Reference Stress Methods: Analysing Safety and Design

Reference Stress Methods

Structural Technology and Materials Group The Structural Technology and Materials Group (STMG) Committee comprises experts representing companies and organizations such as: British Steel plc, Rover Group Limited, University of Nottingham, University of Strathclyde, NAFEMS Limited, University of Wales, Ford Motor Co Limited, Swansea University and, the EPSRC. The STMG Committee serves the membership by organizing relevant seminars and conferences, as well as representing the UK on national and international committees and organizations. 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 of components and of predicting their life in order to achieve minimum cost without compromising integrity; • to provide designers with information on established materials such as steels, aluminium alloys, and fibre-reinforced plastics, and on newer materials such as metal matrix composites and ceramics; • to encourage theoretical and experimental studies on the mechanics of materials forming processes such as rolling, pressing, and extrusion, and the effect that these have on subsequent performance of the component; • to encourage the development of tools for the estimation of stresses, strains, and deformations in structures. Including finite element and boundary element methods, simplified methods, and experimental methods; • to develop computing technology in so far as it is relevant to materials and mechanics of solids; • 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, and stress corrosion; • to ascertain the properties of materials needed for engineering design, including the effect of manufacturing, forming, and joining processes on those properties; • to promote new ideas and publicize new information in a form which practising mechanical engineers can use. More information on the work of the group can be obtained by writing to: Structural Technology and Materials Group Institution of Mechanical Engineers 1 Birdcage Walk London SW1H 9JJ

Reference Stress Methods Analysing Safety and Design Edited by Ian Goodall

Professional Engineering Publishing

Published by Professional Engineering Publishing, Bury St Edmunds and London, UK.

First Published 2003 This publication is copyright under the Berne Convention and the International Copyright Convention. All rights reserved. Apart from any fair dealing for the purpose of private study, research, criticism or review, as permitted under the Copyright, Designs and Patents Act, 1988, no part may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, electrical, chemical, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owners. Unlicensed multiple copying of the contents of this publication is illegal. Inquiries should be addressed to: The Publishing Editor, Professional Engineering Publishing Limited, Northgate Avenue, Bury St Edmunds, Suffolk, IP32 6BW, UK. Fax: +44 (0) 1284 705271.

© 2003 The Institution of Mechanical Engineers, unless otherwise stated.

ISBN 1 86058 362 8

A CIP catalogue record for this book is available from the British Library. Printed by The Cromwell Press, Trowbridge, Wiltshire, UK

The Publishers are not responsible for any statement made in this publication. Data, discussion, and conclusions developed by authors are for information only and are not intended for use without independent substantiating investigation on the part of potential users. Opinions expressed are those of the Authors and are not necessarily those of the Institution of Mechanical Engineers or its Publishers.

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Contents About the Editor

ix

Introduction

xi

Determining the basic parameters Chapter 1 Chapter 2 Chapter 3

Reference stress requirements for structural assessment R A Ainsworth

1

Computational methods for limit states and shakedown A R S Ponter and MJEngelhart

11

Limit loads for cracked piping components

DGMoffat Extending the approach to weldments Chapter 4 Some aspects of the application of the reference stress method in the creep analysis of welds THHyde and WSun Chapter 5

High-temperature creep rupture of low alloy ferritic steel butt-welded pipes subjected to combined internal pressure and end loadings F Vakili-Tahami, D R Hayhurst, and M T Wong

33

57

75

Applications Chapter 6 Chapter 7 Chapter 8 Chapter 9 Index

Code application - below the creep range AR Dowling

113

Code application - within the creep range GA Webster

127

Fracture assessment of reeled pipelines C Arbuthnot and T Hodgson

145

The use of reference stresses in buckling calculations T Hodgson

155 171

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About the Editor Dr Ian Goodall is a member of the Institution's Structural Technology and Materials Group which promoted the Seminar in November 2000 that forms the basis of this volume. He is also a Fellow of the Institution. After University, his experience was principally in the nuclear industry where he spent over 30 years working on structural integrity issues and developing strategic research programmes on other engineering matters. He was responsible, with his colleagues, for bringing together the knowledge base required to produce a document which is now called R5 and entitled An Assessment Procedure for the High Temperature Response of Structures. This procedure uses simplified methods of assessment wherever they are justified. It was required for application to components in both the fast reactor and the advanced gas-cooled reactor where the effects of creep, fatigue, and fracture are important. It is now used throughout the nuclear industry for components operating at elevated temperature. Since leaving the industry he has been working as a Consultant in the structural integrity field working on creep, fracture, and fatigue issues in collaboration with various universities.

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Introduction Reference stress methods and other simplified methods of assessment offer many attractions in the design process and they continue to have considerable value in spite of the rapid improvements in finite element analysis. A significant feature of simplified methods is that they enable rationalization of information both from analysis and from experiment. They are often more intuitive than methods based solely on finite element approaches and allow the sensitivity to input parameters to be assessed rapidly. Consequently they enable design and risk assessments of structural components to be performed very efficiently. The basic principles are set down in Chapters 1, 2, and 3, which define the underlying theory and give advice on the determination of the required parameters. 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 inviting national experts to present papers at a seminar and subsequently to prepare chapters for this volume. At a detailed technical level there are two objectives which are identified in Chapters 1 and 2 and may be summarized as follows. • Firstly to simplify the analysis process, wherever possible, by basing structural assessment on the following: • elastic solutions - with and without defects; • plasticity or limit load solutions - with and without defects; • shakedown solutions for cyclic loading. • Secondly to reduce the influence of detailed variations in material properties by suitable normalization. It transpires that a particularly useful quantity in assessing structures, both with and without defects, is a 'reference stress' which is based on the limit load. This quantity appears frequently in this volume and is defined by the relationship

where aref is the reference stress, F is the applied load, and FL is the rigid-plastic limit load for the structure with a yield stress ery. Chapter 3 gives details of how the limit load may be determined for complex structures such as cracked piping components. in the world of high-speed computing power, most stress analyses can be performed using finite element techniques for both linear and non-linear analysis. There is a need, however, for the provision of underlying theory that enables the analyst to validate his numerical analysis and also to interpret experimental findings. This is a two-way process as detailed results of finite element analysis may be also used to refine estimates of a reference stress. A good example of this is given in Chapters 4 and 5 where the ambition is to extend these concepts to the treatment of the complex situation that exists in weldments.

The real test of any of these approaches is whether they are used by the design engineer, either directly or in developing design codes. In fact, the application of such approaches is widespread and Chapters 6, 7, 8, and 9 in this volume address the application of the techniques to: • code developments, both below and within the creep range; • pipelines; • buckling. Finally, I would like to thank all the authors for their patience with my comments and their efforts in producing this volume on simplified methods. It is, in my view, a very comprehensive introduction to the topic, which will be of value to both design engineers and academics alike.

Ian WGoodall November 2002

1 Reference Stress Requirements For Structural Assessment R A Ainsworth

Abstract The reference stress method is a powerful approximate method for describing the inelastic response of structures. The method has been developed to enable simplified assessment procedures to be produced for both defect-free and defective components. In this Chapter, the background to the reference stress method is briefly described and the accuracy and limitations of the method are discussed. Then specific uses of the technique and their incorporation into structural assessment methodologies to guard against component failure by a number of mechanisms are described.

Notation C* C(t) E E' F F' FL G J K Kp Ks l n

steady-state creep characterizing parameter transient creep characterizing parameter Young's modulus E in plane stress; E /(I - v2) in plane strain load normalizing value of F limit load value of F elastic strain energy release rate characterizing parameter for elastic plastic fracture elastic stress intensity factor value of K for primary loads value of K for secondary loads normalizing length creep stress exponent

2

St t tCD tr uel lif V Eref ec e*s v crret

Reference Stress Methods - Analysing Safety and Design

time-dependent strength time time for failure for continuum damage rupture time elastic displacement steady-state creep displacement rate factor describing the effect of secondary stress strain at reference stress creep strain rate steady-state creep strain rate Poisson's ratio reference stress

<j^.f reference stress used to estimate J cj^f reference stress used to estimate creep rupture ay yield or 0.2 per cent proof stress CTU ultimate stress CT flow stress CTcl,max maximum value of equivalent stress calculated elastically ass max maximum value of equivalent stress in steady-state creep x stress concentration factor

1.1

Background

The essence of the reference stress technique is that the inelastic behaviour of a component under a given loading is related to inelastic materials data at a reference stress defined for the given loading. The method and its background have been described in the book by Penny and Marriott (1) in the context of creep problems. Consider, for example, a component operating under steady load, F, in the creep range for which an estimate of the steady-state creep displacement rate, ii*s, is required. The reference stress estimate is

where 8^(0^) is the secondary creep rate of the material at a reference stress level, tr rcf , and l has dimensions of length. Clearly values for £ and aref are required to use the estimate of equation (1.1). If creep analysis of the component and material of interest were required to derive these values, then there would not be a major advantage in the technique. However, numerical analyses and experimental data suggest that the reference stress can often be estimated with sufficient accuracy from a knowledge of the value of the load F corresponding to plastic collapse (2).

Reference Stress Requirements For Structural Assessment

3

Then

where FL is the plastic collapse load defined for a rigid plastic material with yield strength cr y . Since F L Oy) is directly proportional to 0y, the reference stress of equation (1.2) is independent of cr y ; it is proportional to F and depends on geometry through the term [F L (o y )/a y ]. There then remains a requirement to estimate l in equation (1.1). Suppose an inelastic solution is available for one material (A). Then

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

where E is Young's modulus and iiei is the elastic displacement under load F ensures, for example, that equation (1.1) is accurate for a power-law creeping material in which the stress exponent is unity. Clearly, however, the accuracy will be reduced for non-linear materials and this is discussed in Section 1.2 below. Further examples of reference stress estimates are described in Sections 1.2 and 1.3, below. In essence, the approaches have the following properties: • direct use of inelastic materials data in any convenient form without the need for such data to be described by specific equations such as power-law creep or plasticity; • use of limit load solutions (or shakedown solutions for cyclic loading) which are widely available without the need for inelastic analysis; • can make use of the results of elastic stress analysis; • can be refined to improve accuracy using detailed analysis results where available, or simplified to provide order-of-magnitude estimates where rapid results are needed; • robust for engineering use as they are not sensitive to detailed descriptions of constitutive equations or highly refined analysis.

1.2

Accuracy and limitations

The development of design methods based on reference stress techniques has been discussed by Goodall et al. (3) in the context of creep design of non-defective structures. An important aspect in this is the accuracy of the techniques and any limitations. Energy theorems may be used to demonstrate that, quite generally, the reference stress of equation (1.2) overestimates, on average, the creep energy dissipation rate within a structure (4). However, this does not ensure that any particular estimate of displacement or strain within a component will be safely

4

Reference Stress Methods - Analysing Safety and Design

estimated using approximations such as equations (1.1—1.4). Indeed, at stress concentrations the local stress levels would be expected to be higher than the reference stress of equation (1.2) which essentially describes average component behaviour. However, the energy theorems do give confidence that reliable estimates will be obtained for displacements conjugate to the applied loads. In this section, some specific examples are used to examine the accuracy and limitations of reference stress techniques. Consider, first, creep rupture of structures subjected to steady load, for which the multiaxial creep rupture criterion is similar to the yield criterion used to define the limit load in equation (1.1). In this case, it is possible to show (5) that estimating the time for a structure to fail by the spread of creep rupture damage, ICD, is less than the time-to-rupture obtained from uniaxial stress rupture data at the reference stress of equation (1.2), i.e.

Numerical and experimental data show that the difference between ten and tr (<jref) depends on the magnitude of the stress concentration factor in the component. Defining a stress concentration factor, x, as

where (7el mx is the maximum value of the elastically calculated equivalent stress in the component, then the peak stress in steady-state creep, ass max' is approximately

where n 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 occur when local rupture at the stress concentration occurs, i.e.

However, for creep ductile materials there can be a significant time taken for damage to spread throughout a component after this local damage initiation (3). A pragmatic approach based on numerical and experimental data is to define a rupture reference stress, <j* f , intermediate between aref and ass max. This takes the value

For n > 7, this estimate is over conservative and equation (1.7) is used even for ductile materials with high n. The rupture reference stress, a* f , which may take either the value defined by equation (1.7) or equation (1.9) depending on n and whether or not the material is creep ductile, leads to improved accuracy compared with the simple use of the limit load reference stress. This is illustrated in Fig. 1.1.

Reference Stress Requirements For Structural Assessment

5

As a second example, consider elastic-plastic fracture where an estimate is required for the parameter J which describes the stress and strain fields at the crack tip. In this case, a reference stress estimate of J has been developed as (7, 8)

where

is the elastic value of J which is related to the stress intensity factor K and E' = E in plane stress and E/(l-v 2 ) where v is Poisson's ratio in plane strain. The strain eref is the total (elastic + plastic) strain at the reference stress level. Equation (1.10) is clearly accurate in the elastic regime where sref = o r e f / E . It has also been shown to be accurate in the fully plastic regime by comparison with numerical solutions for fully plastic materials (7). However, it loses accuracy in the small-scale yielding regime where J exceeds G but the reference stress may be less than the limit of proportionality. A convenient correction to improve accuracy in this regime is

This provides a correction at small loads which is phased out as the fully plastic term [the first term on the right-hand side of equation (1.12)] become large. This is a convenient approximate estimate of 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, to define G, and the limit load, to define o ref . These have been collected in compendia for a large number of defective engineering components. This approach has been incorporated into the development of practical flaw assessment procedures such as R6 where the J-estimate is converted into a failure assessment diagram (9) of the type shown in Fig. 1.2. If inelastic analysis of the defective component is available, the accuracy of equation (1.12) may be improved by modifying the reference stress definition as follows. When CTrer = ay, equation (1.12) shows that

where

when ay is defined as the 0.2 per cent proof stress. From the inelastic analysis, the load, F' say, at which J/G equals the value given by equation (1.13) can be identified and then a modified reference stress is given by

6

Reference Stress Methods - Analysing Safety and Design

Use of this reference stress ensures that equation (1.12) is exact at F = F', is exact at low loads where behaviour is elastic, and has the correct dependence on material response under fully plastic conditions. This leads to improved accuracy compared to use of the limit load reference stress and often to reduced conservatism in assessments since F' > FL in many cases.

1.3

Specific uses

In this section, some specific uses of reference stress techniques are listed in terms of their incorporation into structural assessment methodologies to guard against component failure by a number of mechanisms. 1.3.1 Plastic collapse In design codes, plastic collapse may be avoided by satisfying limits on so-called stress intensities or stress resultants. A more accurate approach is to use a plastic collapse solution if available. 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) and has the advantage that the reference stress can be modified subsequently to address creep rupture, deformation limits, and creep fracture. R5 contains procedures for assessing the integrity of components operating at high temperatures and addresses the following failure modes. • • • • •

Excessive plastic deformation due to single application of a loading system. Incremental collapse due to a loading sequence. Excessive creep deformation or creep rupture. Initiation of cracks in initially undefected material by creep and creep-fatigue mechanisms. The growth of flaws by creep and creep-fatigue mechanisms.

Reference stress techniques are used to provide simplified assessment methodologies to guard against all these failure modes and some of these techniques are described in Sections 1.3.2— 1.3.5. For low temperature fracture assessments, a plastic collapse limit is included in R6 (9) which corresponds to

where a is a flow stress to allow for strain hardening beyond yield and often a = X(CTy + <3 'u) whereCTUis the ultimate tensile stress. R6 addresses fracture of components containing defects by both brittle and ductile mechanisms.

Reference Stress Requirements For Structural Assessment

7

It may be noted that the reference stress approach may be extended to cyclic loadings using shakedown concepts rather than plastic collapse concepts. This is addressed elsewhere (11) and is not discussed further here. 1.3.2 Creep rupture As already discussed in Section 1.2, component creep rupture may be avoided by limiting the rupture reference stress of equation (1.7) or equation (1.9) to the time-dependent strength based on uniaxial creep rupture data. This is incorporated in R5 for defect-free components. For defective components or components with sharp stress concentration features, the stress concentration factor x is high and equations (1.7) and (1.9) are over-conservative. In this 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 for crack extension by creep mechanisms (see below). This is the approach adopted in R5 and also in the British Standards document BS 7910 (12). 1.3.3 Creep deformation Average strains in a component during its service life, t, may be limited by imposing the restriction

where St(m%,t) is the stress from isochronous stress-strain data at the temperature of interest for the service life t, for the strain level of m% which is typically one per cent. Peak strains may be similarly limited by imposing a restriction on the steady-state creep stress of equation (1.7). Typically, peak strains are limited to five per cent. In R5 such limits are used. Additionally, creep strains at a reference stress level defined from a shakedown analysis are used to calculated creep damage via a ductility exhaustion approach. The acceptable level of creep damage depends on the level of associated fatigue damage and the margins required in the assessment. 1.3.4 Low-temperature fracture The reference stress J-estimate of equation (1.12) has been converted into a failure assessment diagram approach and is extensively used for defect assessment through R6 (9) and other methods world-wide (12,13). Although equation (1.2) is written for a single primary load, the reference stress method can handle multiple primary loadings by suitable definition of FI . It has also been extended to combined primary and secondary loadings. Recently (14), it has been shown that equation (1.12) may still be used provided the definition of G in equation (1.11) is replaced by

where Kp, Ks are the value of K for the primary and secondary stresses, respectively, and V is a factor. Reference stress methods have been used to evaluate the factor V. For elastic behaviour, V = 1; at low loads plasticity tends to lead to values V > 1 whereas at high loads (aref > ay) plastic relaxation of secondary stresses leads to V < 1 (14).

8

Reference Stress Methods - Analysing Safety and Design

1.3.5 Creep fracture In R5 (4) creep crack initiation and growth are assessed using the creep equivalent, C*, of the low-temperature fracture parameter J. A short-term parameter C(t) is also used for times prior to steady-state conditions. By analogy with equation (1.10), C* maybe estimated from

Modifications similar to those in equation (1.12) for small-scale yielding are used to assess small-scale creep using C(t) (8). Then C* or C(t) are used to assess crack initiation and growth (8,10).

1.4

Closing remarks

This Chapter has briefly described the background to the reference stress techniques. The accuracy of the method and how this may be improved in specific cases has been discussed. Finally, the power of the technique has been illustrated by summarizing a number of practical cases where reference stress methods have been introduced into codes and standards.

Acknowledgement This Chapter is published with permission of British Energy Generation.

References (1)

Penny, R. K. and Marriott, D. L. Design for Creep, Second edition, Chapman & Hall, London (1995). (2) Sim, R. G. Reference stress concepts in the analysis of 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. The development of high temperature design methods based on references stresses and bounding theorems, ASME J. Engng Mater. Technol. 101, 349-355 (1979). (4) Leckie, F. A. and Martin, J. B. Deformation bounds for bodies in a state of creep, ASME J. Appl. Mech. 34, 411-417 (1967). (5) Goodall, I. W. and Cockroft, R. D. H. On bounding the life of structures subjected to steady load and operating within the creep range, Int. J. Mech. Sci. 15, 251-263 (1973). (6) Calladine, C. R. A rapid method for estimating the greatest stress in a structure subject to creep, Proc. IMechE Vol. 178, Part 3L, 198-206 (1964). (7) Ainsworth, R. A. The assessment of defects in structures of strain hardening materials, Engng Fract. Mech. 19, 233-247 (1984). (8) Webster, G. A. and Ainsworth, R. A. High Temperature Component Life Assessment, Chapman & Hall, London (1994). (9) R6, Assessment of the Integrity of Structures Containing Defects, Revision 4, British Energy Generation (2001). (10) R5, Assessment Procedure for the High Temperature Response of Structures, Issue 2, British Energy Generation (1999).

Reference Stress Requirements For Structural Assessment

9

(11) Ponter, A. R. S. Computational methods for limit states and shakedown, Reference Stress Methods - Analysing Safety and Design, Professional Engineering Publishing Limited (2002). (12) BS 7910: 1999, Guide on methods for assessing of the acceptability of flaws in metallic structures, BSi, London (2000). (13) Zerbst, U., Ainsworth R. A., and Schwalbe, K.-H. Basic principles of analytical flaw assessment methods, Int. J. Pres. Ves. Piping 71, 855-867 (2000). (14) Ainsworth, R. A., Sharples, J. K., and Smith, S. D. Effects of residual stresses on fracture behaviour - experimental results and assessment 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 A Ainsworth British Energy Generation Limited, Barnwood, UK

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2 Computational Methods for Limit States and Shakedown A R S Ponter and M J Engelhart

Abstract This Chapter discusses a computational technique, the linear matching method, for the direct evaluation of parameters that determine strength characteristics of a structure subjected to complex histories of loading. Here we discuss limit loads, shakedown limits, and an extended shakedown limit associated with creep rupture. The method consists of the solution of a sequence of linear problems for a constant residual stress field. The solutions provide a monotonically reducing limit load or shakedown limit upper bound. On convergence the method provides the least upper bound associated with the class of displacement rate fields defined by a finite element mesh. The method may be implemented within a conventional finite element code and the solutions discussed here were all generated using the commercial code ABAQUS. The efficiency and accuracy of the method is illustrated through a sequence of solutions of typical life assessment problems.

Notation a s ay

von Mises effective stress von Mises effective strain rate yield stress

£? , £,y

inelastic strain rates

Af ?, A£,J A Pi 8 a-,,

increments of inelastic strain over a loading cycle scalar loading parameter applied load temperature linear elastic stress history

12

pij H, /7

2.1

Reference Stress Methods - Analysing Safety and Design

time-constant residual stress field shear moduli defined in the linear matching method

Introduction

Life assessment methods and design codes were originally developed with the understanding that, at most, only linear elastic solutions were available. These days complete simulation of component performance is possible although there remains the problem of the availability of sufficient material data for constitutive equations and the fact that full analysis is best done at the later stages of a design process. Between these two extremes there exist a number of analysis methods of a simplified nature that provide sufficient information for design or life assessment decisions, based upon less demanding calculations. Among such methods, those based on limit analysis and shakedown analysis provide relevant examples. Such computational methods have the attractive feature of inverting analysis, in the sense that they provide load ranges for which certain types of structural performance occurs, although based upon simple material models. This Chapter discusses the linear matching method, a recent advance in computational methods for shakedown and related problems based upon a particularly useful methodology. The procedure originates from the elastic compensation and related methods (1, 2) where a sequence of linear problems is solved with spatially varying linear moduli. In reference (3) it was demonstrated that the method may be interpreted as a non-linear programming method where the local gradient of an upper bound functional and the potential energy of the linear problem are matched at a current strain rate or during a strain rate history. This interpretation may be used to formulate a very general method for evaluating minimum upper bound solutions. Provided certain convexity conditions are satisfied, it is possible to define a sequence of linear problems where the upper bound monotonically reduces. The sequence then converges to the solution that corresponds to the absolute minimum of the upper bound functional, subject to constraints imposed by the class of strain rate histories under consideration. The theoretical bases for the method and convergence proofs are discussed in references (4, 5). A full discussion of the current range of application of such methods is summarized in reference (6). As a result of these theoretical considerations it is possible to generate limit load and shakedown limits that are the absolute minimum of all upper bound values given by the kinematics of a finite element mesh. Such values are the most accurate that may be obtained within the formal structure of the stiffness finite element method. For the solutions described in this Chapter the general code ABAQUS 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 is concerned with the implementation of the method within a finite element code for limit analysis. This is followed, in Section 2.4, by the solution of two shakedown problems involving variable load and variable temperature. Finally, in Section 2.5, the solution of an unconventional shakedown problem is discussed. The history of load is prescribed and the shakedown limit is required in terms of a minimum creep rupture stress for a maximum creep rupture life. This problem occurs in the life assessment method of British Energy, R5 (7), and demonstrates the flexibility of the method.

