) Yah iaoui K . M oreton D. N. and M offat D. G. Evaluation of limit load data forcracked pipes bends under opening bending a nd compa rison with existing solutions Int. J.Pressure VesselsandPiping Vol.79,2002, 27-36.(19) Spence, J.andFindlay G.E., Limit loads for pipe bends under in-plane b end ing , Proc2n d Int.Conf. on Pressure Vessel Technology San A ntonio 1973,393-399.(20) Calladine C. R. Limit analysis of curved tubes , J Mech. Eng. Sci. Vol. 17, No 2,1974,85-87.(21) Goodall I. W. Lower bound limit analysis of curved tubes loaded by combined internalpressure and in-plane bending mom ent , CE GB Report RD/B/N4360, August 1978.(22) Shalaby M. A. andYounan M. Y. A., Limit loads for pipe elbows with internalpressure under in-plane closing mo me nt , ASME Journal of Pressure VesselTechnology Vol. 120, 1998,35-42.(23) Shalaby M. A. and Younan M. Y. A., Lim it loads for pipe subjected to in-planeopening bending m om ents , ASME Journal of Pressure Vessel Technology Vol. 121,1998,17-23.(24) Chat