Computational Methods for Limit States and Shakedown

13

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

2.2

Shakedown limit for a von Mises yield condition

Consider a body composed of an isotropic elastic-perfectly plastic solid that satisfies the von Mises yield condition

where a = (-fa'~ crj } denotes the von Mises effective stress, o-!. = cr -\Sijakk

the

deviatoric stress, and ay is a uniaxial yield stress. The plastic strain rate, s? , is given by the associated flow rule in the form of the Prandtl-Reuss relationship

where (2.3) denotes the von Mises effective strain rate. Consider the following problem. A body of volume V and surface S is subjected to a cyclic history of load AP^x^t) over ST , part of S, and temperature W(Xj,t) within V, where /I is a scalar load parameter. On the remainder of S, namely Su, the displacement rate ui = 0. The linear elastic solution to the problem is denoted by /lov . In the following we assume that the elastic solutions are chosen so that A > 0. The objective of shakedown analysis is to define a value of A = /l s , the shakedown value, so that for any &<&, shakedown will always occur. The linear matching method is an upper bound approach where, through an iterative process we calculate As or a least upper bound to As. This relies upon two separate theoretical results, the upper bound shakedown theorem and the convergence criteria for the linear matching method, both discussed below. The upper bound shakedown theorem may be expressed in the following form. We define a class of incompressible kinematically admissible strain rate histories, £~, with a corresponding displacement increment fields, A«, c , and associated compatible strain increment

14

Reference Stress Methods - Analysing Safety and Design

The strain rate history, e^, which need not be compatible, satisfies the condition that

In terms of such a history of strain rate an upper bound on the shakedown limit is given by (8-11),

where X^ > A,s, with /ls the exact shakedown limit. In the following we describe a convergent method where a sequence of kinematically admissible strain increment fields, with associated strain rate histories, corresponds to a reducing sequence of upper bounds. The sequence converges to the shakedown limit As, or the least upper bound associated with the class of displacement fields and strain rate histories chosen. The linear matching method relies upon the generation of a sequence of linear problems where the moduli are found by a matching process. For the von Mises yield condition the appropriate class of strain rates chosen are incompressible so the linear problem is defined by a single shear modulus n which varies both spatially and during the cycle. Corresponding to an initial estimate of the strain rate history e~, a history of a shear modulus fj(xt.,t) may be defined by a matching condition

i.e. the effective stress defined by the linear of the material is the same as the yield stress for the e'-. A corresponding linear problem for a new kinematically admissible strain rate history, s~, and a time constant residual stress field, ~pf, may now be defined by

where A. = Xm , the upper bound equation (2.6) corresponding to e^ = e'-. On integrating equation (2.8) over a cycle we obtain

and

Computational Methods for Limit States and Shakedown

15

where p'f denotes the deviatoric components of pf, etc. Note that equation (2.9) defines a linear problem for compatible Asj and equilibrium pf. The convergence proof, given by Ponter and Engelhardt (5), then concludes that

where equality occurs if, and only if, eltj = sf and A.{,B is the upper bound corresponding to £y = e~ • The repeated application of the procedure will result in a monotonically reducing sequence of upper bounds that converge to a minimum when the difference between successive strain rate histories has a negligible effect upon the upper bound. The residual stress field pf from the solution to equation (2.9) also provides a lower bound shakedown limit, /l{fl , as the largest load parameter for which the yield condition is satisfied by the history of stress A^o^. +p~. If the solutions were carried out exactly such lower bounds would themselves be exact, but if a Galerkin definition of equilibrium is used then it is possible to show that the lower bound converges to the least upper bound (3, 12) and provides no additional information, other than an independent check on the accuracy of a finite element implementation. Hence such lower bounds are referred to as pseudo lower bounds. The accuracy of implementation and the role of the pseudo lower bound is discussed below. The choice of the linear problem of equations (2.7)-(2.10) has a simple physical interpretation. For the initial strain rate history, e~, the shear modulus is chosen so that the rate of energy dissipation in the linear material is matched to that of the perfectly plastic material for the same strain rate history. At the same time the load parameter is adjusted so that the value corresponds to a global balance in energy dissipation through equality of equation (2.6). In other words, the linear problem is adjusted so that it satisfies as many of the conditions of the plasticity problem as is possible. The fact that the resulting solution, when equilibrium is reasserted, is closer to the shakedown limit solution and produces a reduced upper bound should be no surprise. However, we need not rely upon such intuitive arguments as a formal proof of convergence exists (3-5,12). A full description of the procedure as a general non-linear programming method with applications to creep problems has been given by Ponter et al. (6).

2.3

Implementation of the method - limit analysis

The method has been implemented in the commercial finite element code ABAQUS. The normal mode of operation of the code for material non-linear analysis involves the solution of a sequence of linearized problems for incremental changes in stress, strain, and displacement

16

Reference Stress Methods - Analysing Safety and Design

in time intervals corresponding to a predefined history of loading. At each increment, user routines allow a dynamic prescription of the Jacobian which defines the relationship between increments of stress and strain. The implementation involves carrying through a standard load history calculation for the body, but setting up the calculation sequence and Jacobian values so 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 then provided to the user routines for the evaluation of the next iteration. In this way an exact implementation of the process may be achieved. The only source of error arises from the fact that ABAQUS uses Gaussian integration which is exact for a constant Jacobian within each element. The condition in equation (2.7) is applied at each Gauss point and results in variations of the shear modulus ft(t) within each element. There is, therefore, a corresponding integration error but the effect of this on the upper bound is small. The primary advantages of this approach to implementation are practical. An implementation can be achieved which is: • easily transferable to other users of the code; • requires fairly minor additions to the basic routines of the code so that a reliable implementation can be achieved; • can be introduced for a wide range of element and problem types. For the case of constant loads the formulation in the previous section reduces to the solution of equation (2.8) or, equivalently, equation (2.9) for a shear modulus distribution defined by equation (2.7). In the upper bound equation (2.6) the time integration is not required. This formulation differs from the formulation given by Ponter and Carter (3) where each solution in the iterative process involves a stress field in equilibrium with an applied boundary load whereas in equation (2.7) the external loads are introduced through the linear elastic solution /1
Computational Methods for Limit States and Shakedown

17

From experience the only source of computational error of any significance that can be identified arises from Gaussian integration of the stiffness matrix of the linear solutions as discussed above. However, there is an indirect check on the magnitude of this error. In the absence of this error; the upper and lower bound load parameters converge to a common value, the least upper bound associated with the class of displacement fields defined by the element structure. Hence differences between the converged bounds give an indirect indication of the significances of this error. This phenomenon is shown in the example shown in Fig. 2.3, where a plate with symmetrically placed cracks is subjected to uni-axial tension. The elements are four noded quadrilateral elements and this mesh would be expected to give a poor solution. This is solved as a limit load problem for a von Mises yield condition. The convergence of the upper and lower bound load parameters is shown in Fig. 2.4. The upper bound converges to the optimal upper bound for this mesh, a value that lies above the known analytic solution, as shown. The difference between the optimal upper bound and the analytic solution is primarily a function of the mesh geometry. The lower bound converges, more slowly, to a value close to but below the optimal upper bound and above the analytic solution. The difference between the two bounds is also a function of the mesh geometry but arises from the error in Gaussian integration. The effect is exaggerated here as the mesh is coarse and the element degrees of freedom are insufficient to capture the rapid changes of strain in 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 reduce for increasing mesh density. In Fig. 2.5 the relative error in the upper bound, defined as Relative error = (optimal upper bound - analytic value)/(analytic value) is shown for uniform meshes for a range of crack problems. The characteristic dimension of a typical element, h, is defined as the diameter 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 that convergence is near linear with h, implying that convergence is primarily concerned with the convergence of equilibrium within the element (rather than between elements). Solutions with errors of less than 1 per cent may be achieved without difficulty. This and other empirically derived information allows the generation of meshes that are likely to yield upper bounds with errors of less than 0.5 per cent. With increasing mesh density the upper and lower bounds more closely approach at convergence and a difference of less than 1 per cent is achieved. The lower bound does not, however, always converge monotonically and it can take a considerable number of cycles for the lower bound to converge. For a convergence criterion that there should be no change in the fifth significant figure of the upper bound load parameter in five iterations (the 5/5 criterion), problems involving the von Mises yield condition converge in about 50 iterations. The method has been used to solve a large number of problems involving structural components with cracks. Accurate limit load solutions for such problems are required for the application of life assessment methods in the power industry (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 the residual stress field p~ was achieved by using hybrid elements with a Poissons ratio of 0.49999. In the convergence proof discussed in references (4, 5) a sufficient condition for

18

Reference Stress Methods - Analysing Safety and Design

convergence is provided by the requirement that the complementary energy surface for the linear material defined by the shear modulus // which touches the yield surface at the 'matching' point, giving rise to equation (2.7), otherwise must either coincide with the yield surface or lie outside it. For the von Mises yield condition the complementary energy surface coincides with the yield surface when Poissons ratio v = 0.5 , but for v < 0.5 it lies within the yield surface and contravenes the conditions of the convergence proof. As this is a sufficient condition for convergence, it is of some interest to observe the effect of introducing volumetric strains into the linear solutions by a fixed value of v < 0.5 throughout the process. Sequences of solutions for a range of values of v have been generated. For the value used in the solutions discussed above further reduction in the difference from the ideal value of 0.5 has a negligible effect on the converged upper bound when compared with other sources of error. However, if a value of v = 0.49 is used for crack problems the upper bound converges to values which lies below the analytic solution for a sufficiently refined mesh. Hence the effect of adopting a value of v significantly less than 0.5 will be to give results which may either be above or below the analytic solution and convergence can not be assured.

2.4

Shakedown solutions

The procedure described by equations (2.7)-(2.10) requires the definition of a shear modulus at each instant during the cycle. There are problems where the distribution of the strain rate history in time during the cycle is unknown in advance but there is an important class of problems where we know ab initio that plastic strains may only occur at certain instants within the cycle. If the loading history consists of a sequence of proportional changes in loads between a set of extreme points, as shown schematically in Fig. 2.6 for a problem involving two loads (Pj, /" 2 ), then the linearly elastic stress history also describes a sequence of linear paths in stress space as shown. For a strictly convex yield condition, which includes the von Mises yield condition in deviatoric stress space, the only instants when plastic strains can occur are at the vertices of the stress history, atj (t,), I = 1 to r. The strain rate history then becomes the sum of increments of plastic strain

and equations (2.8)-(2.10) become

Computational Methods for Limit States and Shakedown

19

The implementation of the method involves the following sequence of calculations: an initial solution 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. As a result of this initial solution, the iterative method described in equations (2.13)-(2.15) is applied. The plastic strain components at instants where there is no strain in the converged solution then decline in relative magnitude until they make no contribution to the upper bound. In the following solutions the iterative method was continued until there was no changed in the fifth significant figure in the computed upper bound for five consecutive iterations. The number of iterations required depended upon the nature of the optimal mechanism. For reverse plasticity mechanisms the number of iterations required could be quite high, in excess of 100, whereas for mechanisms where all the plastic strains occurred at a single instant at each point in the body (although not necessarily the same instant) the number of iterations required was significantly less and 50 iterations was a typical number. For a less exacting convergence criteria a significantly smaller number of iterations are required and variation of the upper bound with iteration numbers shown in Fig. 2.2 is typical of both limit load and shakedown solutions. Figure 2.7 shows the symmetric section of a finite element discretization for a plane stress plate, with a circular hole, subjected to biaxial tension. The shakedown limit has been evaluated for the two histories of (Pl (t),P 2 (t)) shown in Fig. 2.8. The interaction diagram of the shakedown limit evaluated by the method are shown in Fig. 2.9 together with the limit load for monotonic increase in (P1, P 2 ). The elastic limit is also shown, i.e. the highest load levels for which the elastic solutions just lie within the elastic domain for the prescribed yield stress and also for a yield stress of 2ay . It may be observed that in all cases the shakedown limit is given either by the limit load for the loads at some point in the cycle or at the elastic limit for 2cry . As the initial loading point is zero load, this later condition corresponds to the variation of the elastic stresses lying within the yield surface if superimposed upon an arbitrary residual stress. This is a well known result and arises from the fact that the mechanism at the shakedown limit corresponds to a reverse plasticity condition at the point of stress concentration in the elastic solution, on either the major or minor axis of the hole surface. Figure 2.10 shows the classic Bree problem where either a plate or a tube wall thickness is subjected to axial stress and a fluctuation temperature difference, A0, across the plate or through the wall thickness. The problem has been solved both as plane stress plate problem, where curvature of the plate due to thermal expansion is restrained, and as an axisymmetric cylinder. The two solutions for a temperature independent yield stress are both shown in Fig. 2.11 in terms of a,, the maximum principal thermoelastic stress due to A0. The plate solution coincides with the classic Bree solution for a Tresca yield condition (the problem is essentially one dimensional) whereas the solution for the axisymmetric problem lies outside the classic Bree solution to a maximum ^extent of 15 per cent, the maximum difference between the Tresca and von Mises yield condition. The reverse plasticity solution, which corresponds to a, = 2ay in both cases, is overestimated by both computed solutions. This is due to the way reverse plasticity limits are evaluated. The optimizing strain rate history consists of increments of strain which result in a zero accumulation of strain over the cycle. The contribution of a single Gauss point (or in this problem a row of Gauss points) dominates

20

Reference Stress Methods - Analysing Safety and Design

over the contribution of all other Gauss points and the limit is governed by the variation of the elastic stress at that point. In Fig. 2.11 we adopt for
2.5

An extended shakedown limit

Consider the following problem. A body is subjected to a known cyclic history of loading. The corresponding elastic solution is given as before by /lav and the particular load history under consideration corresponds to /I = 1. The load history contains a significant component which derives from variations in temperature. The yield stress is assumed to vary with temperature and may have two values. At lower temperatures the yield stress equals a constant valuecr^ r . At higher temperature it is replaced by a creep rupture stress < r c ( t f , 9 ) which depends upon the time to creep rupture tf which is understood as a property of the structure as a whole, and the local temperature 9. We require the largest creep rupture time t, for which the prescribed loads remain within the shakedown limit for this definition of the yield stress. This problem is posed within the methodology of the life assessment method R5 (7) as a means of assessing the remaining creep rupture life of the structure. Here we treat it as a novel shakedown problem where the parameter we wish to optimize, the creep rupture time tj , is included in the definition of the shakedown problem through the definition of the yield stress at each point in the body and each instant during the cycle when plastic strains can occur. Hence the yield stress at each point of the body at time tm is defined by

We assume the following form for ac (t f, 9)

and R ( 1 ) = g(1) = 1 so that ac ( t f , 6) = a^T when tf =t0 and 6 = 9T. Hence we wish to compute the value of R for which the shakedown limit is given by A = 1. For a prescribed mechanism of deformation at some stage in an iterative process with this definition of the yield stress there will be contributions to the volume integral of the plastic energy dissipation which originate from aLJ and ac. If we denote by L>" and Def the contributions to the total dissipation Dp given by those volumes and those times where the low temperature and creep stress operate

Computational Methods for Limit States and Shakedown

21

then we can derive, from equations (2.17) and (2.18), the following relationships between small changes in A.UB and R for a particular mechanism of deformation

where

This relationship forms the basis for an iterative process which converges to the value of R and hence the rupture time tf corresponding to the shakedown for /I = 1. We begin by choosing an initial value of R = R0 and tf so that the shakedown limit in the converged solution is expected to be /I < 1. For fixed R0 the iterative process is allowed to converge until the k-th iteration yields the first upper bound value of the load parameter which satisfies J^UB <1. The value of R is then changed according to equation (2.19) at each iteration so that AUB returns to the preassigned value of A. = 1. i.e.

Hence

and the process is repeated. At each iteration the value of R increases and converges, from below, to the value for which the shakedown limit is given by A = 1. As there is a dynamic change in the yield value at each point in the structure and time during the cycle, the solution method does not match the convergence criteria for straightforward shakedown and some care needs to be taken in the use of this technique. In the following example we adopt the following simple form for the temperature dependence of ac

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Reference Stress Methods - Analysing Safety and Design

In Fig. 2.12 the solutions for the Bree problem are shown with 6> 7 .=400°C and other material constants 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 parameter corresponding to the prescribed yield conditions given by equation (2.16) although it is worth noting that the dependency of the yield stress on temperature causes a change in the yield stress at each iteration. For the case ofR = 1.5, ay = a" throughout the volume. In Fig. 2.13 we show the inverse problem where the load parameter A is prescribed and the value of R is evaluated corresponding to a shakedown state. The two solutions shown in Fig. 2.13 corresponding to points A and B in Fig. 2.12 for R = 0.1. The two phases of the process can be seen where the initial value of R0 = 0.05 is maintained constant for the first few iterations until /LkUB <1, when R is allowed to increase according to the relationships (2.21) and (2.22) until convergence takes place. Convergence is slower in the case corresponding to point B in Fig. 2.12, where, in the converged solution, a reverse plasticity mechanism operates. This problem demonstrates the potential flexibility of the method. Traditionally, shakedown analysis has been seen as a method of defining a load parameter for a prescribed distribution of material properties and load history. It is clear from this example that the shakedown problem may be posed in other ways; in this particular problem the quantity optimized concerns a material property which enters the problem in only part of the volume and only during part of the load cycle. It is clearly possible, using the type of technique discussed in this section, to pose a variety of optimization problems depending upon the needs of the problem. This introduces possibilities for shakedown analysis which have not previously been available.

2.6

Conclusions

The Chapter discusses a method of evaluating shakedown limits by a convergent non-linear programming method which has its origins in the 'elastic compensation' method (1, 11). The existence of convergence proofs allows an implementation in a commercial finite element code which is both efficient and robust. The set of examples of both limit loads and shakedown limits given here demonstrate its numerical stability and the ability to approach the analytic solution from above through mesh refinement. The last example involves the optimization of a creep rupture stress so that, for a prescribed load history, the body lies within a shakedown limit. Such problems occur in situations where both low temperature plasticity and high temperature creep properties limit the load capacity of the structure. It is clear from this example that convergent methods can be devised for such novel shakedown problems, thereby introducing a degree of flexibility into numerical methods for shakedown analysis which have not previously been available.

Acknowledgements The work in this Chapter was supported by the University of Leicester and British Energy Limited. The authors gratefully acknowledge this support. The calculations for Fig. 2.5 were carried out by S Hentz of the Ecole Normal Superier de Cachan, Paris.

Computational Methods for Limit States and Shakedown

23

References (1)

Ponter, A. R. S. and Carter, K. F. 'Limit state solutions, based upon linear elastic solutions with a statially varying elastic modulus'. Comput. Methods Appl. Mech. Engrg., Vol. 140 (1997) pp 237-258. (2) Ponter, A. R. S. and Carter, K. F. 'Shakedown State simulation techniques based on linear elastic solutions', Comput. Methods Appl. Mech. Engrg., Vol. 140 (1997) pp 259-279. (3) Ponter, A. R. S., Fuschi, P., and Engelhardt, M. 'Limit Analysis for a General Class of Yield Conditions', European Journal of Mechanics, A/Solids, Vol. 19 (2000), pp 401^21. (4) Ponter, A. R. S. and Engelhardt, M. 'Shakedown Limits for a General Yield Condition', European Journal of Mechanics, A/Solids, Vol. 19 (2000), pp 401^21. (5) Ponter, A. R. S., Chen, H., Boulbibane, M., and Habibullah, M. The Linear Matching Method for the Evaluation for Limit Loads, Shakedown Limits and Related Problems, Keynote Lecture, Proceedings of the Fifth World Congress on Computational Mechanics (WCCMV), July 7-12, 2002, Vienna, Austria, Editors: Mang, H. A.; Rammerstorfer, F. G.; Eberhardsteiner, J., Publisher: Vienna University of Technology, Austria, ISBN 3-9501554-0-6, http://wccm.tuwien.ac.at. (the paper may be downloaded from this site until 2008). (6) Koiter, W. T. 'General theorems of elastic-plastic solids', in Progress in Solid Mechanics, eds. Sneddon, J. N., Hill, R., Vol. 1, pp 167-221, 1960. (7) Gokhfeld, D. A. and Cherniavsky, D. F. 'Limit Analysis of Structures at Thermal Cycling', Sijthoff\& Noordhoff. Alphen an Der Rijn, The Netherlands, 1980. (8) Konig, J. A. 'Shakedown of Elastic-Plastic Structures', PWN-Polish Scientific Publishers, Warsaw and Elsevier, Amsterdam, 1987. (9) Polizzotto, C., Borino, G., Caddemi, S., and Fuschi, P. Shakedown Problems for materials with internal variables, Eur. J. Mech. A/ Solids, Vol. 10, pp 621-639, 1991. (10) Seshadri, R. and Fernando, C. P. D. 'Limit loads of mechanical components and structures based on linear elastic solutions using the GLOSS R-Node method', Trans ASMEPVP,Vo\. 210-2, San Diego, pp 125-134,1991. (11) Mackenzie, D. and Boyle, J. T. 'A simple method of estimating shakedown loads for complex structures', Proc. ASMEPVP, Denver, 1993. (12) Goodall, I. W., Goodman, A. M., Chell, G. C., Ainsworth, R. A., and Williams, J. A. 'R5: An Assessment Procedure for the High Temperature Response of Structures', Nuclear Electric Limited, Report, Barnwood, Gloucester, 1991.

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Reference Stress Methods - Analysing Safety and Design

Fig. 2.1 Finite element mesh for plane strain model of indentation of half space - eight noded quadrilateral elements were used

Fig. 2.2 Convergence of the upper bound to the limit load for the indentation problem of Fig. 2.9. The lower curve corresponds to a mesh where each element shown in Fig. 2.9 has been divided into 16 elements. The analytic Prandtl solution is shown for comparison - note that the yield stress is here denoted by
Compulational Methods for Limit States and Shakedown

25

Fig. 2.3 Double edge cracked plate subjected to uniaxial tension. The elements are four noded quadrilateral plane stress elements. The limit load solution, expressed as an applied pressure, is independent of the width of the plate

Fig. 2.4 Convergence of upper and lower bounds for the problem shown in Fig. 2.3 for a yield stress of
26

Reference Stress Methods - Analysing Safety and Design

Fig. 2.5 Variation of the relative error of a range of crack problems with differing element types with the element size h compared with the crack ligament size a

Fig. 2.6 A history of load which describes a straight line path in load space (a) produces a history of elastic stresses which describes straight lines in stress space, (b) - as a result it is known a priori that plastic strains only occur at the r vertices during the cycle: r = 4 in the Figure

Computational Methods for Limit States and Shakedown

Fig. 2.7 Finite element mesh for plane stress problem of Fig. 2.8

Fig. 2.8 Loading histories for the shakedown solutions shown in Fig. 2.9

27

28

Reference Stress Methods - Analysing Safety and Design

Fig. 2.9 Limit load and shakedown limits for the geometry and loading histories shown in Fig. 2.8 and mesh shown in Fig. 2.7 - note that the shakedown limit is identical to the least of the limit load or the reverse plasticity limit - D denotes the plate thickness

Computational Methods for Limit States and Shakedown

Fig. 2.10 Bree problem - a plate or axisymmetric tube is subjected to fluctuating temperature differences and axial stress - mesh geometries of eight noded quadrilateral elements

29

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Reference Stress Methods - Analysing Safety and Design

Fig. 2.11 Shakedown limits for the Bree problem of Fig. 2.10 modelled as both a plane stress plate and an axisymmetric thin walled tube - the solid line is Bree's solution for a Tresca yield condition

Fig. 2.12 Shakedown limits for the problem discussed in Section 2.5 for prescribed values of R. The diamonds correspond to the Bree solution which coincides with the computed solution for R = 1.5. Points A and B refer to the solutions in Fig. 2.13 for R = 0.1

Computational Methods for Limit States and Shakedown

31

Fig. 2.13 Convergence of R to R = 0.1 for the extended shakedown method discussed in Section 2.5. Curves labelled A and B correspond to the convergence to the corresponding points in Fig. 2.12. The slow convergence of case B is due to the dominance of a reverse plasticity mechanism A R S Ponter Department of Engineering, University of Leicester, UK M J Engelhart Airworthiness and Structural Integrity Group, QinetiQ, Farnborough, UK

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3 Limit Loads for Cracked Piping Components D G Moffat

Abstract This Chapter summarizes recent work on the influence of cracks on the limit loads of piping elbows and branch junctions subjected to internal pressure and moment loading.

Notation a A B d D ML M'L ML MLO MP PL PL P"L PLO R r T t Y

crack depth constant constant branch pipe mean diameter elbow or branch run pipe mean diameter limit moment limit moment/plain pipe limit moment limit moment of cracked component/limit moment of uncracked component single load case limit moment twice-elastic-slope plastic load limit pressure limit pressure/plain pipe limit pressure limit pressure of cracked component/limit pressure of uncracked component single load case limit pressure elbow radius pipe mean radius run pipe thickness for branch junctions branch pipe thickness for branch j unctions or elbow thickness uniaxial yield stress

34

Reference Stress Methods - Analysing Safety and Design

P 0 a A,

elbow circumferential defect half angle elbow axial defect subtended angle branch junction crack half angle bend characteristic tR/r2 or branch junction characteristic (d/D) (D/T)0'5

3.1

Introduction

Since the late 1970s, structural assessment procedures for cracked components have evolved and the introduction of the 'two-criteria' approach (1) has matured into the R6 method (2) now used internationally. The latter requires the defected structure to be assessed by a failure assessment diagram (FAD), which is a two-dimensional plot of a fracture mechanics measure against a plastic collapse measure. The former aspect of the problem is not considered here. For the evaluation of the latter measure, knowledge of both applied load and the limit load of the defected structure are required. Limit load solutions for a variety of cracked structures already exist, especially for simpler geometries such as plates and cylinders (3, 4). These were obtained by analytical stress analyses procedures using the Lower Bound Theorem [e.g. (5)]. However, for more complex components such as piping elbows and branch junctions (tees), such data are sparse. Over the past few years, work has been underway on piping system components at The University of Liverpool with the aim of providing data on elbows and branch junctions that will help to fill this gap. An FE parametric study was conducted on short-radius (elbow radius/pipe radius, R/r = 2) piping elbows with internal axial cracks at the crown or circumferential cracks at the intrados, the loading being internal pressure, opening-bending, or a combination of the two (6). The terminology used in the elbow study is shown in Fig. 3.1. In order to provide confidence in the elbow FE modelling procedures, an experimental investigation was conducted on a series of 13 short-radius elbows with cracks and subjected to opening-bending, the latter being defined as shown in Fig. 3.1. The global results are presented in (7) and the local results in (8). The terms 'global' and 'local' will be defined below. During the elbow study, some difficulties were experienced in modelling components with large, deep cracks (a/t = 0.75). To throw some light on this, a study was conducted on axially loaded, plain pipes with fully circumferential, internal, part-penetrating cracks. The FE results are presented in (9) where it was shown that a focused mesh was the most accurate method of evaluating the limit load with a biased, standard mesh a good economic alternative. Some experimental results from good quality components (10) confirmed the accuracy of the full non-linear FE analyses (as distinct from limit load analysis). The tests also indicated that, for these cylindrical, fully circumferentially, cracked components, there was a factor of about 1.6 between the measured, ultimate load-carrying capacity and the calculated FE limit load for all crack depths tested.

Limit Loads for Cracked Piping Components

35

The next stage of the work was concerned with assessing limit loads for cracked, welded piping branch junctions. Reference (11) presents the results from an FE parametric study on d/D = 0.5 junctions subjected to internal pressure, branch out-of-plane bending, or a combination of the two. The FE results have been extended to d/D = 0.95 and experiments have been conducted on two uncracked and three cracked tees. These latter results are presented 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 the experimental work the 'cracks' have been produced using the electric discharge machining (EDM) process. The remainder of this Chapter will focus on work at Liverpool University on elbows and welded branch junctions and will present sample data from these studies. First, however, an attempt will be made to clarify the procedures used to define limit loads and plastic loads.

3.2

Definitions of limit loads and plastic loads

In the ASME III design code (13) and the CEN TC54 unfired pressure vessel draft standard prEN13445, Part 3 on Design (14), there are 'design-by-analysis' (DBA) sections which provide alternative procedures to the 'design-by-formula' (DBF) routes for designing pressure vessel components. In each case the DBA procedures rely on the concept of limit analysis techniques to assess limit loads for components, derived from load versus displacement ('displacement' here meaning any relevant measure of change of shape) plots using numerical analysis procedures. In both design codes the term 'limit load' is used to denote the calculated maximum load found (by analytical or numerical procedures) assuming elastic/perfectly plastic material properties and using small displacement analysis. In ASME III the twice-elastic-slope (TES) method is used to determine what is referred to as a 'collapse load' from a load versus displacement plot determined using full non-linear analysis (i.e. material and geometry) or experimental data. Gerdeen (15) has appealed against the use of the term 'collapse load' since, for pressure vessel structures, the true collapse load is often much higher than the TES load. Gerdeen recommended that the term 'plastic load' be used to define loads obtained using the TES procedure (or equivalent) and this term will be used herein. Gerdeen also recommended that the area under the load versus displacement plot should represent energy (e.g. moment versus rotation or pressure versus volume change) but recognized that this would not always be possible since, for example, volume change is difficult to assess, especially for cracked components. ASME III warns that 'particular care should be given to ensure that the strains or deflections that are used are indicative of the load carrying capacity of the structure,' but does not specify the criteria upon which this decision should be based.

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Reference Stress Methods - Analysing Safety and Design

In the CEN TC54 draft standard (14) the tangent intersection (TI) method is recommended for determining limit loads, as defined above, from load versus displacement plots. In an attempt to remove the subjectivity involved in the TI method, the draft standard states that, 'If there is no maximum (in the load versus displacement plot) in the region of principal strains less than ±5 per cent, the greatest tangent intersection value shall be used with one tangent through the origin, the other through a point where the maximum principal strain does not exceed +5 per cent.' Limit loads determined using the TI method, can be referred to as 'TI limit loads'. The term 'TI plastic load' can be used to define a plastic load obtained from a full non-linear analysis load versus displacement plot and using the TI method. Figure 3.2 illustrates the constructions used to determine the TES and TI plastic loads, LTES and LTI. In Fig. 3.2(b) the load L5 is the load at which the maximum principal strain is 5 per cent. A tangent is drawn to the load versus displacement curve at load L5 and the point at which this intersects the extended elastic line is the TI plastic load LTI or limit load LLTI, as the case may be. In a recent publication (16), the Liverpool group have attempted to assess the merits of these different definitions of plastic load and have confirmed that a range of limit loads and plastic loads can be obtained from the same basic analysis, depending on the definition adopted. This uncertainty concurs with the 1999 EPERC Design by Analysis Manual (17) where, '...the difficulty of extracting meaningful plastic design loads from elastic-plastic finite element analysis' is emphasized. The conclusions in (16) were that, (a) the ASME III TES procedure gives a reasonable estimate of plastic loads but, unfortunately, does not give a unique value for any one component and, (b) the 5 per cent principal strain TI method in CEN TC54 does give a unique value of plastic load but can significantly underestimate limit load plateau values. In this Chapter on cracked components, limit loads are obtained as indicated above and using the 'fifteen-times' or the 'five-times' elastic slope criterion to ensure that a representative plateau value is obtained from the limit analysis load-displacement plot (see Fig. 3.3 as an example of this using the fifteen-times elastic slope criterion). Plastic loads are obtained from the full non-linear FE analysis, or from experimental load-displacement plots, using the ASME III TES procedure [Fig. 3.2(a)], notwithstanding the concerns expressed above. In the R6 manual (2), the terms 'global' limit load and 'local' limit load are used. The former is obtained from load versus displacement plots where the 'displacement' selected is representative of the overall behaviour of the component - for example, diametral growth or nozzle rotation. The term 'local' limit load is only relevant to components with partpenetration cracks and is defined in R6 as, 'the load needed to cause plasticity to spread across the remaining ligament....' In the work on elbows to be presented in the next section, an attempt was made to measure experimental local plastic loads using electrical resistance strain gauges (8). However, the conclusion reached was that this was not likely to be feasible and therefore the only viable method of assessing local limit loads is considered to be FE analysis where the spread of plasticity through the ligament can be assessed. In what follows, any reference to limit loads or plastic loads will imply the adjective global rather than local.

Limit Loads for Cracked Piping Components

3.3

37

Elbows

3.3.1 Defect free elbows The Liverpool group have recently (18) summarized the available literature on limit load assessment of defect free piping elbows. Reference is made to the theoretical work by Spence and Findlay (19), Calladine (20), and Goodall (21) which has led to the development of closed form expressions for the determination of limit loads for undefected pipe bends under in-plane bending. The combined loading case of internal pressure and in-plane bending has been addressed by Goodall (21). More recently, Shalaby and Younan (22, 23) used the FE technique to investigate limit loads of pressurized, defect-free piping elbows under both closing and opening in-plane bending. The limit load was found to increase then decrease with increasing pressure for all elbows for both closing and opening modes of bending. The work by Chattopadhyay et al. (24) involved elastic/strain-hardening FE analyses to evaluate the plastic moments of six elbows (A, = 0.24 to 0.6) under the effect of combined inplane closing/opening bending with a varying level of internal pressure. Plastic moment data was obtained by the TES method from moment versus end rotation plots. Curve fitting was applied to the results to produce two closed form equations for the closing and opening plastic moments. For some of the cases where the elbows were unpressurized, the results were noted to be higher than those predicted from the formulae given in (19, 20, or 21). This can be attributed to the stiffening effect of the connecting straights, material strain hardening, and non-linear geometric effects (see below). In contrast with these analytical assessments, extensive experimental work by Greenstreet (25) was carried out in the US at Oak Ridge National Laboratory. Greenstreet performed room temperature tests on twenty, 6-inch (152 mm) nominal diameter, schedule 40 (7.11 mm thick), 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 (ASTMA106B) and the remaining four were stainless steel (A312-304L). The limit moment expressions mentioned above from Refs (19-21) share the same form and can 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 1.04 (21). If A is taken to be 0.94 and n as 2/3, then it is shown in (18) that equation (3.1) under predicts the available experimental data for opening bending (by Greenstreet (25), Griffiths (26), and Yahiaoui et al. (7) in the range 20-50 per cent, depending on the definition used for the experimental plastic moments. A significant outcome of the above analytical and experimental work on plastic loads (as distinct from limit loads) is that elbows subjected to closing in-plane bending have a lower plastic moment than nominally identical elbows under opening in-plane bending. This is caused by the non-linear ovalizing of the elbow cross-sections where the ovalizing effect weakens or strengthens the elbow depending on whether the applied moment is closing or opening respectively. While the above is true for uncracked elbows, for elbows with circumferential cracks at the intrados or axial cracks at the crown, closing bending causes the

Reference Stress Methods - Analysing Safety and Design

38

cracks to close rather than open as has been observed in reference (26). For this reason, the cracked elbows considered in the next section have been subjected to opening bending. 3.3.2 Cracked elbows The terminology adopted for the Liverpool cracked elbow work is as shown in Fig. 3.1. ABAQUS 20-noded brick elements (C3D20R) were used in the FE study. Four elements through the thickness were found to be adequate for most cases but for long, deep (a/t = 0.75) cracks, additional mesh refinement was needed (6). Figure 3.4 illustrates the deformation mode for an elbow under opening bending with an internal axial crack at the crown (0 = 75°, a/t = 0.75). The results of the elbow limit load parametric study are summarized in Figs 3.5-3.8. Figures 3.5 and 3.6 are for opening bending and pressure respectively for internal crown axial cracks. (Note the false zero on these two plots.) Figures 3.7 and 3.8 are for the same loads for internal intrados circumferential cracks. The conclusion from this data is that part-penetrating defects up to half the thickness deep have little influence on opening bending and pressure limit loads. It is only when the defects are deep or through-wall that their presence significantly reduces limit load levels. Table 3.1 indicates the models used for the experimental elbow work under opening bending loading. The experimental arrangement is shown in Fig. 3.9 and the FE model simulated this set-up. For comparison with the experimental data, the FE models were re-run using full nonlinear analysis (non-linear geometry plus true stress versus strain curve). The true stress versus 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 diameter 88.9 mm; thickness 5.49 mm, bend radius 76.2 mm). Table 3.1 Test elbow details Test elbow identification

E0 (note 2) E1 E2 (note 3) E2A (note 4) E3 E7 E8 E9 E4 E6 E10 E12 E11 Notes: 1. 2. 3. 4.

Defect Type (note 1) A A A A A A A C C C C C

Length (degrees) 0

a/t

2/7

-

-

46 46 120 120 120

0.5 0.5 0.5 1.0 0.5 0.75 1.0 0.5 1.0 0.5 0.75 1.0

21 21 21 21 75 75 75

A = axial (crown), C = circumferential (intrados) Defect free elbow Duplicate test of E1 Duplicate test of E1

Limit Loads for Cracked Piping Components

39

For reasons that will not be explained here, there were three, nominally identical, opening bending tests with a short internal axial crack at the crown (6 = 21 degrees, a/t = 0.5). Figure 3.11 shows the inevitable scatter in the experimental moment versus crosshead displacement data with the FE data passing through the middle of the latter. Figure 3.12 gives the moment versus crosshead displacement data for an intrados crack having 2p = 46 degrees, a/t = 1.0. There is a good agreement up to a moment of about 10 kNm when the test model crack started to grow, after which the experimental moment falls off. Figure 3.13 presents the moment versus displacement plots for an intrados crack with 2(3 = 120 degrees, a/t = 0.75. In this case, the agreement is good up to a moment of about 8 kNm at which stage the ligament failed and the load dropped off. The data in Figs 3.11-3.13 are generally representative of the comparison between the test data and the full, non-linear FE predictions. The fact that there is good 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 Figs 3.5-3.8 was acceptable thus giving confidence in this data. In Figs 3.11-3.13 moment versus displacement curves the twice-elastic-slope FE plastic load is indicated by MP and the FE limit load by MI., the latter being obtained using the yield stress of 328 MPa from Fig. 3.10. In each case the limit moment ML is a conservative estimate of the TES plastic load. This was true of all elbows tested (7). Figure 3.14 presents a summary of the experimental data which again emphasizes how tolerant these short radius piping elbows are to cracks, insofar as plastic/limit loads are concerned.

3.4

Welded branch junctions

3.4.1 Defect free junctions The main concern in the Liverpool work was to assess limit loads of cracked branch junctions. However, prior to embarking on the parametric study of cracked components, a study was completed on uncracked branch junctions subjected to pressure loading. The range of parameters in the study was as follows, with an additional set of models (not listed here) to study the effect of branch/run pipe thickness ratio t/T:

The results of this work will be presented in reference (27) which includes a procedure, based on the results obtained, for calculating the limit pressure of any branch junction in the range defined above. In 1968, Cloud and Rodabaugh (28) produced an analytical expression for estimating the limit pressures PL of uncracked branch junctions. This was presented using a factor, p* , to adjust the Tresca plain pipe limit pressure equation:

40

Reference Stress Methods - Analysing Safety and Design

This compares quite well with the Liverpool data but is somewhat conservative for thickwalled, equal pressure strength (t/T = d/D) junctions. 3.4.2 Cracked branch junctions The terminology used in the cracked branch junction work and the location of the cracks investigated are shown in Fig. 3.15. Sample results from reference (11) are presented here. The branch junctions studied had a branch/run pipe mean diameter ratio d/D = 0.5, a thickness ratio t/T = 1.0, and diameter/thickness ratios D/T of 10, 20, or 30. One of the FE models is shown in Fig. .16 where it can be seen that the FE mesh has been biased towards the crack tip. Only through-wall cracks were investigated and the models had three elements through the thickness in the junction region. Loading was either internal pressure or branch out-of-plane bending, the latter being applied in the direction that would cause crack opening (Fig. 3.17). The global limit load results were obtained from moment versus nozzle end rotation curves. Limit moments were normalized to the branch pipe limit moment d2 tY and limit pressures were normalized to the run pipe limit pressure 2TY/D. These normalized values are referred to here as M'L and P'L, . The normalized limit loads versus crack angle 2a are presented in Fig. 3.18 for the three D/T ratios 10, 20, and 30. Alternatively the limit loads could be normalized on the basis of the uncracked branch junction values taken from the ordinate of Fig. 3.18. These are referred to as M"L and P"L and are presented in Fig. 3.19 where it can be seen that the points collapse nicely on to two lines, representing the two load categories. Not surprisingly, Fig. 3.19 indicates that large, through-wall cracks have a significant weakening effect 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 by varying the ratio of the loads applied. In Fig. 3.20 the non-dimensional parameters ML/MLO and PL/PLO (where MLo and PLO represent the single load case limit moment and pressure) are used to indicate the interaction behaviour for the D/T = 20 case. It is clear that for the uncracked model 20A the interaction diagram may be reasonably assumed to be circular. For model 20D with the longest crack (2a = 140 degrees), the interaction relationship tends towards linear, with the intermediate cracked models (20B, 2a = 49 degrees and 20C, 2a = 95 degrees) distributed evenly between the uncracked case and the 2a = 140 degrees case. Similar results were obtained for D/T = 10 and 30 and for d/D = 0.95 branches (12) .

3.5

Conclusions

Leaving aside the possibility of failure by fast fracture, short radius piping elbows are capable of sustaining quite large defects without dramatic reductions in their limit load capacity. Partpenetrating defects up to half the thickness deep have little influence on pressure and opening bending limit loads. It is only when the defects are deep or through-wall that their presence significantly reduces limit load levels.

Limit Loads for Cracked Piping Components

41

For the d/D = 0.5 piping branch junctions with through-wall cracks at the flank, subjected to internal pressure or out-of-plane branch bending, limit loads are significantly influenced by the crack length. If the limit loads are normalized on the basis of the uncracked limit loads and plotted against crack length, the results converge to single lines for each of the two loads. Limit load interaction diagrams for pressure versus out-of-plane branch bending show that, for the uncracked cases, circular interaction is relevant, but that for increasing crack length, there is a clear trend towards linear interaction.

3.6

Acknowledgements

The work presented here was part sponsored by British Energy on behalf of IMC (British Energy and BNFL) and the UK Engineering and Physical Sciences Research Council. Their support is gratefully acknowledged. My grateful thanks to colleagues Kadda Yahiaoui, Mike Lynch, and Dave Moreton for all their hard work in producing this data - this Chapter is really a summary of their work.

References (1)

Dowling, A. R. and Townley, S. H. A., The effect of defects on structural failure: A two 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, Vol. 1-3, Electric Power Research Institute, EPRINP-6301-D/N14, Palo Alto, California, USA, 1989, 1990, and 1991. (5) Calladine, C. R., Plasticity for Engineers, Ellis Horwood, 1985. (6) Yahiaoui, K., Moffat, D. G., and Moreton, D. N., 'Piping elbows with cracks - Part 1 : A parametric study of the influence of crack size on limit loads due to pressure and opening bending', J. Strain Analysis, Vol. 35, No 1, 2000, 35-46. (7) Yahiaoui, K., Moffat, D. G., and Moreton, D. N., 'Piping elbows with cracks - Part 2: Global finite element and experimental plastic loads under opening bending', Journal of Strain Analysis, Vol. 35, No 1, 2000, 47-57. (8) Yahiaoui, K., Moreton, D. N., and Moffat, D. G., 'Local finite element and experimental limit loads of flawed piping elbows under opening bending', Strain, J. of British Soc. For Strain Measurement, Vol. 36, No 4, 2000, 175-186. (9) Lynch, M. A., Moffat, D. G., Moreton, D. N., and Ainsworth, R. A., 'Limit loads for cylinders with fully circumferential cracks in tension: Comparison of analytical and finite element data', Procs 9th Int. Conf. on Pressure Vessel Technology, Sydney, 2000. (10) Lynch, M. A., Moreton, D. N., and Moffat, D. G., 'Limit loads for axially loaded cylinders having full circumferential cracks: Experimental, analytical and numerical studies', submitted to STRAIN, BSSM. (11) Lynch, M. A., Moffat, D. G., and Moreton, D. N., 'Limit loads for a cracked piping branch junction under pressure and branch out-of-plane bending', Int. J. Pressure Vessels and Piping, Vol. 77, No 4, 2000, 185-194. (12) Lynch, M. A., 'Limit loads of piping branch junctions with cracks', PhD Thesis, The University of Liverpool, 2001

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Reference Stress Methods - Analysing Safety and Design

(13) ASME Boiler and Pressure Vessel Code, Section III, 1995. (14) Draft European Standard, prEN 13445-3, Unfired pressure vessels - Part 3: Design, annex B, section B9, CEN, 1999. (15) Gerdeen, J. C., 'A critical evaluation of plastic behaviour data and a unified definition of plastic loads for pressure components', PVRC Welding Research Council Bulletin 254, section 16, ASME, 1979. (16) Moffat, D. G., Hsieh, M. F., and Lynch, M., 'An assessment of ASME III and CENTC54 methods of determining plastic and limit loads for pressure system components', J. Strain Analysis, IMechE, 2001 Vol. 36, No 3, 301-312. (17) The Design-by Analysis Manual, European Pressure Equipment Research Council, European Commission Joint Research Centre, EUR 19020 EN, Section 1.3, 1999. (18) Yahiaoui, K., Moreton, D. N., and Moffat, D. G., 'Evaluation of limit load data for cracked pipes bends under opening bending and comparison with existing solutions' Int. J. Pressure Vessels and Piping, Vol. 79, 2002, 27-36. (19) Spence, J. and Findlay, G. E., 'Limit loads for pipe bends under in-plane bending',Proc 2nd Int. Conf. on Pressure Vessel Technology, San Antonio, 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 internal pressure and in-plane bending moment', CEGB Report, RD/B/N4360, August 1978. (22) Shalaby, M. A. and Younan, M. Y. A., 'Limit loads for pipe elbows with internal pressure under in-plane closing moment', ASME, Journal of Pressure Vessel Technology, Vol. 120, 1998, 35-42. (23) Shalaby, M. A. and Younan, M. Y. A., 'Limit loads for pipe subjected to in-plane opening bending moments', ASME, Journal of Pressure Vessel Technology, Vol. 121, 1998,17-23. (24) Chattopadhyay, J., Nathani, D. K., Dutta, B. K., and Kushwaha, H. S., 'Closed form collapse moment equations of elbows under combined internal pressure and in-plane bending moment', ASME Journal of Pressure Vessel Technology, Vol. 122, No 4 November 2000,431-436. (25) Greenstreet, W. L., 'Experimental study of plastic responses of pipe elbows', ORNLNUREG Report 24, 1978. (26) Griffiths, J. E., 'The effect of cracks on the limit load of pipe bends under in-plane bending: an experimental study', Int. J. Mech. Sci., Vol. No 2,1979, 119-130. (27) Lynch, M. A., Moreton, D. N., and Moffat, D. G., 'A parametric study of limit loads for piping branch junctions', To be submitted. (28) Cloud, R. L. and Rodabaugh, E. C., 'Approximate analysis of the plastic limit pressure of nozzles in cylindrical shells', Trans. ASME, J. Engng Power, 90, 1968, 171-176.

Limit Loads for Cracked Piping Components

Fig. 3.1 Notation for cracked piping elbows

Fig. 3.2 Constructions used to determine TES and TI plastic loads

43

44

Reference Stress Methods - Analysing Safety and Design

Fig. 3.3 Limit analysis load-displacement plot

Fig. 3.4. Deformation mode of elbow under bending with axial crack at the crown (9 = 75 degrees, a/t = 0.75)

Limit Loads for Cracked Piping Components

Fig. 3.5 Parametric plot of limit moments for elbows with crown axial cracks: opening bending

Fig. 3.6 Parametric plot of limit pressures for elbows with crown axial cracks: internal pressure

45

46

Reference Stress Methods - Analysing Safety and Design

Fig. 3.7 Parametric plot of limit moments for elbows with intrados circumferential cracks: opening bending

Fig. 3.8 Parametric plot of limit pressures for elbows with intrados circumferential cracks: internal pressure

Limit Loads for Cracked Piping Components

Fig. 3.9 Test arrangement

Fig. 3.10 True stress strain curve for elbow material

47

48

Reference Stress Methods - Analysing Safety and Design

Fig. 3.11 Moment versus displacement plots for elbows El, E2, and E2A with internal axial crack at the crown ((6 = 21 degrees, a/t = 0.5)

Fig. 3.12 Moment versus displacement plots for elbow E6 with circumferential through-wall crack at the intrados (2p = 46 degrees, a/t = 1.0)

Limit Loads for Cracked Piping Components

Fig. 3.13 Moment versus displacement plots for elbow El2 with circumferential through-wall crack at the intrados (2p = 120 degrees, a/t = 0.75)

49

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Reference Stress Methods - Analysing Safety and Design

Fig. 3.14 Summary of experimental data: (a) axial defects at the crown; (b) circumferential defects at the intrados

Limit Loads for Cracked Piping Components

Fig. 3.15 (a) Branch junction terminology and (b) weld details

51

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Reference Stress Methods - Analysing Safety and Design

Fig. 3.16 Mesh for cracked model 10D (crack angle = 140 degrees)

Fig. 3.17 Crack opening under OPB loading (model 10C - crack angle = 99 degrees)

Limit Loads for Cracked Piping Components

Fig. 3.18 Variation of normalized limit loads, M'L and PL with increasing crack angle

53

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Reference Stress Methods - Analysing Safety and Design

Fig. 3.19 Variation of limit load ratios, ML" and PL", with increasing crack angle

Limit Loads for Cracked Piping Components

55

Fig. 3.20 Interaction diagram for combined pressure and moment loading for D/T = 20

D G Moffat Department of Engineering, University of Liverpool, UK

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4 Some Aspects of the Application of the Reference Stress Method in the Creep Analysis of Welds T H Hyde and W Sun

Abstract This chapter describes some aspects of the application of the reference stress technique in determining the creep properties for weld materials and in the prediction of the steady-state creep deformations of and stresses in welds. The application of the technique in impression creep testing, which is particularly suitable for determining the creep properties of the HAZ material in welds, is described. A two-bar structure and a three-material, thick-walled, pipe weld are used to illustrate how reference parameters are determined for multi-material structures. These examples are used to demonstrate the general applicability of the technique for the creep deformation analysis of multi-material components. Some limitations of the reference stress approach, for multi-material components, are also described.

Notation A1,A2 areas of bars b, d indenter length and width B, n constants in Norton creep law dim non-dimensional functions of dimensions D, D0 reference multiplier and normalized reference multiplier DN normalized deformation rate (= D0 when a = a^) f, g functions of stress indices and dim h, w, (j> dimensions of impression specimens h0, Ri, T, w0 weld dimensions L1, L2 lengths of bars

58

Reference Stress Methods - Analysing Safety and Design

pi, aax

internal pressure and mean axial stress

p P u' u nom u0

mean indenter pressure force deformation rate and nominal deformation rate radial direction deformation rate of plain pipe on the outer surface

UrO

radial direction deformation rate at position C

a, OR P, r| Ac, Ac sc, e°

stress scaling factor and reference stress parameter conversion factors creep deformation and creep deformation rate creep strain and creep strain rate

s^in sa CT, anom

minimum creep strain rate constant in Norton's law stress and nominal stress

4.1

Introduction

A specific feature of welds is the material inhomogeneity in the weld region. A typical weld in a component consists of parent material (PM), heat-affected zone (HAZ), and weld metal (WM), Fig. 4.1; each of the material zones may have different creep properties from the others. Due to this material mismatch, welds may be particularly susceptible to failure during service at elevated temperature. Welds are metallurgical complex due to the local effects of heat from the welding process and the subsequent effect of this heat on the local metallurgical structures. In addition, welds are often formed with variable dimensions and highly irregular shapes. Furthermore, the effect of system load on the failure behaviour may be significant. For instance, within the low temperature HAZ regions of welds in main steam pipelines, cracking is directly influenced by the local microstructural properties and is usually associated with excessive axial or bending stresses, caused by the system loading. Therefore, the assessment of the high temperature performance and creep rupture behaviour of welds is a multi-disciplinary activity which requires knowledge of service conditions, metallurgical studies, laboratory testing, numerical modelling, etc. In general, the strategy proposed for predicting the failure of welded component is to determine the positions with the most critical stresses within each of the material zones. When the critical stress values are used in conjunction with the damage or rupture data for each specific zone, the position and time at which failure will initiate can be identified. For example, a simple failure prediction method, based on a steady-state approach, uses the peak stress within the failure dominant material zone, with the creep rupture data for each zone in order to determine which gives the lowest failure time in order to determine the failure position and time. During the assessment of high temperature welds, it is necessary to determine both the stresses and the deformation of the various material zones. Because of the complex nature of the problem, analytical solutions can not be obtained for predicting the stresses and deformations under creep conditions, within multi-material components, such as welded structures, and accurate determination of the local stresses, within welds, near the free surfaces

Some Aspects of the Application of the Reference Stress Method in the Creep Analysis of Welds

59

at the material boundaries, is difficult due to the theoretical stress singularity which exists. If accurate material properties are available for each of the material zones, detailed stress analyses, including damage mechanics analyses (1), can be performed in order to determine the stresses, deformations, and failure times for the welded components, using, for example, the finite element (FE) method. If accurate material properties do not exist for all material zones, then the reference stress method is an attractive alternative for estimating the deformation and life. This chapter describes how the reference stress technique can be used to determine some of the important creep properties for the weld materials and in performing steady-state creep deformation analyses of welds. In particular, the application of the reference stress technique in the interpretation of impression creep test data is described; this is particularly useful for determining the creep properties of HAZ materials in welds. A two-bar structure and a threematerial, thick-walled, pipe weld are used to illustrate how the reference parameters are determined for multi-material structures. These examples are used to demonstrate the general applicability of the technique for the creep deformation analysis of multi-material components.

4.2

Mackenzie's method for determining reference parameters

For a component, made of a single material, obeying Norton's power law creep, of the form,

subjected to steady loading, the stress distribution within the component will approach a steady state and the displacement rate, at a point of interest, will approach a steady-state value, A s s . For some components, it is possible to obtain analytical expressions for Ass [e.g. (2, 3)]. These show that the general form of the solution is:

where F1 (n) is a function of the stress index, n, in equation (4.1), F2 (dim) is a function of the component dimensions, and anom is a conveniently determined nominal stress for the component geometry and loading. By introducing an appropriate scaling factor, a, for the nominal stress, equation (4.2) can be rewritten as:

n

Choosing a (= OCR) so that FI (n)/(ctR) is independent (or approximately independent) of n, then equation (4.3) becomes

60

Reference Stress Methods - Analysing Safety and Design n

where OCR is the reference stress parameter, D is the reference multiplier (D = (Fi (n)/(aR) )F2 (dim)) and ec (CTR) is the creep strain rate obtained from a uniaxial creep test at the reference stress,
4.3

Application of the reference stress method to the determination of the creep properties of HAZ material using the impression creep test method

In weld situations, since creep properties are functions of position and the HAZs are narrow (2 to 4 mm), the determination of the material creep properties in these regions, using conventional uniaxial tests, is problematic. Therefore, non-standard test methods have been devised in order to obtain the HAZ creep properties. One of these methods is the so-called impression creep testing technique (7, 8). Impression creep testing involves the application of a steady load, via a flat-ended indenter placed on the surface of a material, at elevated temperature. The displacement-time record from such a test is related to the creep properties of a relatively small volume of material in the immediate vicinity of the indenter. The indenter can be cylindrical or rectangular, and for these types of indenters it has been shown (7, 8) that the reference stress method can be used to convert the mean pressure under the indenter, p, to the corresponding uniaxial stress,CT,i.e.

and the creep displacement, Ac, can be converted to the corresponding uniaxial creep strain, sc, i.e.

where r\ and p are conversion factors (reference parameters) and d is the diameter of the cylindrical indenter or width of rectangular indenter, respectively. The method, which was used to determine the reference parameters r| (= aR) and p (= D/d), for different specimen and

Some Aspects of the Application of the Reference Stress Method in the Creep Analysis of Welds

61

indenter dimensions, is based on MacKenzie's method. The method of determining reference parameters has been fully described previously (7, 8). Using the deformation results from a series of the impression creep tests, subjected to a range of loads, the material properties B and n in equation (4.1) can be conveniently determined. A schematic diagram of the impression creep test using a rectangular indenter and the typical test samples are shown in Fig. 4.2. It should be noted that the impression creep technique is only strictly applicable when the impression deformations obtained during the tests are relatively small, compared with the indenter width and the specimen thickness. Care must be taken to ensure that the contact area between the indenter and the test material is large enough, compared to metallurgical features (e.g. grain size), to ensure that characteristic, bulk properties are obtained. For this reason, a long, rectangular indenter, as indicated in Fig. 4.2(a), rather than a cylindrical indenter, is preferable. As well as the obvious benefit of increasing the contact area, the resultant increase in the applied load levels and the reduction of constraint are of benefit and hence reasonably accurate results are obtained, even when relatively large deformation, up to about 0.2 times the indenter width, are obtained. Experimental results have shown that accurate creep properties can be obtained from impression creep tests for a number of metallic materials (9-12). Figure 4.3 shows examples of the minimum creep strain rate data obtained from uniaxial and impression creep tests for 316 stainless steel at 600 degrees C (9-10) and a 2-l/4CrlMo weld metal at 640 degrees C (11), using a rectangular indenter and test samples of the type shown in Figs 4.2(b) and 4.2(c). It can be seen that the data obtained from the two types of creep tests are in good agreement.

4.4

The reference stress method for multi-material components

4.4.1 Creep deformation of a two-bar structure The analytical solution, derived for a two-material, two-bar structure, Fig. 4.4, is used to illustrate the application of MacKenzie's method for determining the creep deformations in multi-material components for which analytical solutions exist. Assuming that the two materials have the same n-value, but different B values in Norton's law, equation (4.1), the steady-state load point deformation rate is given by (13)

where A0 = A1 + A2. equation (4.6) can be rewritten, i.e.

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Reference Stress Methods - Analysing Safety and Design

where onom = P/A0. According to the reference stress method, if a is chosen so that the right hand side of equation (4.7) is independent of n, then a = OCR and the right hand side of equation (4.7) is related to the reference multiplier, D. In order to assess the applicability of the reference stress method, to multi-material situations, the normalized deformation rate, DN (= A/L ] e' : (ao„„„,)), was plotted against n, with different B1/B2 and L1/L2 ratios, for a range of a-values. Examples for some specific material and dimension combinations are shown in Figs 4.5(a) to 4.5(d). It can be seen from Fig. 4.5 that, in general, approximately linear relationships between log (DN) and n exist, with the non-linearity becoming greater when the term B1L1/B2L2 becomes significantly greater or less than unity, see Figs 4.5(c) and 4.5(d). It can be inferred from Fig. 4.5 that in all of the cases presented, OCR « 1, and therefore, A/L^e 0 (aR
4.5

General formulation of multi-material components

Analytical solutions for the creep of multi-material components can be derived for a number of relatively simple cases, e.g. (13-15). For example, steady-state creep solutions exist for bar structures, compound beams in pure bending and compound thin and thick cylinders under internal pressure (15). For convenience, the Norton creep law of the following form is used:

where eoi and ni (which can be different for each material) are material constants and anom is a conveniently chosen nominal stress. Based on these analytical solutions, a general formulation for the deformations of and stress in multi-material components has been proposed (15). At a position in material i of a component consisting of p materials, the general forms of the solutions for deformation rate, u., and stress, (jj, are:

where g1, g2, ..., gp and f1, f2, ..., fp are functions of n1, n2, ..., np, respectively, and the nondimensional component dimensions, dim, and unom and
Some Aspects of the Application of the Reference Stress Method in the Creep Analysis of Welds

63

When n1 = n2 = ... = np = n, equations (4.9a) and (4.9b) can be rewritten as:-

when a = OCR, the reference stress parameter, then the right hand side of equation (4.10a) is related to the reference multiplier, D. Details of the derivation and application of the general formulation, equation (4.9), have been reported previously (15). In this paper, the simpler form, equation (4.10a), is used to demonstrate the applicability of the reference stress method in predicting the creep deformation of multi-material components. An example of a threematerial pipe weld is used for this purpose.

4.6

Application of the reference stress method for determining the creep deformations of pipe welds

An axisymmetrical thick-walled, main steam, pipe weld, shown schematically in Fig. 4.1, is used to illustrate the applicability of MacKenzie's method to practical, multi-material, welded components. The pipe weld consists of three-material zones, the parent material (PM), heataffected zone (HAZ), and weld metal (WM), and is subjected to an internal pressure, pi, and a mean axial stress, o ax . The steady-state creep deformation rates in the radial direction at the weld centre on the outer surface (position C in material 3, Fig. 4.1) are used to illustrate the application. The dimensions of the pipe weld are: T = 63.5 mm, Rj = 114.3 mm, h0 = 4 mm, and w0 = 23 mm, with a weld interface angle of 15 degrees, see Fig. 4.1. Assuming that the n-values of the three materials are the same, the gi in equation (4.10a) are simply functions of n, and dim. Then, the normalized deformation rate, DN (=u ro /(ct"u 0 )), at the position of interest can be written as

in which gi (i = 1, 2, and 3) are the unknown functions to be determined for the geometry and stress index, n, of the component studied. Knowing the gi values for particular n-values, the Devalue which renders the right hand side of equation (4.11) independent of n, i.e. a = (XR, and hence the normalized reference multiplier, D0 (= u r o /(a R n u o )) can be determined for any

64

Reference Stress Methods - Analysing Safety and Design

£oi / eoj ratios. Then the deformation rates, u r o , under these specific conditions, can be directly determined from equation (4.11). In this particular case, the unom in equation (4.10a) is defined as the radial deformation rate of the plain pipe (PM) on the outer surface, ii o . The g1, g2, and g3 for a range of n-values (3-9) obtained from FE analyses (15) are shown in Table 3.1. By curve fitting to the data in Table 3.1, gi (n) functions can be accurately determined. Thus, using these gi (n) functions in equation (4.11), the normalized deformation rates at this particular position, for any combinations of n-values, and £03 / sa and £03 / £o2 ratios, can be easily determined. Hence, the effects of n-value and £oi ratios on the iiro value can be easily assessed.

Table 4.1 gi values for different n, obtained for the iiro at position C in the three-material pipe weld (15) _N 3 5 7 _9

g1 -0.1135 -0.1641 -0.1501 -0.1545

0.0387 0.0402 0.0356 0.0376

g2

g3

1.0651 1.1237 1.1168 1.1171

Examples of the variations of the normalized deformation rate, DN (=u ro /(ct"u 0 )) with n, for a range of a values, for the three-material pipe weld, with different combination of £o3 / ea and £03 / £02 ratios, are shown in Figs 4.6(a) to 4.6(f). It can be seen from Fig. 4.6 that, in general, approximately linear relationship between log (DN) and n exist, similar to those obtained for the two-bar structure and in all other cases for which analytical solutions exist (15), an a value which renders DN approximately independent of n is obtained; in this case, aR » 1. When the parent and weld materials are the same, very good linearity is obtained, Figs 4.6(c) and 4.6(d). However, as with the two-bar structure, the results show that the relationships become more non-linear as the material mismatch becomes more significant, as shown in Figs 4.6(e) and 4.6(f). The accuracy of the deformation rates, obtained using equation (4.11), can be assessed by comparing the results with the corresponding solutions obtained from FE analyses (using the interpolated gi values). The uro /ii o results, at point C, obtained for n = 4 and 6 for the pipe weld, with two so3 I sol and eal / so2 ratios, are presented in Table 2 [C*R = 1 in equation (4.11) was used for all cases]. In Table 4.2, the uro /uo values were obtained directly from the FE analyses using the corresponding n-values and £03 / sol and £03 / so2 ratios in Table 4.2, while the D0 values were obtained by equation (4.11) using the interpolated gi values, Table 4.1, which were obtained with different n-values and (any) different £03 / eal and s0} I sa2 ratios. It can be seen that the deformation rates are accurately predicted by using equation (4.11).

Some Aspects of the Application of the Reference Stress Method in the Creep Analysis of Welds

65

Table 4.2 Comparison of uro /upvalues at position C obtained from FE analyses and equation (4.11)

D0 n 4 4 6 6

4.7

^03 / £„!

2.0 0.5 2.0 0.5

^03 1 ^02

0.5 2.0 0.5 2.0

(Equ. (4.11)) 1.178 0.898 1.164 0.882

uro/uo (FE)

1.165 0.894 1.155 0.889

Discussion and conclusions

This chapter describes some aspects of the application of the reference stress method, based on MacKenzie's method, for determining the creep properties for welds, and in the prediction of the steady-state creep radial deformation rates of welds. The use of the technique in interpreting the data from impression creep tests has proved to be beneficial for the determination of some of the important creep properties for HAZ materials in welds. Examples, including a two-bar structure and a three-material thick-walled pipe weld, have been used to illustrate the general applicability of the reference stress method for multimaterial components, where, for simplicity, it was assumed that the stress indices in the Norton's law are the same for all of the materials. The results presented can be useful both in gaining a better fundamental understanding of the application of the reference stress method and in practical analyses of deformations of and stresses in multi-material components, such as welds. The results presented show that, in general, the reference parameters can be reasonably accurately obtained if the mismatch between the material properties of a multi-material component are not too large, i.e. a good linear relationship between log (DN) and n can be obtained and therefore, accurate reference parameters can be determined. For the cases with large material mismatches, the results obtained indicate that predictions obtained using the reference stress method may become less accurate. One of the major requirements for the study of high temperature welds is the accurate determination of the creep properties for each of the material zones of the weld. In this respect, the determination of the creep and damage material constants for the HAZ material is particularly difficult. If known, these material constants can be used in conjunction with the FE method to obtain the stresses and deformations in welds and to predict the failure life of welds. In this context, the use of the impression creep test technique is useful. Using the reference approach, some of the material properties can be easily and reasonably accurately obtained by converting impression creep test data to the corresponding uniaxial data. Although material mismatches exist within welds, in many practical cases, efforts are made to minimize the material mismatch by choosing similar weld metal, performing post weld heat treatment, etc. For thick-walled main steam l/2Crl/2Mol/4V: 2 l/4CrlMo pipe welds, the nvalues for the parent, weld, and HAZ materials are generally around 4, and differences in Bvalues, in equation (4.1), within the three materials, are generally less than 20 times [e.g.

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Reference Stress Methods - Analysing Safety and Design

(17)]. This ensures that reasonably accurate predictions can be obtained, using the reference stress method, for the creep deformation of these welds. The general formulations for the stresses in and deformations of simple multi-material components can be directly applied to weld situations. This is particularly useful for simplifying and presenting the results of parametric analyses of welds. The formulation allows the effects of the material properties to be easily assessed and the results of parametric analyses to be presented in compact and easily manageable forms (14). Further investigation is necessary, to take account of situations in which the materials have different n-values, in order to gain a better understanding of the range of applicability of the reference stress method, for multi-material components. Based on the results obtained from the two-bar structure and the three-material pipe weld used in this paper, it can be concluded that: (1) (2)

The reference parameters for multi-material components can be reasonably accurately determined if the material property mismatch is not too great. Using carefully selected nominal stresses, CTnom, and deformation rates, u n o m , the reference stress parameters, UR, were found to be close to unity, for the two cases considered.

Acknowledgement The authors wish to acknowledge EPSRC, British Energy, PowerGen, and Innogy for their support of the research described in this chapter.

References (1) (2)

(3) (4) (5) (6) (7)

Hall, F. R. and Hayhurst, D. R., Continuum damage mechanics modelling of high temperature deformation and failure in a pipe weldment, Proc. R. Soc. London, A443 (1991), pp. 383-403. Anderson, R. G., Gardener, L. R. T., and Hodgkings, W. R., Deformation of uniformly loaded beams obeying complex creep laws, J. Mech. Engng. Sci., Vol. 5 (1963), pp. 238-244. Johnsson, A., An alternative definition of reference stress for creep, IMechE. Conf. Pub. 13 (1973), pp. 205.1-205.7. MacKenzie, A. C., On the use of a single uniaxial test to estimate deformation rates in some structures undergoing creep, Int. J. Mech. Sci., Vol. 10 (1968), pp. 441-453. Sim, R. G., Reference stress concepts in the analysis of the structures during creep, Int. J. Mech. Sci., Vol. 12 (1970), pp. 561-573. Sim, R. G., Evaluation of reference stress for structures subjected to creep, Int. J. Mech. Sci., Vol. 13 (1971), pp. 47-50. Hyde, T. H., Yehia, K. A., and Becker, A. A., Interpretation of impression creep data using a reference stress approach, Int. J. Mech. Sci., Vol. 35 (1993), pp. 451-462.

Some Aspects of the Application of the Reference Stress Method in the Creep Analysis of Welds

(8)

(9) (10) (11)

(12)

(13) (14)

(15)

(16)

(17)

67

Hyde, T. H., Sun, W. and Becker, A. A., Analysis of the impression creep test method using a rectangular indenter for determining the creep properties in welds, Int. J. Mech. Sci., Vol. 38 (1996), pp. 1089-1102. Hyde, T. H., Creep crack growth in 316 stainless steel at 600° C, High Temperature Technology, 6 (1988), pp. 51-61. Sun, W., Creep of service-aged welds, PhD Thesis (1996), University of Nottingham. Hyde, T. H., Sun, W., Becker, A. A., and Williams, J. A., Creep continuum damage constitutive equations for the parent, weld and heat-affected zone materials of a serviceaged l/2Crl/2Mol/4V: 2 l/4CrlMo multi-pass weld at 640° C, J. of Strain Analysis, 32 (1997), pp. 273-285. Hyde, T. H., Sun, W., and Williams, J. A., Creep behaviour of parent, weld and HAZ materials of new, service-aged and repaired l/2Crl/2Mol/4V: 2 l/4CrlMo pipe welds at 640" C, Material at High Temperatures, 16 (1999), pp. 117-129. Hyde, T. H., Yehia, K., and Sun, W., Observation on the creep of two-material structures, J. Strain Analysis, Vol. 31 (1996), pp. 441-461. Hyde, T. H., Sun, W., Tang, A., and Budden, P. J., An inductive procedure for determining the stresses in multi-material components under steady-state creep, J. Strain Analysis, Vol. 35 (2000), pp. 347-357. Hyde, T. H., Sun, W., and Tang, A., A general formulation of the steady-state creep deformation of multi-material components, Proc. of the 4th Int. Conf. on Modern Practice in Stress and Vibration Analysis, September (2000), Nottingham, UK, pp. 481492. Hyde, T. H., Sun, W., and Tang, A., A parametric analysis of stresses in a thick-walled pipe weld during steady-state creep, Proc. 5th Int. Colloquium on Ageing of Materials and Methods for the Assessment of Lifetimes of Engineering Plant, Cape Town, April (1999), pp. 231-246. Coleman, M. C., Parker, J. D., and Walters, D. J., The behaviour of ferritic weldments in thick section l/2Crl/2Mol/4V pipe at elevated temperature, Int. J. Pres. Ves. & Piping, Vol. 18 (1985), pp. 277-310.

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Reference Stress Methods - Analysing Safety and Design

Fig. 4.1 Three-material thick-walled pipe weld model

Fig. 4.2 Impression creep testing and test specimens

Some Aspects of the Application of the Reference Stress Method in the Creep Analysis of Welds

Fig. 4.3 Minimum creep strain rate data obtained for 316 stainless steel at 600 degrees C and 2-l/4CrlMo steel at 640 degrees C

Fig. 4.4 Two-bar structure

69

70

Reference Stress Methods - Analysing Safety and Design

Fig 4.5(a) Variation of the normalized deformation rate, DN, with n, for the two bar structure, for a range of a-values, with B1/B2 = 5 (Ai/Ao = 0.3, A2/A0 = 0.7, and L/L2 = 0.5)

Fig. 4.5(b) Variation of the normalized deformation rate, DN, with n, for the two bar structure, for a range of a-values, with B1/B2 = 0.2(A1/A0 = 0.3, A2/Ao = 0.7, and L1/L2 = 2)

Some Aspects of the Application of the Reference Stress Method in the Creep Analysis of Welds

Fig. 4.5(c) Variation of the normalized deformation rate, DN, with n, for the two bar structure, for a range of a-values, with B1/B2 = 20 (A1/A0 = 0.3, A2/A0 = 0.7, and L1/L2 = 0.5)

Fig. 4.5(d) Variation of the normalized deformation rate, DN, with n, for the two bar structure, for a range of a-values, with B1/B2 = 0.05 (A1/Ao = 0.3, A2/A0 = 0.7, and L,/L2 = 2)

71

72

Reference Stress Methods -Analysing Safety and Design

Fig. 4.6(a) Variation of the normalized deformation rate, DN, with n, for the three-material pipe weld, for a range of a-values, with £o3/ £0] = 5 and £03/£02 = 0.2

Fig. 4.6(b) Variation of the normalized deformation rate, DM. with n, for the three-material pipe weld, for a range of a-values, with £03/ £„, = 0.2 and £o3/ £o2 = 5

Some Aspects of the Application of the Reference Stress Method in the Creep Analysis of Welds

Fig. 4.6(c) Variation of the normalized deformation rate, DN, with n, for the three-material pipe weld, for a range of a-values, with £03/ £0i = 1 and £«3/ £o2 = 0.2

Fig. 4.6(d) Variation of the normalized deformation rate, DN, with n, for the three-material pipe weld, for a range of a-values, with £03 / £0i = 1 and £o3/ £o2 = 5

73

74

Reference Stress Methods - Analysing Safety and Design

Fig. 4.6(e) Variation of the normalized deformation rate, DN, with n, for the three-material pipe weld, for a range of a-values, with £o3/£ol = 20 and £o3/ Eo2 = 0.05

Fig. 4.6(f) Variation of the normalized rate, DN, with n, for the three-material pipe weld, for a range of a-values, with 8o3/£ol = 0.05 and 8o3/£02 = 20 T H Hyde and W Sun University of Nottingham, UK

5 High-temperature Creep Rupture of Low Alloy Ferritic Steel Butt-welded Pipes Subjected to Combined Internal Pressure and End Loadings F Vakili-Tahami, D R Hayhurst, and M T Wong

Abstract Constitutive equations are reviewed and presented for low alloy ferritic steels which undergo creep deformation and damage at high-temperatures. Finite element continuum damage mechanics (CDM) studies have been carried out using these constitutive equations on buttwelded low alloy ferritic steel pipes subjected to combined internal pressure and axial loads at 590 °C and 620 °C. Two dominant modes of failure have been identified: firstly, fusion boundary failure at high stresses; and, secondly, Type IV failure at low stresses. The stress level at which the switch in failure mechanism takes place has been found to be associated with the relative creep resistance and lifetimes, over a wide range of uni-axial stresses, for parent, heat affected zone (HAZ), Type IV, and weld materials. The equi-biaxial stress loading condition (mean diameter stress equal to the axial stress) has been confirmed to be the worst loading condition. For this condition, simple design formulae are proposed for both 590 °C and 620 °C.

Notation A B C D Di G

material constant material constant material constant diameter and creep energy dissipation function constants (i = 1,5) material constant

76

h H H* kc M m n P r s t t T a x 8 e <)> <J> F v a a* a co *P

Reference Stress Methods - Analysing Safety and Design

material constant state variable representing primary creep strain hardening material constant material constant material constant time exponent of Norton's power-law stress exponent of Norton's power-law internal pressure radial distance deviatoric stress time pipe thickness temperature multi-axial stress rupture parameter stress exponent in damage rate and rupture time equations Kronecker delta creep strain material constant creep damage state variable due to ageing energy dissipation rate potential function for power-law formulation multi-axial stress-state sensitivity exponent stress break stress knee stress creep damage state variable due to cavitation energy dissipation rate potential function for power-law formulation

Subscripts 1 principal values of stress e effective or Von Mises value f rupture value int internal value m mean or average value mdh mean diameter hoop value out external value

5.1

Introduction

Design codes [BS 806: 1993 (1), BS 1113: 1989 (2), and BS 5500: 1991(3)] for hightemperature, pressurized welded pipes are based on the uni-axial stress rupture properties of the parent pipe material, and do not account for the effects of the multi-axial stress states which arise from the different mechanical properties of the parent material, weld material, and the associated phases. In these codes, the pipe wall thickness, t, is determined from the thin pressure vessel formulae for the mean diameter hoop stress

High-temperature Creep Rupture of Low Alloy Ferritic Steel Butt-welded Pipes Subjected to Combined Internal Pressure and End Loadings

77

where Dm (= D i n +t) is the mean diameter and, Din and Pint are internal diameter and pressure respectively. Due to these limitations, creep failure in weldments can occur earlier than the expected rupture time. In an attempt to overcome this shortcoming, Hall and Hayhurst (4) developed a CDM-based finite element (FE) solver, DAMAGE XX, which incorporates the physics of the creep deformation and rupture of the individual phases of the weld materials. This approach has been shown to predict successfully the deformation, damage, and failure history of the fullsize pressure vessel weldment tests of Coleman et al. (5). The research highlighted the important role of the difference in the creep characteristics of the weld metal, the heat affected zone (HAZ) material, and the parent material. It was shown that the mismatch between material phases results in a marked redistribution of stress from the weld metal into the HAZ and the parent material. It was also shown that the location of the maximum axial and hoop stresses shifts from the inner surface to the outer surface in all micro-structural regions of the weld during primary-secondary creep. This leads to the initiation and evolution of creep damage in the outer third of the weld metal and the HAZ, that is in the region known as fusion boundary (FB). It was also observed that during the secondary creep stage, the maximum effective stress occurs at the inner surface of the pipe and, the maximum principal stress occurs at the outer surface. Hence, the multi-axial stress rupture criterion, a, has an important role in further stress redistribution, the determination of the damage distribution, and the predicted lifetime. And, depending upon the value of the multi-axial stress rupture criterion, the weldment can show either strengthening or weakening relative to the behaviour of the material phase in which failure occurs. Hayhurst, Dimmer, and Morrison (6) have also observed this phenomenon in the estimation of the creep rupture times for notched bars. In order to investigate the effect of the material properties on the rupture time of butt-welded pressure vessels, Wang and Hayhurst (7) have used the FE CDM solver, DAMAGE XX, to carry out studies on the weldment lifetimes for different combination of mechanical properties of the weld and HAZ materials. At least forty different low-alloy, ferritic-steel, butt-welded pressure vessels were analysed for a constant internal pressure and temperature with the same parent material but with different creep characteristics for the weld and HAZ materials. To this end, a set of normalized material parameters were introduced to define the creep behaviour of the weld metal and the HAZ material relative to the parent metal. The introduction of the normalized material parameters permitted the determination of the constants in the constitutive equations (7). From these studies, it was concluded that an optimal set of weld and HAZ material properties exists which results in an improvement in the lifetime prediction of 30 per cent over that obtained for the initial material property data set. Hence, CDM-based FE methods have been found to be a valuable tool for the analysis of the creep rupture behaviour of the pipe weldments. They may also be used to establish a methodology for the design of weldments. To reduce cost and improve speed, approximate methods have also been proposed by Perrin, Hayhust, and Ainsworth (8) for the analysis of butt-welds. This method is based on the modal method developed by Leckie and Hayhurst (9), which is used to compute the creep rupture lifetime of kinematically determinate structures; this category includes butt-welded internally pressurized pipes. Lifetimes predicted using the modal method have been found to be conservative, on average by 14 per cent, when compared to lifetimes determined using CDM

78

Reference Stress Methods - Analysing Safety and Design

analyses. In addition, the modal method accurately predicts the regions of intense damage. It assumes: • kinematical determinacy; • no damage is accumulated in reaching a stationary state; and • the time to achieve a stationary state is not a significant fraction of the component lifetime. The above assumptions limit the application of the method up to the time at which the first failure occurs. Inspite of this, the method provides a fast, accurate, and simple means for the estimation of the rupture lifetimes for weldments. Since this method accounts for stress redistribution within the weldment due to the different creep properties of the weld and HAZ materials, and to the multi-axial stress rupture criterion of each material, it gives more reliable results when compared with other approximate methods which are based on the reference stress analysis techniques. Hayhurst and Miller (10) have used CDM-based constitutive equations to model the behaviour of the weld, HAZ, Type IV, and parent material in a low alloy ferritic steel weldment operated at 590 °C in a welded pressure vessel connection. They used the equations in DAMAGE XX, to model the creep deformation and multi-axial rupture behaviour of weldments in the axi-symmetric equivalents of the crotch and flank sections of pressurized pipe-work branch connections. It was shown that the crotch section fails in the Type IV region adjacent to the branch, while the flank section fails in the Type IV zone close to the sphere (main pipe). This is due to the low effective stress and high first stress invariant in these regions which themselves are a result of stress redistribution caused by the mismatch in the creep properties of the weld, HAZ, Type IV, and parent materials. The results of these CDM analyses have yet to be compared with the results of vessel tests which are currently in progress. In recent years, a number of experimental studies have investigated the high-temperature creep failure mechanisms for the weldments in ferritic steel steam pipework. These investigations have shown that most high-temperature creep failures take place by the growth of circumferential creep cracks in the HAZ region, adjacent to the parent material. This zone, known as the Type IV region (11, 12), experiences the lowest temperatures during the welding process which are in the range 850-900 °C. At these temperatures, significant coarsening and incomplete dissolution of the carbide precipitates takes place together with the refinement of grain size which, in turn, leads to a significant loss of creep resistance of the metal. CDM techniques have been used by Perrin and Hayhurst (12) to model the deformation and the failure of weldments which fail within the Type IV region. They analysed the creep behaviour of homogenous and circumferencially welded notched bars to determine the values of the multi-axial stress rupture criterion of the Type IV material. The resulting mechanismbased constitutive equations were used in the CDM-based FE solver, DAMAGE XX, to predict the deformation and rupture of a uni-axially loaded crossweld specimen. The computational predictions were shown to be, quantitatively and qualitatively, in good agreement with the experimental results (12). CDM-based creep analyses of the butt-welded, low-alloy, ferritic-steel pipes have been carried out by Hayhurst and Perrin (13) to overcome two major shortcomings of the pioneering work carried out by Hall and Hayhurst (4). These are firstly, the neglect of the Type IV region of the weldment; and, secondly the deficiencies of using a single damage state

High-temperature Creep Rupture of Low Alloy Ferritic Steel Butt-welded Pipes Subjected to Combined Internal Pressure and End Loadings

79

variable. Hence, a set of constitutive equations was introduced which incorporated two state variables: the first, ra, to model the creep cavitation; and, the second, O, to model the coarsening of carbide precipitates. This set of constitutive equations was then used to perform FE creep CDM analyses to predict the failure of the butt-welded pipe subjected to the combined internal pressure and independent end loads at the operating temperature of 620 °C. It was shown that the presence of the end load tends to invoke Type IV failure. Hayhurst and Perrin (13) considered two loading conditions: firstly, a pure internal pressure, which generates stress on the end caps of the vessel with the ratioCTmdh/CTiixiai= 2; and, secondly, a combined internal pressure and additional end load, which results in the ratio of amdh/o'axiai = 1 • In the remainder of this chapter reference is made to the former condition, <Jmdh/o~axiai = 2, for tubes subjected to internal pressure only. This statement is valid for thin tubes; but, is a close approximation for thick tubes. The latter case, o~mdh/o~axiai = 1, was selected since it is an extreme condition considered in design codes. The magnitudes of these loadings were selected to give the same lifetime of 60 000 h in each case when the pipe is constructed entirely of the parent material (no weldment). Hayhurst and Perrin (13) showed that for the welded pipe under pure internal pressure (amdh/CTaxiai = 2), creep damage initiates and evolves along the fusion boundary as a macro-crack in a time of 50 000 h. Hall and Hayhurst (4) have previously observed this type of failure for similar conditions. For the second loading condition (amdi/c^axiai =1), it was observed that creep damage initiates and grows from outer surface along the Type IV region, and forms a creep crack in a lifetime of 27 000 h. The research led to the conclusion that there is a need to accurately characterize the different failure modes and their interplay with respect to the different loading conditions in low-alloy, ferritic-steel, butt-welded pressure pipes. Hence, the objectives of this Chapter are: firstly, to understand and to predict the creep behaviour and failure mechanisms of low-alloy, ferriticsteel weldments for different loading and temperature conditions; and, secondly, to characterize the interplay between different failure modes with respect to the operating temperature, loading conditions, and stress level. An understanding of such conditions will enable the proposal of a set of simple and easy to use design equations which will provide a conservative lifetime for welded pipes as a function of internal pressure and end loadings.

5.2

Outline of the investigation

In this Chapter the creep deformation and failure of a low-alloy, ferritic-steel, butt-welded, pressure steam pipe has been analysed using the FE CDM-based solver, DAMAGE XX. A low alloy steel combination of O.5Cr 0.5Mo 0.25V pipe welded with 2.25Cr IMo weld metal, which have been used extensively in steam pipework in fossil fired and nuclear power stations, have been selected for this study. Two operating temperatures of 590 °C and 620 °C have been selected to take into account the effect of the temperature level on the creep behaviour of the different material phases in the weldment. In addition, to understand the interplay between the different failure modes, a wide range of internal pressures and independent end loads have been used: 0 < (<Jmdh/o"axiai) - OT> which include two design code conditions of practical interests: o-mdh/o-axiai = 2 and, amdh/CTaxiai = 1. Two different sets of CDM-based constitutive equations have been employed to describe the creep behaviour of the low-alloy, ferritic-steel parent material and of each material zone of the weldment at the two operating temperatures. For 590 °C, the hyperbolic-sine stress sensitivity equations (12) have been used. And at 620 °C, the Norton Power Law stress sensitivity

80

Reference Stress Methods - Analysing Safety and Design

equations has been used (5, 13). Each set of constitutive equations incorporate two state variables: one which models the creep constrained cavitation, o>; and the other which models the coarsening of the carbide precipitates, . The objectives of this Chapter are to: • review the creep constitutive equations used for low-alloy, ferritic-steels and provide a framework for the deformation rate potentials used in the equations; • categorize those loading conditions which determine Type IV and fusion boundary failures; • understand the interplay between these two failure modes; • appreciate the underlying/controlling physics which dictate each particular failure mode; • predict the creep lifetimes of weldments using CDM; and, • propose or justify simplified design/analysis methods which provide a reasonably conservative estimation of the lifetime as a function of internal pressure. In the following section, the CDM-based constitutive equations will be presented which will be used to characterize the creep deformation and rupture behaviour of each distinct material region.

5.3

Constitutive equations for creep without damage

5.3.1 Uni-axial relations Following Bailey (14, 15) and Odqvist (16), it will be assumed that high-temperature creep takes place without change in volume of the material. Two uni-axial laws are considered here; firstly the n-power creep law due to Norton (17):

where G and n are materials constants at a given temperature and m is the index introduced by Andrade (18, 19) to describe time hardening encountered in primary creep. The second creep law is the hyperbolic-sine law (20)

where A and B are materials constants at a given temperature. The parameter H is a state variable used to model the change of dislocation density during primary creep. When the material enters secondary creep, H takes its saturation value H*. Equation (5.2) is used to model creep behaviour over narrow ranges of stress while equation (5.3) is frequently used to model creep behaviour from stresses close to zero, to stress levels in excess of the first, timeindependent, yield stress. 5.3.2 Multi-axial relations 5.3.2.1 n-Power law

Odqvist (21) has generalized equation (5.2) to multi-axial conditions using the scalar potential function

High-temperature Creep Rupture of Low Alloy Ferritic Steel Butt-welded Pipes Subjected to Combined Internal Pressure and End Loadings

81

where the effective stress, cre> is given by cre2 = (3 Sij Sij/2), Sij is the deviatoric stress tensor given by Sij- = crij-5y Skk/3, and 8ij is the Kronecker delta and Skk obeys the summation connection. The strain rate equation then becomes

and the energy dissipation rate is given by

5.3.2.2 Hyperbolic-sine law Othman, Hayhurst, and Dyson (22) have generalized equation (5.3) to multi-axial conditions using the scalar potential function

The strain rate equation (5.3) then becomes

and the energy dissipation rate is given by

5.4

Deformation potential functions in the presence of damage

In the presence of damage, the energy dissipation potentials can be rewritten by forming the product of the potential functions for the no-damage conditions, and the values of the relevant state variables. This has been justified by Leckie and Hayhurst (9) who have shown that, to a close first approximation, the damage state variables equally affect all components of the strain rate tensor. Hence for the n-power creep law, equation (5.4), the potential function becomes

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where co is the creep-constrained, cavity growth-state variable, and, O is the ageing state variable; and, for the hyperbolic-sine law (5.7), the potential function becomes

The state variable evolution equations and their stress-state sensitivities, given in later sections, have been derived from a knowledge of the physical mechanisms and empirical formulations based on extensive experimental data.

5.5

Constitutive equations

In this section, constitutive equations are presented for 590 °C hyperbolic-sine stress sensitivity has been used in preference to This selection has been motivated entirely by the availability of by the results of studies carried out at 620 °C by Hayhurst and subjected to combined internal pressure and end load.

and 620 °C. At 590 °C, the the Norton power law form. calibrated equation sets; and Perrin (13) on welded pipes

5.5.1 Material behaviour at 590 °C In low-alloy, ferritic-steels, the Norton power law stress exponent, n, is assumed to be constant over limited ranges of stresses and temperatures. This issue has been addressed by Dyson and McLean (23), and Kowalewski, Hayhurst, and Dyson (24) who have shown that the hyperbolicsine stress function is capable of describing the strain rate behaviour of these materials over a much wider range of stress. Since the material behaviour will be modelled over a wide range of stress, the latter formulation has been used for 590 °C. The set of CDM-based constitutive equations model: hardening mechanism of creep, and two softening mechanisms, which include ageing and cavity initiation and growth. Hence the model incorporates three state variables. The first state variable, H, is used to represent the strain hardening effect attributed to primary creep. Initially, H is zero and, as strain accumulates, it increases to a limiting value of H*. The second state variable, O, describes the coarsening of the carbide precipitates, and is defined from the physics of ageing to vary from zero to unity. The coarsening of the carbide precipitates or ageing leads to a progressive loss in the creep resistance of particle hardened alloys such as ferritic steels. The third state variable, co, represents inter-granular creep constrained cavitation damage and is chosen to vary from zero for the virgin state of metal to o>f at failure (25). The multi-axial form of this set of constitutive equations is

High-temperature Creep Rupture of Low Alloy Ferritic Steel Butt-welded Pipes Subjected to Combined Internal Pressure and End Loadings

83

where a1 is the maximum principal stress. N = 1 when a1 > 0 and, N = 0 whenCTI< 0. The material parameters: A, B, C, h, H*, and kc are constants to be determined from the uni-axial creep behaviour. The parameter v is the multi-axial stress sensitivity index. Hayhurst and Miller (10) using the methods due to Perrin and Hayhurst (26, 27) have determined the material parameters for constitutive equations (5.12) at the temperature of 590 °C and these values are presented in Table 5.1. The approach followed is now briefly explained.

Table 5.1 Material parameters for CDM-based constitutive equations (5.12) for each of the weldment materials of low alloy ferritic steel, O.5Cr O.SMo 0.25V welded with 2.25Cr IMo weld metal at 590 °C Material Parameter

Parent

Weld

HAZ

Type IV

A(h- 1 ) B(MPa -1 ) C(-) h (MPa) H *(-) kc(h-1) vf(-) «,,(-)

2.1618 x 10-9 0.205 24 1.8537 2.4326 x 105 0.5929 9.2273 x 10-5 2.8 1/3

2.5289 x 10-10 0.191 06 1.8537 3.9070 x 104 0.4625 2.7289 x 10-4 2.8 1/3

2.1618 x 1Q-9 0.205 24 1 .8537 2.4326 x 105 0.5929 9.2273 x 10-5 2.8 1/3

6.8568 x 10-8 0.10414 4.5550 1.7441 x 104 0.6500 -4 7.2572 x 10 2.8 1/2

Material constants for the parent material have been determined by using numerical optimization methods to fit the model to experimental data. For this purpose, a functional has been defined and minimized, which is based on the error between the predicted and experimental creep data. The experimental creep data for the cast Ml: O.5Cr 0.5Mo 0.25V ferritic steel used by Perrin and Hayhurst (26) has been collected at constant load over a temperature range of 615-690 °C. These data have been supplemented by uni-axial creep test data reported by Flewitt et al. (28). After determination of the associated activation energies for deformation and damage evolution, the material parameters for parent material at 590 °C have been obtained by extrapolation (26,10). The normalized global creep property ratios (27) have been used to determine the constitutive constants of the non-base weld and Type IV materials. Examination of the HAZs of buttwelds prepared for in-service operation shows that the microstructures vary from fine to coarse grain, with a predominance of the refined structure. This is recognized in the R5 assessment procedure (29) which makes provision for the definition of the ratio of coarse to fine grain microstructure. The purpose of this Chapter is to examine the transition between Type IV and fusion boundary cracking, and only one HAZ microstructure will be considered. The HAZ is assumed to have the same properties as the parent material, since real mixed HAZ microstructures are expected to behave more like fully refined than fully coarse structures. This is justified on two grounds: firstly, the work of Perrin and Hayhurst (12) used

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the assumption that the HAZ and parent materials have the same material properties to accurately predict the results of experiments on cross-weld testpieces; and secondly, the work of Hall and Hayhurst (4) accurately predicted the damage evolution and lifetime of a buttwelded pipe from a knowledge of the uni-axial laboratory data and the multi-axial stress rupture criteria, the material behaviour for the HAZ and parent materials, assessed in terms of minimum creep rates and lifetimes, were within a factor of less than two. Figure 5.1 shows the variation with uni-axial stress of creep rupture lifetimes and the minimum creep strain rates for the low-alloy ferritic-steel at 590 °C predicted using constitutive equations (5.12) together with the material data presented in Table 5.1. It can be seen that for the stress level of below 80 MPa, the Type IV material shows the weakest behaviour both in terms of the damage evolution [cf. Fig. 5.1 (a)] and minimum creep rate [cf. Fig. 5.1(b)]. Perrin and Hayhurst (26) have determined a value for the multi-axial stress state index, v = 2.8 [cf. equation (5.12d)], using the lifetimes, the failure strains, and the distributions of creep damage recorded from multi-axial creep rupture tests carried out by Flewitt et al. (28) at 675 °C. Since, the multi-axial stress state index, v, is not expected to vary significantly with temperature, it is assumed to have the same value, v = 2.8, at 590 °C. In the absence of any multi-axial creep rupture data for the weld materials at 590 °C, the stress index, v, has been assigned the same value as for the parent material (10). These assumptions have been justified using the experimental data recorded by Flewitt et al. (28) and Cane (30). 5.5.2 Material behaviour at 620 °C Coleman et al. (5) have shown that Norton's power law adequately describes the creep behaviour of the low-alloy ferritic-steel. Since the conventional single damage state variable theory (4, 7, 31) is not capable of predicting softening due to carbide coarsening in low-alloy ferritic steels, an additional state variable,
where m, n, G, x, M, (j>, and kc are material constants and a is the multi-axial stress rupture criterion (33). The power law stress sensitivity form of the constitutive equations is an approximation to the hyperbolic-sine stress sensitivity equations (5.12). It can, therefore, only be expected to be valid over a restricted range of stress defined by the laboratory data, and is not appropriate for extrapolation over long times. However, to explain the creep behaviour over a wider stress range, two sets of material constants are used to describe the bi-linear behaviour of the material.

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The values of the material constants for both the low- and high-stress regimes, which are summarized in Table 5.2, have been obtained using the optimization and extrapolation methods developed by Perrin and Hayhurst (34). These methods are based on, and verified using the experimental data for the cast Ml ferritic steel O.SCr 0.5MO 0.25V reported earlier (34). Although most of the data was collected at 635 °C, it covers a wide temperature range of 615-690 °C. Based on these data, Perrin and Hayhurst (34) determined the material constants at 620 °C using extrapolation techniques. The multi-axial stress rupture parameter, a [cf. equation (5.13c)], for the parent material and the HAZ were determined by Perrin and Hayhurst (35) and, Hall and Hayhurst (4) respectively. The multi-axial stress rupture parameter, a, for the Type IV material has been assigned the same value of a as the parent material (36). Figure 5.2 shows the creep rupture lifetime and the minimum creep strain rates for the lowalloy ferritic-steel at 620 °C for different stress levels, obtained using CDM-based constitutive equations (5.13) together with the material data of Table 5.2. It can be seen that for stresses below the level of 40 MPa, the Type IV material is the weakest. However, at this temperature, as may be observed from Fig. 5.2b, the weld material is the weakest in terms of minimum creep rate, regardless of the stress level. Table 5.2 Material parameters for CDM-based constitutive equations (5.13) for each of the weldment materials of low alloy ferritic steel, O.5Cr O.5Mo 0.25V welded with 2.25Cr 1Mo weld metal at 620 °C. The units are stress in MPa, strain in percent and time in hours Weld material

Parent material (7>CT

CT
a m n G X M a i|>

kc

100

-0.2769 4.3523 1.4718 x 4.8237 1.5819* 0.15 4.0741 3.1659 x

10-11 10-13 10-4

100 -0.2881 7.4695 8.8830 x 10-18 7.0673 2.5752 x 10-18 0.15 7.8750 3.1659 x 10-4

a m n G X M a kc

100 -0.2769 4.3523 1.4718 x 4.8237 1.5819 x 0.15 4.0741 3.1659 x

(T>6

10-11 10-13 10-4

a >CT 90 -0.3515 6.1870 3.5085 x 5.0312 4.5729 x 0.4298 6.0354 4.441 3 x

10-9 10-9 10-5

10-14 10"13

10-5

Type IV material

HAZ material

a
a
100 -0.2881 7.4695 8.8830 x 10-18 7.0673 2.5752 x 10-18 0.15 7.8750 3.1659 x 10"

a
CT>CT

120

10-11 10-12 10-3

-0.2852 6.4371 1.5651 x 5.8702 1.1776x 0.15 6.7131 1.0225 x

10-15 10-15 10-3

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Comparison of Figs 5. la and 5.2a indicates that, in respect of lifetimes, the behaviour at the two temperatures show similar relative strengths. However, Figs 5.1b and 5.2b indicate different behaviour patterns. In both cases the parent is the most creep resistant while at 590 °C the Type IV material is the weakest; and, at 620 °C the weld material is the weakest. This different pattern of behaviour is a consequence of the selected constitutive equations; and, the way in which the data has been extrapolated. It is the subject of a more detailed current investigation; however, for the purpose of this chapter both equations will be used to test the sensitivity of the welded pipe behaviour to Type IV and to fusion boundary failure. It is worth pointing out that at both 590 °C and 620 °C, the HAZ is in the fully refined form (36), and can be expected to have the same behaviour as the parent material. Consequently, the HAZ material has been assigned the same material parameters as the parent material. In the next section the geometry of the butt-welded pipe is introduced, together with the internal pressure and end loading conditions.

5.6

Pipe geometry and loading cases

The geometry of the butt-welded steam pipe is shown in Fig. 5.3 together with the different material zones. The welded pipe is constructed from hot drawn O.5Cr O.5Mo 0.25V parent material in the normalized and tempered condition, using 2.25Cr 1Mo weld material. The welds are stress relieved, and are assumed to have zero residual stresses. The deformation and damage evolution of the pressurized pipe has been modelled using the FE CDM solver, DAMAGE XX, and the constitutive equations (5.12) and (5.13). An axi-symmetric representation of the thickwall welded pipe is used, with the geometry given in Fig. 5.3. By virtue of the symmetry, only half of the welded pipe is modelled, and the FE discretization has been achieved using constant strain axi-symmetric triangular elements. The FE mesh has 4392 degrees of freedom and 4202 elements. The weld model is composed of four material regions: the parent, the weld, the heat affected (same properties as parent material), and the Type IV materials (cf. Fig. 5.3). The boundary conditions for the model are specified in Fig. 5.3. The total axial stress, aaxial, which is applied as a boundary condition to the end of the pipe/mesh, is composed of two components given on the right-hand side of the equation

where Dout, and Din are external and internal diameters of the pipe respectively. The first term on the right-hand side in equation (5.14) is due to the effect of the internal pressure acting on the end cap of the pipe, and the second term is due to an independently applied end stress, aend. Four different loading conditions are considered: (a) a uni-axially loaded pipe, aaxial, with amdh = 0; (b) mean diameter hoop stress, amdh, with aaxial = 0. The influence of the internal pressure on the end caps is neutralized by the application of a compressive end load, aend; (c) internal pressure stressing, amdh/<7axial = 2; and, (d) equi-biaxial tensile stressing, amdh/caxial = 1 • This is achieved by application of an additional tensile end stress, cyend.

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87

The loading conditions have been selected to represent a broad spectrum of design conditions; but, it is recognized that the equi-biaxial tension condition (d) is the most severe, and is therefore the worst case considered in the design codes and assessment routes.

5.7

Computational techniques

The CDM-based FE solver, Damage XX, has been used to carry out creep analyses of the butt-welded pipe for the different loading conditions. Because of the features of the fine mesh used, small element-to-element variations in the field variables resulted in an increased stiffness of the governing equations. This, in turn, required increased solution times. In order to reduce the solution times, elemental values of the damage variable, co, were homogenized (Appendix A5.1). The homogenization of the damage values associated with each element, smoothes the damage gradients between adjacent elements. In this way, the stiffness of the differential equations is reduced and consequently, larger time-steps can be used for the time integration. Run-times being typically reduced by a factor of 12. Application of the homogenization algorithm results in creep rupture times which are a factor of 14 per cent longer than the lifetimes obtained without use of the algorithm. The results obtained using the homogenization techniques for each of the loading cases presented have been corrected using at least three unhomogenized solutions spanning the range of the applied stresses. Although lifetimes for the two solution techniques are different, no perceptible differences were observed in the field variables except for damage being smoother and more continuous.

5.8

Presentation of results

Results are presented in this section; firstly, to show the effect of loading condition on the welded pipe lifetime; and, secondly to show how the uni-axial data for Type IV and weld materials influence the lifetimes of the welded pipes. 5.8.1 Effect of loading condition on pipe lifetime 5.8.1.1 Behaviour at 590 °C

Rupture times for the butt-welded pipe at 590 °C are presented in Fig. 5.4. It can be seen that the loading case
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Reference Stress Methods - Analysing Safety and Design

For theCTmdh,only loading case (aaxial = 0), the pipe lifetimes are almost a factor of ten longer than those for the amdh/aaxial = 1 loading case. For stresses below 80 MPa, the lifetimes are similar to those given in Fig. 5.1 (a) for the uni-axial parent material data. 5.8.1.2 Behaviour at 620 °C Rupture times for the butt-welded steam pipe at 620 °C are presented in Fig. 5.5 for different loading conditions. The same pattern of behaviour may be observed as that given for 590 °C in Fig. 5.4. The order of severity of loading cases beingCTmdh/aaxial= 1, 0, 2, <x> with amdh only, i.e. amdh/Oaxial -> °°, having the longest lifetime. For stresses below 40 MPa, the maximum ratio of lifetimes with the lifetime in the amdh/CTaxial = 1 case is approximately four; a figure which is low when compared to the factor often for 590 °C. Comparison of Fig. 5.5 with the corresponding uni-axial data of Fig. 5.2(a) does not reveal the same conditions as for the temperature of 590 °C except that for stresses below 40 MPa, the lifetimes for the loading condition cjmdh only are approximately the same as the parent uniaxial values. 5.8.1.3 Design The data presented in Figs 5.4 and 5.5, for 590 °C and 620 °C respectively, clearly show that the loading condition crmdh/CTaxial = 1 is the worst or least conservative condition. This confirms a procedure which is based on this fact and is well established in design codes. In the following discussions attention will be focused on this loading condition, both for brevity and for its significance in design. 5.8.2

The influence of uni-axial material properties of Type IV and weld materials on lifetimes of welded pipes The effect of the uni-axial mechanical properties of the Type IV and weld materials on welded pipe lifetimes is now examined for both 590 °C and 620 °C. 5.8.2.1 Behaviour at 590 °C Presented in Fig. 5.6(a) is a comparison of uni-axial data for Type IV and weld materials with predicted pipe lifetimes for the loading condition amdh/CTaxial = 1. Comparison of the data leads to the conclusion that the weakening effect shown in the Type IV material data below the stress level of 80 MPa causes a sharp bend in the lifetime-(Tmdh curves. Above
High-temperature Creep Rupture of Low Alloy Ferritic Steel Butt-welded Pipes Subjected to Combined Internal Pressure and End Loadings

89

the weakening effect in the Type IV data below the stress level of 40 MPa causes a bend in the lifetime-amdh curves at approximately amdh = 33 MPa. Above amdh = 33 MPa, the vessel lifetimes are significantly lower than for the weld material data; this is not only because the weld material has the shortest lifetimes, but, also because the minimum creep rates are higher [cf. Fig. 5.2(b)] than for any other material phase. This is analogous to the low stress behaviour at 590 °C where the Type IV material has the shortest lifetimes and largest minimum creep rate. Below CTmdh — 33 MPa, the vessel lifetimes are higher than those given by the Type IV material data. However, the material with the highest minimum creep rates is the weld material, cf. Fig. 5.2(b). The softer weld material clearly causes stresses to redistribute away from the Type IV region, so producing a stress state which causes the Type IV material to have increased lifetimes. Hence, a strengthening effect can be seen at low stress levels [(Fig. 5.6(b)]. For example, the creep lifetime for the butt-welded pipe at 10 MPa with amdh/Oaxial = 1 is 314 475 h whereas the uni-axial rupture time for Type IV material at 10 MPa is estimated to be 178 550 h. 5.8.2.3 Design implications - materials selection It is clear that if either the Type IV or weld materials have the shortest lifetimes and highest minimum creep rates, then this is sufficient to produce stress-states and stress redistribution which results in a weakening of the welded pipe below that which might be expected from the corresponding material uni-axial data. When selecting materials at the design stage, such property combinations should be avoided wherever possible.

5.9

Interplay between fusion boundary and Type IV failure

From Figs 5.1-5.6, one would expect the failure zone to shift from the fusion boundary region at high stresses to the Type IV zone at low stresses. To understand the interplay between fusion boundary and Type IV failure, predicted damage distributions obtained from the FE CDM analyses of welded pipes are presented in Figs 5.7 and 5.8. 5.9.1 Behaviour at 620 °C Figure 5.7(a) shows the damage, 03, distribution close to failure for the pipe at 620 °C with c mdh =10 MPa, forCTmdh,/aaxial= 1• Failure can clearly be seen to have taken place in the Type IV region. Initially the stress components are large at the inner bore of the pipe. Damage rates are therefore high in this region, consequently, stresses redistribute from the inner bore, radially outwards to the outer surface producing more uniform damage distribution across the section. In addition, the difference in the material properties causes more stress redistribution from the weld material into the HAZ and Type IV regions. Hence, the peak stress shifts from the inner bore to a location near to the outer surface in the Type IV zone. This leads to the initiation and growth of damage along the Type IV region from the outer surface inwards. The damage field, co, for the pipe at 620 °C is presented in Fig. 5.7(b) withCTmdh= 40 MPa, for Omdh/CTaxial = 1, close to failure. Comparison of Figs 5.7(a) and 5.7(b) shows that failure shifts from the Type IV region at lower stresses to the fusion boundary zone at higher stresses. For

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Reference Stress Methods - Analysing Safety and Design

the loading condition of crmdh/cJaxial = 1, the transition stress levels are found to be 30-35 MPa for the operating temperature of 620 °C. 5.9.2 Behaviour at 590 °C The behaviour at 590 °C is similar to that at 620 °C except that the transition stress level from fusion boundary failure to Type IV failure is increased to 80-100 MPa, cf. Fig. 5.6(a). 5.9.3 Mixed mode fusion boundary and Type IV failure Figure 5.8 shows the mixed Type IV and fusion boundary failure for both operating temperatures. It may also be observed that additional tensile axial stress promotes the Type IV failure; Fig. 5.8(a) shows dominant Type IV failure at amdh = 20 MPa while Fig. 5.8(b) shows predominant fusion boundary failure at amdh =100 MPa. It must be stressed that the kinematical determinacy of the pipe means that once advanced damage occurs at the outer surface of the pipe in the Type IV region, then failure takes place relatively quickly with only a small fraction of life spent in damage evolution in the fusion boundary region. Figure 5.9 is presented to describe/characterize, in broad terms, the synergistic behaviour of the two failure modes with respect to the loading conditions and the operating temperature. This figure shows the variation of the creep rupture time with respect to the mean diameter hoop stress, CTmdh, and the axial stress,CTaxial,for a butt-welded pipe subjected to a constant combined uniform internal pressure and end load at the two different operating temperatures: 590 °C and 620 °C. In this figure the vertical axis represent the creep lifetime and the colour contours represent the type of weld failure mode. The value of 100 per cent for 0 denotes complete Type IV failure and, the value of 0 per cent for & denotes complete fusion boundary failure. Figure 5.9(a) shows that at 590 °C, below the stress level of 80 MPa, the failure is a complete Type IV mode; and, beyond this level up to 100 MPa, there is a mixed Type IV and fusion boundary failure [also see Figs 5.7(a) and 5.8(a)]. These observations are consistent with the discussion presented in the previous section. The pronounced weakening effect of the Type IV material is the major reason for the dominance of this failure mode at 590 °C. This weakening effect is a result of the susceptibility of the Type IV material to both damage evolution and creep deformation (cf. section 5.8 and Fig. 5.1). On the contrary, as can be seen in Fig. 5.9(b), at 620 °C, complete Type IV failure is limited to stress levels below 15 MPa. Complete fusion boundary failure occurs at stresses above 35 MPa. Between these two stress levels, a mixed fusion boundary and Type IV failure can be observed [cf. Fig. 5.8(b)].

5.10 Effect of multi-axial stress rupture criterion of Type IV material Although the mismatch in the uni-axial secondary creep rates of the constituent materials of the weldment has a major role in the relative behaviour of the weldments at 590 °C and 620 °C (cf. Sections 5.8 and 5.9), some part of the differences between the creep behaviour of the welded vessel at 590 °C and 620 °C can be attributed to the different multi-axial stress rupture criteria employed for each material zone. At 590 °C for the hyperbolic-sine constitutive equations, where v defines the multi-axial stress rupture criterion, each material zone has been assigned the same value of v = 2.8, which corresponds to a value of a = 0.4 0.5, depending on the stress level (27). At 620 °C for the power-law constitutive equations, where a defines the multi-axial stress rupture criterion, the value of a = 0.15 (bias towards the effective stress control for damage growth) has been taken for the parent, HAZ and Type

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91

IV materials, whereas the value of a = 0.4298 has been assigned for the weld material (balance between maximum principal stress and effective stress control for damage growth). The effect of different multi-axial stress rupture criteria for the Type IV material will now be investigated for the loading case Omdh/tfaxial = 1 • It can be seen in Fig. 5.6(a) that at 590 °C, the weldment lifetimes fall below those of the uniaxial Type IV lifetimes, with the difference in lifetimes increasing as the stress reduces. Since the Type IV region with high creep deformation rate [cf. Fig. 5.1(b)] is highly constrained by the adjacent parent and HAZ zones, then the effective stress will be less than maximum principal stress i.e. ai/a e > 1.0 in this region. This fact reveals the important role of the a1/ae term particularly when it is raised to the power of v = 2.8. As the applied load decreases, the Type IV region creeps proportionally faster than other material zones [cf. Fig. 5.1(b)] and this causes higher values of the ratio (J1/ae. Hence, the computed creep lifetimes diverge progressively from those for the Type IV uni-axial data as the stress level reduces. This effect is shown in Fig. 5.6. It can be seen in Fig. 5.6(b) that at 620 °C different behaviour is exhibited, partly because the weld metal creeps the most rapidly (although the Type IV material has the lowest uni-axial lifetime at low stress levels), and partly because the multi-axial rupture criteria for the weld metal is much more biased toward maximum principal stress (a = 0.4298) than that for the Type IV region (a = 0.15). Based on the above reasoning, if a for the Type IV material were increased to a similar value to that used at 590 °C, then the relative lifetimes would be more like those for the 590 °C, especially at low stress level where failure occurs in the Type IV zone. To see the effect of higher values of the multi-axial stress rupture criteria, a, at 620 °C, its value was increased from 0.15 to 0.4298 (i.e. the same value as for the weld metal, which corresponds approximately to v = 2.8). The creep lifetimes obtained using a = 0.4298 for the Type IV material are compared in Table 5.3 with those for the previous solutions. It can be seen that as umdh decreases, with failure taking place in the Type IV region [cf. Fig. 5.9(b)], then the computed creep lifetimes for a = 0.4298 diverge progressively from the lifetimes obtained for lower value of a = 0.15; and, they fall below the uni-axial rupture times for the Type IV material [cf. Fig. 5.6(b)]. Hence, this confirms the assertion made earlier. Table 5.3 Creep lifetime obtained using different values of the multi-axial stress rupture criteria, a, for the Type IV material zone at 620 °C for the loading case o mdh ,/a axial =1• The a value for the weld, the HAZ, and the parent material remained unchanged Lifetime (h) O-axial = Vmdh (MPa)

10 20

40 60

a = 0.4298

a = 0.15

155989 35846 4486 504

314475 67110 4475 505

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Reference Stress Methods - Analysing Safety and Design

These results highlight the importance of the multi-axial stress rupture criteria of all phases of the weldment; and, underscore the need for accurate experimental data.

5.11 Failure at high internal pressures At high pressures for both 590 °C and 620 °C, failure occurs in the parent material due to the large radial creep deformation in the weldment. For brevity, the associated mechanisms will now be discussed for pipes at 590 °C only. The failure mode is shown in Fig. 5.10 where, damage, co, and normalized effective creep strain profiles are presented for a butt-welded pipe at 590 °C with (jmdh = 140 MPa for the loading condition of amdh/aaxial = 2. The superimposed profile denotes the deformed state of the pipe. It can be seen from Fig. 5.10(a) that the damage is localized in the parent pipe. Figure 5.10(b) shows the effective creep strain field with a concentration of strain in the weld region. The material data for 590 °C given in Fig. 5.1, shows that for stresses greater than 100 MPa the parent material has lower lifetimes and higher minimum creep rates than the Type IV material. This means that failure will either take place in the weld or parent material. The weld material zone deforms to take up a more favourable load carrying geometry, and stresses are redistributed from the weld to the parent pipe. Failure takes place in the parent pipe on a 45-degree through-thickness surface where the maximum principal tension stress is high. For vessels operated at 620 °C, the same mechanisms operate and the same explanations hold, except that the stress levels and pressures are different. The failure mechanism reported for higher stress/pressures are mainly of academic interest, since design pressures for pipes used in plant are low enough for lifetimes to be in excess of 105 h, where failure takes place by Type IV failure. For this reason the parent pipe failure mode has been omitted from Fig. 5.9.

5.12 Approximate design formulae The objective of this section is to provide a set of simple design equations which provides a conservative approximation for the creep lifetime of the butt-welded pipe as a function of the internal pressure. This procedure will be presented in two parts, for the operating temperatures: 590 °C and 620 °C. 5.12.1 Butt-welded pipes operated at 590 °C The hyperbolic-sine form of the CDM-based constitutive equations (5.12), which have been employed to describe the uni-axial creep behaviour of the low-alloy ferritic-steel at 590 °C, can not be integrated in closed form. However, the equation may be simplified by assuming H = H* and KC = 0, then after integration (Appendix A5.2) the following approximate relationship may be derived

High-temperature Creep Rupture of Low Alloy Ferritic Steel Butt-welded Pipes Subjected to Combined Internal Pressure and End Loadings

93

where Di (i = 1- 5) are constants which may be evaluated by fitting equation (5.15) to uni-axial experimental data. However, it is assumed that equation (5.15) may be used to describe the mean diameter hoop stress-lifetime behaviour of the welded pipe under the condition Cmdh/cJaxial = 1, which gives the minimum lifetime in comparison with the other loading conditions (cf. Fig. 5.4). The implication is that this loading condition can be used to provide conservative lifetime predictions for all loading cases. In this way the approach will be appropriate for use in design codes. The results of the CDM FE analyses, cf. Sections 5.8 and 5.9, for the welded pipe with amdh/Oaxial = 1 have been used to evaluate the constants D1 (i =1-5) in equation (5.15). These constants are given in Table 5.4. The open circles of Fig. 5.11 show that a satisfactory fit can be made using equation (5.15) over a wide range of stresses, which includes lifetimes of relevance in design of up to 8 x 104 h.

Table 5.4 Coefficients of equation (5.15) used to describe the lifetime in hours as a function of mean diameter hoop stress in MPa for the welded pipe under the loading condition ofCTmdh/<Jaxial= 1 D1

D2 5

2.014 38 x 1Q

D3 4

-5.48 x 10

2

5.97 x 10

D4

D5

-0.7219

1.80 x 104

Since there is a linear relationship between the mean diameter hoop stress crmdh (=P int Dm /2t) and the internal pressure, Pint, of the pipe, equation (5.15) may be rewritten as

where the coefficients Gi (i = 1,5) are given in Table 5.5. Table 5.5 Coefficients of equation (5.16) used to describe the lifetime in hours as a function of internal pressure in MPa for the welded pipe under the loading condition of CTmdh/aaxial =1 G1 1.53083X10 5

G2 -5.48x10 4

G3 1.443X10 3

G4

G5

-4.216

3.082 x 103

Examination of the open circles of Fig. 5.11 and the broken line of Fig. 5.6(a) shows that for stresses amdh ^ a* = 80 MPa the coefficients in equation (5.15) correspond to those for the weld material. For this domain, it may also be shown that a good approximation can be made by using equation (5.15) with D5 = 0. In physical terms this corresponds to fusion boundary failure, cf. Section 5.8 and 5.9; and, for higher values of amdh, to failure in the parent pipe which is controlled by creep deformation in the weld material, cf. Section 5.11.

94

Reference Stress Methods - Analysing Safety and Design

For stress Cmdh< a*, the description provided by equation (5.15) and shown in Fig. 5.11 may be simplified without loss of accuracy by setting D4 = 0. As discussed in Sections 5.8 and 5.9, the physical mechanism of failure is creep constrained cavity growth in the Type IV material region. Because of the stress-state effect created by the different stress level sensitivities of creep rates for the parent and Type IV materials, cf. Fig. 5.1(b), this gives an increasingly weaker weldment as amdh decreases [cf. Fig. 5.6(a)]. It is, therefore, not possible to relate the coefficients Di in equation (5.15) to either parent or Type IV material rupture behaviour, given in Fig. 5.1 (a). As a consequence, it is necessary to carry out the complete creep continuum damage mechanics analysis for the pipe weldment to determine lifetimes. Five data points are required to calibrate the coefficients D1-D5, or, if D4 is neglected for low stresses, then four data points are required. The representation of the data presented in Fig. 5.11 for 590 °C can be achieved using equation (5.16), which can also be used for design and for extrapolation to longer lifetimes. Since it may be possible to carry out the design of a welded pipe by doing less than three CDM analyses, it maybe unnecessary to use equations (5.15) or (5.16). Instead, it will suffice to perform the calculations directly for the cases of interest. However, the power of using these equations is in interpolation and extrapolation involving many cases. 5.12.2 Butt-welded pipes operated at 620 °C The power law form of the CDM-based constitutive equations (5.13), which have been employed to describe the creep behaviour of the low-alloy ferritic-steel at 620 °C, implies that the associated design equations, which relate lifetime to internal pressure, should be of power law form. To enable one to explain the different failure modes, two sets of equations have been proposed for the stresses below and above the break stress level of a* = 33 MPa (cf. Sections 5.8 and 5.9). As was the case for 590 °C, the data for the loading condition of fJmdh/CTaxial = 1 are used as a worst case for the lifetime assessment. These results are used to establish the following set of approximate design relationships

and

or

and

High-temperature Creep Rupture of Low Alloy Ferritic Steel Butt-welded Pipes Subjected to Combined Internal Pressure and End Loadings

95

where
5.13 British Energy R5 assessment procedure For the purpose of weld assessments in Cr Mo V pipe work components, the R5 assessment procedure (29) Volume 1 considers four distinct weldment zones or constituents: (i) (ii) (iii) (iv)

Cr Mo V parent material (unaffected by welding); (ii) parent Type IV zone; (iii) parent heat affected zone (HAZ); and (iv) 2Cr Mo weld metal.

96

Reference Stress Methods - Analysing Safety and Design

In the present case, the properties of the HAZ are assumed to be the same as the parent material, i.e. it is assumed that a refined grain structure is present in the HAZ. Consequently only three zones require consideration. To accomplish a weld assessment using the R5 procedure (29), the following information is required for these zones: • material creep rupture properties; • stress redistribution factor, k, which ensure that the long term compatibility of circumferential creep strains is maintained. For geometries and loadings considered here, where the creep reference stress is dominated by the hoop stress, it is necessary to modify the reference stress (8) by a factor k to ensure creep strain compatibility. This factor accounts for the stress redistribution (e.g. off-loading from weaker weld constituents to the stronger ones) across the weldment. The values of the k factor for hoop and axial stress dominated conditions at 565 °C for all material constituents is taken as unity, with the exception of the case for the hoop stress in the weld constituent for which the k factor is taken as 0.7. Reference to Figs 5.1(b) and 5.2(b) shows that the weld metal creeps faster than the parent material at both 590 °C and 620 °C. This is in line with the R5 assessment procedure which indicates that at 565 °C the weld metal creeps faster than the parent material hence the k factor is 0.7. The above observations on creep deformation may be relevant in providing an explanation of the creep rupture response of the pipes, which are given in Figs 5.6(a) and 5.6(b). At 620 °C [cf. Fig. 5.6(b)], the rupture times for the pipes are broadly in line with the advice given in the R5 assessment procedure; failure times follow the lower of the rupture line for the weld metal and the Type IV region. Note that in the R5 assessment procedure, for amdh/craxial = 1, the value of the axial stress in the weld metal determines the weldment lifetime. At 590 °C, cf. Fig. 5.6(a), the rupture times for the pipes are hard to explain by using the design assessment route R5 in that they fall well short of the rupture line for the Type IV region. There are two possible reasons for this: • the stresses in the Type IV region are much higher due to the lower creep rates in the parent and weld metals; this seems plausible; and, • the multi-axial stress rupture criterion for the Type IV material at 590 °C has been taken as v = 2.8 (approximately equivalent to a = 0.43) whereas in reality it is probably closer to the value of a = 0.15 selected for the temperature of 620 °C (cf. Section 5.8). The studies reported in Section 5.10 clearly highlight the importance of an accurate knowledge of the multi-axial stress rupture criterion.

5.14 Conclusion A review has been presented of the constitutive equations used to model creep deformation and rupture of low-alloy ferritic-steels in the temperature range 590-620 °C. A thermodynamic framework has been provided for the deformation potential theory used in these equations.

High-temperature Creep Rupture of Low Alloy Ferritic Steel Butt-welded Pipes Subjected to Combined Internal Pressure and End Loadings

97

CDM analyses have been performed using the FE CDM-based solver, DAMAGE XX, to predict the high-temperature creep behaviour of low-alloy, ferritic-steel butt-welded steam pipes. The analyses show that the failure mode is strongly dependent upon the operating temperature, loading condition, and the stress level. At high stress/pressure failure takes place in the fusion boundary; and, at low stress/pressure failure occurs in the Type IV region. A failure mechanisms diagram has been generated to enable the determination of the mechanism as a function of stress state, stress level, and temperature. The reason for the switch in failure mechanism at both 590 °C and 620 °C from fusion boundary failure at high stress to Type IV failure at low stress can be traced to the log stresslog lifetime and log stress-log minimum creep strain rate curves for the uni-axial data for all the material phases of the weldment: Parent, HAZ, Type IV, and weld materials. It is the change with stress level of the relative strengths which determines the overall behaviour of butt-welds. When a material phase has both the shortest lifetimes and largest minimum creep rates relative to the other material phases of the weldment then failure occurs in that phase producing a weakening relative to the uni-axial data for corresponding stress levels. The importance is stressed of an accurate knowledge of the multi-axial stress rupture criteria of the constituent phases of the weldment. The constitutive equations with hyperbolic sine stress level sensitivity, used for 590 °C, are appropriate for extrapolation over a wide range of stresses, since they are based on the physics of the governing processes. But, the power law equivalent set of equations, used for 620 °C, is an approximation over a restricted stress range, and care should therefore be exercised in extrapolating between the two equation sets for different temperatures, over a wide range of stresses. Design equations are proposed which are based on the critical equi-biaxial loading condition: CTmdh/CTaxial = 1', their use provides conservative designs. Their formulation and proposed use is based on the physical understanding provided by the detailed results of the CDM analyses.

Acknowledgement The research has been carried out as part of an EPSRC-ERCOS programme under grant GR/M44941; and funding has been provided by British Energy, Barnwood. Interactions with Professor B. F. Dyson, Dr D. W. Dean, Dr D. A. Miller, and Dr I. W. Goodall are gratefully acknowledged. The help of Dr I. J. Perrin in the early planning stages of the research is gratefully appreciated. The authors gratefully acknowledge the work of Mr. Teo Hoon Hong in producing Fig. 5.9 of this Chapter.

98

Reference Stress Methods - Analysing Safety and Design

Appendix A5.1 Homogenization of the damage state variable To reduce the run-times for the DAMAGE XX, elemental values for the damage variation, ct/e), have been homogenized. In this way, the damage gradient between adjacent elements is reduced. For this purpose, an average nodal damage value, COi, for each node (for example node i) is determined by

where co(q) is the elemental damage value for the qth constant stress, strain, and damage element, and N is the total number of elements which are connected to node i. Then an average elemental value, co (q) , is obtained for each element by using three average nodal values associated with that particular element

where COi, COj, and C0k are the average nodal damage values for the three nodes: i, j, and k of the element q. The algorithm is executed each iteration, and the homogenized values of elemental damage values are used to compute the current strain rates. In all other respects the algorithm remains unchanged.

Appendix A5.2 Approximate integration of the constitutive equations at 590 °C To integrate the constitutive equations (5.12) in closed form, the following assumptions have been made. Firstly, the primary creep hardening state variable, H, is assumed to take its steady state value H*; and, secondly, the role of the damage state variable, O, which represents the coarsening of the carbide precipitates, or ageing, is neglected by assuming kc = 0. Therefore, uni-axial form of the constitutive equations (5.12) has been integrated neglecting the coupling with state variables H and . The resultant equation will be approximate, since the coupling between variables can be significant; however, this equation will be used only to identify a suitable functional form required to describe lifetimes. While this assumption is not strictly valid, since ageing takes place over long periods at low stresses, the creep constrained cavity growth parameter, co, dominates; hence, the assumptions permit a good approximation to true lifetimes to be obtained. Therefore, the constitutive equations (5.12) can be rewritten in uniaxial form as

Integration of the above equation between initial and final values of to = 0 at t = 0, and co = cc>f at t = tf respectively, will give the creep rupture time as a function of stress level, a. To

High-temperature Creep Rupture of Low Alloy Ferritic Steel Butt-welded Pipes Subjected to Combined Internal Pressure and End Loadings

99

execute the integration, the transformation u = Q/(l-o>) is introduced where Q (=Ba(l-H*)) is a function of stress. Hence, lifetime, tf, will be

where R (= ACN) is a material constant. The above integral can be rewritten as

where B2k (=^C'2kBi,) are the Bernoulli numbers. The truncation of the above series after the i=o

third term gives

where £ is the truncation error. For convenience, equation (A5.2.4) can be rewritten as a function of stress

where A; (i = 1- 4) are constants. Evaluation of the error term, £, can be achieved by calibration against the experimental data. It was observed that this could be best achieved by describing £, using a term which is linear in stress, this results in the accurate determination of creep lifetimes. Hence, the following form of the equation is used

where Di(i = 1- 5) are constants.

References (1) (2) (3)

British Standards Institution 1993 Specification for design and construction of ferrous piping installations for and in connection with land boilers, BS 806:1993. British Standards Institution 1989 Specification for design and manufacture of watertube steam generating plant (including superheaters, reheaters and steel tube economizers), BS 1113:1989. British Standards Institution 1991 Specification for unfired fusion welded pressure vessels, BS 5500:1991.

100

(4) (5) (6) (7)

(8) (9) (10) (11)

(12) (13) (14) (15) (16) (17) (18) (19) (20) (21) (22)

Reference Stress Methods - Analysing Safety and Design

Hall, F. R. and Hayhurst, D. R. 1991 Continuum damage mechanics modeling of hightemperature deformation and failure in a pipe weldment. Proc. R. Soc. Lond., A 433, 383. Coleman, M. C., Parker, J. D., and Walters, D. J. 1985 The behaviour of ferritic weldments in thick section O.5Cr O.SMo 0.25V pipe at elevated temperature. Int. J. Press. Vessels & Piping, 18, 277. Hayhurst, D. R., Dimmer, P. R., and Morrison, C. J. 1984 Development of continuum damage in the creep rupture of notched bars. Phil. Trans. R. Soc. Lond., A 311, 103. Wang, Z. P. and Hayhurst, D. R. 1994 The use of supercomputer modeling of hightemperature failure in pipe weldments to optimise weld and heat affected zone materials property selection. Proc. R. Soc. Lond., A 446, 127. Perrin, I. J., Hayhurst, D. R, and Ainsworth, R. A. 2000 Approximate creep rupture lifetimes for butt welded ferritic steel pressurised pipes. European J. Mechs A/Solids, 19, 223. Leckie, F. A. and Hayhurst, D. R. 1974 Creep rupture of structures. Proc. R. Soc. Lond., A 340, 323. Hayhurst, D. R. and Miller, D. A. 1998 The use of creep continuum damage mechanics to predict damage evolution and failure in welded vessels. IMechE S539/008,117. Gooch, D. J. and Kimmins, S. T. 1987 Type IV cracking in O.5Cr O.SMo 0.25V/2.25Cr IMo weldments. Editors: Wilshire B., Evans R. W. Proceedings of the Third International Conference on Creep and Fatigue of Engineering Materials and Structure, Swansea. Perrin, I. J. and Hayhurst, D. R. 1999 Continuum damage mechanics analyses of Type IV creep failure in ferritic steel crossweld specimens. Int. J. Press. Vessels & Piping, 76, 599. Hayhurst, D. R. and Perrin, I. J. 1995 CDM analysis of creep rupture in weldments. The ASCE Engineering Mechanics Conference, Boulder, Colorado, USA, May 1995. Published in .Proc. ASCE, 1, 393. Bailey, R. W. 1929 Creep of steel under simple and compound stress and the use of high initial temperature in steam power plants. Trans World Power Conference Tokyo, 3, 1089. Bailey, R. W. 1935 The utilisation of creep test data in engineering design. Proc. IMechE, 131, 260. Odqvist, F. K. G. 1974 Mathematical theory of creep and creep rupture. Oxford Mathematical Monographs, 2nd Edition, Clarendon Press, Oxford. Norton, F. H. 1929 Creep of steel at high temperatures. Me Graw-Hill, New York. Andrade, E. N. da C. 1910 The viscous flow in metals and allied phenomena. Proc. R. Soc. Lond., A 84, 1. Andrade, E. N. da C. 1914 The flow in metals under large constant stresses. Proc. R. Soc. Lond., A 90, 329. Garofalo, F. 1963 Emperical relation defining stress dependence of minimum creep rate in metals. Trans., AIME, 227, 351. Odqvist, F. K. G. 1934 Creep stresses in a rotating disc. Proc. IV International Congress for Applied Mechanics, Cambridge, 228. Othman, A. M., Hayhurst, D. R., and Dyson, B. F. 1993 Skeletal point stress in circumferentially notched tension bars undergoing tertiary creep modelled with physically based constitutive equations. Proc. R. Soc. Lond., A 441, 343.

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(23) Dyson, B. F. and Mclean, M. 2000 The role of micromechanisms quantification in developing creep constitutive equations. Proc. 5th IUTAM Symp. On Creep in Structures, Nagoya, Japan, Kluwer, Academic Press, 14 pages. (24) Kowalewski, Z. L., Hayhurst, D. R., and Dyson, B. F. 1994 Mechanisms based creep constitutive equations for an aluminium alloy. Journal of Strain Analysis, 29, No 4, 309-316. (25) Dyson, B. F. and Gibbons, T. B. 1987 Tertiary creep in nickel-based superalloys: analysis of experimental data and theoretical synergies. Acta Metall., 35, 2355. (26) Perrin, I. J. and Hayhurst, D. R. 1996 A method for the transformation of creep constitutive equations. Int. J. Press. Vessels & Piping 68, 299. (27) Perrin, I. J. and Hayhurst, D. R. 1996 Creep constitutive equations for a O.5Cr O.5Mo 0.25V ferritic steel in the temperature range 600-675 °C. J. Strain Anal., 31, 299. (28) Flewitt, P. E. J., Browne, R. J., Lonsdale, D., Shammas, M. S., and Soo, J. N. 1989 Multiaxial stress in relationship to creep life: evaluation of testing procedures, preliminary assessment of multiaxial stress criterion and strategy for testing. CEGB Report OED/STB(S)/88/0037/R. (29) British Energy Generation Ltd. 1999 Assessment procedure for the high temperature response of structures, R5 Issue 2. (30) Cane, B. J. 1981 Creep fracture of dispersion strengthened low alloy ferritic steels. Acta Metall., 29, 1581. (31) Hayhurst, D. R., Dimmer, P. R., and Chernuka, M. W. 1975 Estimates of the creep rupture lifetime of structures using the finite element method. J. Mech. Phys. Solids, 23, 335. (32) Perrin, I. J. and Hayhurst, D. R. 1994 Physically based creep constitutive equations for a O.5Cr O.5Mo 0.25V ferritic steel at 635°C. UMIST Dept. Mech. Eng. Internal Report DMM.94.23. (33) Hayhurst, D. R. 1972 Creep rupture under multi-axial states of stress. J. Mech. Phys. Solids, 20, 381. (34) Perrin, I. J. and Hayhurst, D. R. 1994 A method for the extrapolation of creep constitutive equations to represent the behaviour at different temperatures and of different materials within the same domain. UMIST Dept. Mech. Eng. Internal Report DMM.94.24. (35) Perrin, I. J. and Hayhurst, D. R. 1994 Computational damage modelling of multi-axial creep tests to determine the multi-axial stress criteria for a O.5Cr O.5Mo 0.25V ferritic steel. UMIST Dept. Mech. Eng. Internal Report DMM.94.25. (36) Perrin, I. J. 1995 Computer-based Type IV creep CDM design of low alloy ferritic steel weldments. Ph.D. Thesis. UMIST.

102

Reference Stress Methods - Analysing Safety and Design

Fig. 5.1 (a) Variation in uni-axial stress with lifetime for low-alloy ferritic-steel, O.5Cr O.5Mo 0.25V, parent, Type IV, and weld material of2.25CrlMoat590°C

Fig. 5.1(b) Variation in uni-axial minimum creep strain rate (% h-1 ) with stress level for low-alloy ferritic-steel, O.5Cr O.5Mo 0.25V, parent, Type IV, and weld material of 2.25Cr 1Mo at 590 °C

High-temperature Creep Rupture of Low Alloy Ferritic Steel Butt-welded Pipes Subjected to Combined Internal Pressure and End Loadings

Fig. 5.2(a) Variation in uni-axial stress with lifetime for low-alloy ferritic-steel, O.SCr O.SMo 0.25V, parent, Type IV, and weld material of2.25Cr!Moat620°C

Fig. 5.2(b) Variation in uni-axial minimum creep strain rate (% h ) with stress level for low-alloy ferritic-steel, O.SCr O.SMo 0.25V, parent, Type IV, and weld material of 2.25O IMo at 620 °C

103

104

Reference Stress Methods - Analysing Safety and Design

Fig. 5.3 Dimensions and loading conditions for the low-alloy ferritic-steel steam pipe, 0.5Cr O.5Mo 0.25V, welded with 2.25Cr 1Mo fall dimensions are in mm)

High-temperature Creep Rupture of Low Alloy Ferritic Steel Butt-welded Pipes Subjected to Combined Internal Pressure and End Loadings

Fig. 5.4 Variation of lifetime with stress level for a steam pipe, O.5Cr O.5Mo 0.25V, welded with 2.25Cr 1Mo at 590 °C

Fig. 5.5 Variation of lifetime with stress level for a steam pipe, O.5Cr O.5Mo 0.25V, welded with 2.25Cr 1Mo at 620 °C

105

106

Reference Stress Methods - Analysing Safety and Design

Fig. 5.6 Comparison of the variation of lifetime of butt-welded pipes with the mean diameter hoop stress, omdh, for the loading condition CTmdh/Oaxial = 1, with uni-axial data for the Type IV and weld materials. The solid and broken lines denote the uni-axial lifetime for the Type IV and weld materials respectively (a) open circles denote the predicted pipe lifetimes at 590 °C; and, (b) solid circles denote the predicted pipe lifetimes at 620 °C

High-temperature Creep Rupture of Low Alloy Ferritic Steel Butt-welded Pipes Subjected to Combined Internal Pressure and End Loadings

Fig. 5.7(a) Damage field, o, for a weldment at 620 °C withCTmdh= 10 MPa, and amdh/aaxial = 1, close to failure at 314 467 h showing Type IV failure

Fig. 5.7(b) Damage field, CD, for a weldment at 620 °C with amdh = 40 MPa, and omdh/CTaxial = 1, close to failure at 4475 h showing fusion boundary failure

107

108

Reference Stress Methods - Analysing Safety and Design

Fig. 5.8(a) Damage field, to, for a weldment at 620 °C with omdh = 20 MPa and tr mdh /O axial = 1, close to failure at 67 110 h showing mixed Type IV and fusion boundary failure

CO

Fig. 5.8(b) Damage field, co, for a weldment at 590 °C with (Tmdh = 100 MPa and amdh/cvaxial = 1, close to failure at 2375 h showing mixed Type IV and fusion boundary failure

« a.

—<
aill

« S* $

J

•S'-Ss* 411® * " s?

§•« ~ "• O

S a 3 w isW5 SC« "t2° «r *-

Hli

•= 2 O > ^ « "8 °o ^

1 IS S3 1 H

.a za *-^ «>» •B h a " -a « •§ S &^ S & S E3 ftU o ji

J3 _ °

o ^

U

B E

u •" ^

a ^3 o *> a S S ^ £ -S

E |§ |

2|«© *•

S

B

Tr,

s 1512 £ 2 is 2

-

E

c *5 r o

•S^ "«

3

1*1 eg ^

>• i"§ a

WD S I £

«! ^bs S 2 >r> " « •M U w ft,

E £ S o

109

110

Reference Stress Methods - Analysing Safety and Design

(a)

(b)

Fig. 5.10 Damage field, o>, (a) and, normalised effective creep strain, X e (= Ee/e0) (b), for a pipe at 590°C, with amdh = 140 MPa, for loading condition amdh/A*axial = 2, close to failure at 1629 h. The superimposed profile denotes the deformed state of the pipe. Figure shows that the localized damage field (a) and, the effective creep strain field (b) are compatible with this profile

High-temperature Creep Rupture of Low Alloy Ferritic Steel Butt-welded Pipes Subjected to Combined Internal Pressure and End Loadings

111

Fig. 5.11 Comparison of the variation of the predicted lifetimes of butt-welded pipes, symbols, with the mean diameter hoop stress,
© The Royal Society 2002

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6 Code Application - Below the Creep Range A R Dowling

Abstract This Chapter outlines the influence that reference stress techniques have had on the development of defect assessment methods for structures operating below creep temperatures. Early developments of low-temperature defect assessment procedures were largely empirical. Although this produced relatively straightforward assessment processes, the theoretical underpinning was limited. Reference stress methods provided a bridge between the complex technical descriptions of the fracture process and the necessarily simplified practical procedure. Their introduction also allowed for a confident expansion of the conditions which could be realistically considered in an assessment, especially the treatment of residual and thermal stresses.

Notation COD E J Je Kr KIP KI s Kjc Kmat Lf Lk Lr Lu

crack opening displacement Young's modulus J integral, elastic plastic crack tip characterizing parameter linear elastically calculated value of J fracture parameter in R6, proximity to fracture elastic stress intensity factor due toCTPstresses elastic stress intensity factor due to as stresses fracture toughness material fracture toughness failure parameter in two-criteria approach fracture proximity parameter in two-criteria approach failure parameter in R6, proximity to collapse collapse proximity parameter in two-criteria approach

114

Reference Stress Methods - Analysing Safety and Design

Sr V 8 ey sref o a1 aref ay op orefS os p

failure parameter in R6 up to Revision 2 plasticity correction for op stresses crack opening displacement yield strain reference strain stress limit stress reference stress yield stress stress arising from loads which contribute to plastic collapse reference value of os stress stress arising from loads which do not contribute to plastic collapse plasticity correction for as stresses

6.1

Introduction

The phrase 'reference stress techniques' has enough mystique associated with it to scare off a large number of practising engineers. However, ask those same engineers about design codes and they will quite happily talk about yield and ultimate stresses and the derived design and allowable stresses. This is the simplest type of reference stress technique and it has been used for many decades in many spheres of engineering. However, when it comes to defect assessment procedures, the main topic of this Chapter, the idea of a reference stress method was only knowingly introduced a significant time after the procedures had been widely applied. The idea of defect assessment in engineering structures evolved from linear elastic fracture mechanics that was developed to explain the observed behaviour of brittle materials. The ruling material property in this case was fracture toughness and this could be related to a function involving loading (stress) and defect size. It was not related to any material reference stress. Thus developers of defect assessment methods applicable to more ductile materials were not minded to connect reference stress techniques to the behaviour of structures containing defects. In this Chapter, the early development of post yield fracture mechanics procedures will be described, indicating that there were already some clues to possible relevant reference stresses. These early developments were primarily empirical and it was only when more sophisticated theoretical underpinning was attempted that more specific reference stress techniques were adopted in R6 Revision 3 (1). The current state of the most influential defect assessment procedures will be outlined and the Chapter will conclude by relating the expected future direction.

6.2

PD6493 the predecessor of BS7910

PD 6493: 1980 (2) was derived from the COD (crack opening displacement) design curve developed at the Welding Institute (3). The interest was in moving away from linear elastic fracture mechanics. Most structural metals did not exhibit brittle behaviour in service, in reality they were designed not to do so. The stimulation for the Welding Institute work was to investigate the effects of post-weld heat-treatment on fracture behaviour. It had long been

Code Application - Below the Creep Range

115

known that welds were the most likely sites of any significant defects and that expensive and time consuming post-weld heat-treatment improved structural behaviour. Linear elastic fracture mechanics was unable to explain this adequately so a controlled experimental programme of wide plate tests was commissioned. For these wide plate tests, shown schematically in Fig. 6.1, steel plates around a metre in width were welded together. Different geometries of defect were introduced into the weld. Some of the assemblies were subject to stress-relief heat-treatment, some were tested as received. The intention was to quantify the effect that residual stresses had on the load carrying capacity of the assembly. No attempt was made to categorize the behaviour in terms of residual stress distribution or magnitude, except to assume that yield level stresses were present. The outcome of this was the development of a COD design curve, Fig. 6.2, where the fracture parameter was the COD and the severity of load was measured in terms of applied strain, normalized to the material strain at the yield point. This was, in effect, a reference strain which could be related to the material yield stress if linear elastic conditions were assumed. The particular conditions to be assessed could in this way be related to the required toughness, in COD terms, which would be sufficient to prevent fracture due to the particular defect of interest. It should be noted that some particular locations and conditions have been highlighted on Fig. 6.2. These give some indication of the trends in toughness required to prevent failure, for example, the loading conditions at a nozzle are more aggressive than in a plain shell region and the use of stress relief improves the inherent integrity of a welded structure.

6.3

Two-criteria approach the predecessor of R6

The COD design curve was of little additional help in the electricity supply industry at the time it was developed since all major safety critical components were fully stress relieved. Nevertheless it was of prime importance to understand the effect that defects would have on structural integrity. Another empirical study was put in place, not this time by commissioning new tests but by studying the literature to see if structural behaviour, rather than simply material properties, was influential in describing the observed fracture behaviour of structures operating with relatively ductile materials The structures analysed all contained known defects and for all of them it was clear that failure was initiated at the location of the defect. The analysis was undertaken on the assumption that failure was governed either by the linear elastic fracture mechanics behaviour relevant to brittle materials or that the role of the defect was to reduce the plastic collapse load of the structure. It was clear that structural behaviour was not quite as simple as that and that there was a transition between those extreme behaviours. In order to present the information, the ultimate collapse load of the defective structure was used to normalize both axes of the figure. Thus was spawned the two-criteria approach (4). Figure 6.3 is taken from that reference. It is a very short step to relate the ultimate collapse load to a specific stress, related to the material ultimate tensile stress. Again without realizing it a reference stress was being introduced into defect assessment methods. The two-criteria approach evolved of course, into the first R6.

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The significant breakthrough for R6 was the development of a practical and useful failure assessment diagram. This was achieved by providing, in calculational terms, a complete separation of linear elastic behaviour from plastic collapse behaviour while retaining the requirement to assess both, hi doing this the axes were specified as K r and Sr where

and

and the failure assessment diagram was formulated as

This curve represented the boundary within which failure avoidance conditions could be demonstrated. Thus in redefining the axes the role of a reference stress was obscured. It was not until the development of Revision 3 of R6 (1) that the phrase 'reference stress techniques' became inextricably linked to defect assessment methods for by then it played a fundamental role in the theoretical understanding.

6.4

The birth of R6 Revision 3

By the time Revision 3 of R6 was being considered, the J fracture parameter had reached widespread acceptance as a material parameter capable of use as a measure of resistance to ductile fracture (5). The two world leading post-yield fracture mechanics assessment methods described above were empirically based. It was therefore important to assess these for their compatibility with the new J parameter. The relationship of J to K under elastic conditions was well known and J could also be related to COD through a geometrical parameter. The usefulness of J directly as a fracture parameter was severely hampered by the complicated calculations necessary for its determination in anything other than simple loadings in simple structures. GE-EPRI attempted to overcome this difficulty by assembling a handbook of reference solutions (6) for a number of common geometries under well-defined loadings. Note that these were reference solutions, nothing to do with reference stresses. The main problem then was relating practical problems to these relatively few reference conditions. Ainsworth (7) identified a way through the difficulties, proposing a method of adapting R6 to become the first defect assessment method based fundamentally on reference stress methods. The GE-EPRI scheme was founded on the premise that material stress strain behaviour could be well described by a power law equation, usually referred to as a Ramberg-Osgood

Code Application - Below the Creep Range

117

formulation. This was not always a good fit. Ainsworth was able to reformulate the relevant equations so that the actual material stress strain data could be used. In addition he was able to employ simple approximations that significantly reduce the geometrical dependence of the J solutions. These approximations were adopted to err on the side of safety in the sense that J was overestimated. The result was a new failure assessment diagram which could be related to the existing R6 diagram but which could be formulated as

sref is a reference strain obtained from the true stress strain curve for the material at a reference stress of aref = LrOy

CTP are the stresses contributing to plastic collapse. The material reference point was therefore the yield stress, but the relevant reference stress was a function of the parameter L-. Other minor refinements were made to the above equation to provide a smooth transition between elastic, small scale plastic, and fully plastic behaviours so that the preferred Option 2 curve in the new R6 became formulated as

Equations (6.1-6.3) represent the limiting failure criterion in terms of Kr which is directly equivalent to (Je/J)'/2 in terms of J. Equation (6.3) is shown in Fig. 6.4. There was now, for the first time, a practical defect assessment tool that was well supported by theoretical studies as well as having a large experimental validation pedigree. A major advance made possible by the introduction of reference stress techniques was the ability to deal properly with combinations of secondary and primary stresses. It was already appreciated that these had different influences on gross structural behaviour, but the differences were exaggerated when defects were present. True, secondary stresses do not influence plastic collapse, but they can and do affect the onset of plasticity in a structure. This means that equation (6.3) which is a function only of material tensile properties, and Lr can not properly describe the limiting condition for a structure with and without residual and/or thermal stresses. This is because such stresses do not affect Lr but their presence can certainly affect the build up of strain towards sref. Reference stress techniques were, however, able to establish an effective mechanical load which gave the same J or K as any combination of thermal and residual stresses and mechanical load. Figure 6.5 shows a typical variation in J for a structure with mechanical load both with and without secondary loads. Since the secondary stresses do not affect Lr, the effective mechanical load can be effectively realized as a vertical shift in the J curve or the associated failure assessment diagram. In the latter case this shift is shown as p in Fig. 6.5.

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Reference Stress Methods - Analysing Safety and Design

Figure 6.5 is a simple diagrammatic representation of the principle behind the use of p as a plasticity correction factor. When accurate calculations are performed for different relative magnitudes of o p and os stresses over the full range of L, values, a family of possible 'shifts' or differences between the appropriate failure assessment diagrams, is derived as shown in Fig. 6.6. Within R6 it was considered not to be very helpful to reproduce the family of curves so the p factor was simplified to overestimate the resulting differences between failure assessment diagrams. The recommended p factor is therefore represented by a bounding dashed line as in Fig. 6.6. The p factor is used by modifying the very simple definition of Kr presented earlier so that it becomes

where Kj" and K^ are the elastic stress intensity factors due to stresses which do and do not contribute to collapse respectively and Kmat is the material fracture toughness.

6.5

Current European defect assessment procedures

The fundamental principles behind the strain-based COD approach and the stress based R6 approach were clearly closely related both through the J formulation and in consequence by the consideration of reference stress concepts. The first major revision of PD 6493 was issued in 1991 and included a number of R6 principles including the failure assessment diagram concept. This trend has continued in the latest version that has been issued as BS 7910 (8). This now includes most of R6 including a number of its appendices while retaining the COD approach adapted to fit the failure assessment diagram concept. Following the issue of Revision 3, R6 continued to be developed, primarily by the publication of appendices. These refined the application of the procedure and allowed advantage to be taken of some of the conservatism inherent in the generalized procedure. In the wider European context there are a number of national approaches, many based on BS 7910 (8) or R6 and some also referring to GE-EPRI, ASME, or a direct crack driving force parameter such as J. A Brite-Euram initiative formed a part European Union funded project between parties in UK, France, Sweden, Germany, Spain, Netherlands, Ireland, Finland, and Belgium with the objective of developing a unified European defect assessment procedure. The project was known as SINTAP and it reported in 1999 (9). The international pedigree of SINTAP has focused attention on its findings. For example it introduced a number of new failure assessment diagrams to assist users when specific data is limited. It explicitly presents a crack driving force method alongside the failure assessment diagram method and it has suggested using a multiplying parameter, V, to allow for plasticity effects in the presence of secondary stresses as an alternative to p.

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119

These improvements in SINTAP coupled with the long period since the last major revision of R6 stimulated a new Revision of R6 (10). This adopted, from SINTAP, the new failure assessment diagrams, including a reformulation of the default Option 1 diagram, and the alternative V formulation for assessing defects in the presence of secondary stresses. The alternative V formulation for assessing defects in the presence of secondary stresses provides for calculating Kr as

This formulation provides for a better rationalization of the effects of thermal and residual stresses in that V applies directly to the secondary stress term and only that term. Whether V is greater or less than one immediately indicates whether the effect of the secondary stress is enhanced or relaxed by local plasticity. Nevertheless the values of V and p are derived from the same calculations so the recommended values in the SINTAP and R6 procedures are fully consistent and compatible. The major current procedures are therefore R6, BS 7910, and SINTAP. There is a certain amount of overlap among them and each has something the others do not. All use some form of failure assessment diagram of a form introduced by R6 and hence all can be underpinned by referring to reference stress methods.

6.6

The way forward

R6 development is continuing targeted primarily at reducing remaining conservatism in the procedures, extending the valid regime of alternative methods, and improving and extending validation. The re-ordering of the document for Revision 4 into five main chapters provides a more practical basis for including future modifications since the number of cross links within the document has been significantly reduced. The opportunity was taken with Revision 4 to include the main reference compendia, e.g. stress intensity factors, limit load solutions, residual stress distribution within the document. These can be updated as required. CEN Technical Committee 121, Working Group 14 has proposed that BS 7910 be adopted as the basis of a European Technical Report as a precursor to the formation of a Work Item for the development of a European Standard. This proposal has not yet been accepted by the parent committee and the timetable for acceptance is currently unknown. As a result of this uncertainty the British Standards Committee decided to initiate a programme to refine and revise BS 7910 over a target five-year period to take account of SINTAP and R6 developments specifically and any other relevant developments in the technical area. The target date for a revised document is 2006. A recent initiative in the standardization of defect assessment procedures has seen the formation of a European network, FITNET. The principal objective is the development of a European defect assessment standard suitable for a variety of industries. The network is expected to report in 2005.

120

6.7

Reference Stress Methods - Analysing Safety and Design

Conclusion

Reference stress techniques have assisted the development of defect assessment procedures by providing a theoretical foundation to empirical methods. By providing a common basis for comparison they have also assisted in the harmonization of different methods so that there is now widespread commonality in the treatment of defects in structures and quantifying their effect on integrity.

Acknowledgement This Chapter is published with the permission of BNFL Magnox Generation Business Group.

References (1)

R6 Revision 3, Assessment of the integrity of structures containing defects, British Energy Generation Ltd, Amendment 11, August 2000. (2) BS PD 6493: 1980, Guidance on some methods for the derivation of acceptance levels for defects in fusion welded joints, British Standards Institution, London, 1980. (3) Burdekin, F. M. and Dawes, M. G., Practical use of linear elastic and yielding fracture mechanics with particular reference to pressure vessels, Inst. Mech. Engrs, Paper C5/71, 1971. (4) Bowling, A. R. and Townley, C. H. A., The effects of defects on structural failure: a two-criteria approach, Int. J. Pres. Ves. Piping Vol. 3, 1975. (5) Hutchinson, J. W. and Paris, P. C., Stability analysis of J-controlled crack growth in elastic-plastic fracture (eds Lander, J. D., Begley, J. A., and Clarke, G. A.) ASTM STP 668,31(1979). (6) Kumar, V., German, M. D., and Shin, C. F., An engineering approach for elasticplastic fracture, EPRI Report NP1931, 1981. (7) Ainsworth, R. A., The assessment of defects in structures of strain hardening material, Engng Fract. Mech., Vol. 19, 1984. (8) BS 7910: 1999, Guide on methods for assessing the acceptability of flaws in fusion welded structures, British Standards Institution, London, 1999. (9) SINTAP (Structural Integrity Assessment Procedures for European Industry), Project No BE95-1426, Final Procedure, November 1999. (10) R6 Revision 4, Assessment of the integrity of structures containing defects, British Energy Generation Limited, April 2001.

Code Application - Below the Creep Range

Fig. 6.1 Schematic of wide plate tests

Fig. 6.2 Crack opening displacement design curve from reference (3)

121

Fig. 6.3 Two -criteria failure diagram from Reference (4)

Fig. 6.4 R6 revision 3 failure assessment diagram

Fig. 6.5 Schematic illustration of the R6 plasticity correction term

Code Application - Below the Creep Range

Fig. 6.6 The variation of p with different loading combinations A R Dowling BNFL Magnox Generation Business Group, Berkeley, UK

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7 Code Application - Within the Creep Range G A Webster

Abstract Methods of estimating the failure due to creep of engineering plant which may contain defects are discussed. The mechanism of the cracking process is described and it is shown how the reference stress concept can be applied to predict failure. A worked example is considered to illustrate the assessment procedure and a further example included to demonstrate the sensitivity of the calculations to choice of reference stress formula.

Notation a a B,H, W c,l A,D C C*, C(t) C' d Dc E G h In K AX KIC

crack depth creep crack growth rate plate/pipe dimensions half crack length co-efficients in creep crack growth law co-efficient in creep law creep fracture mechanics parameters co-efficient in fatigue crack growth law pipe internal diameter creep damage elastic modulus elastic strain energy release rate non-dimensional function of crack size normalizing factor stress intensity factor stress intensity factor range fracture toughness

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Reference Stress Methods - Analysing Safety and Design

Kr Lr m n N P PLC q r rc ti tR(ref) tT £c, s s ref' £ref £ij

non-dimensional stress intensity factor ratio applied to yield load ratio power exponent in fatigue crack growth law power exponent in creep law number of fatigue cycles load limit load of cracked body power exponent in creep crack growth law distance ahead of crack tip creep process zone size incubation time rupture life at reference stress transition time creep strain, strain rate creep strain, strain rate at reference stress creep strain rate tensor

sa, cr0

co-efficients in creep law

e^ (
normalizing factors

£f,s'f 9 p cr,(Tij aref, Oy

uniaxial, multiaxial creep failure strain angle at crack tip plasticity interaction term stress, stress tensor reference, yield stress

Subscripts p s

primary secondary

7.1

Introduction

Reference stress, aref, concepts may be employed for describing failure in the creep range. In this presentation it will be shown how this technique can be applied to predict failure by crack growth, net section rupture or a combination of both processes in high temperature plant containing defects which may experience creep during operation. Several codes make use of this approach (1-4); these include the British Energy Generation R5, French appendix A16, and British Standards published document BSPD 6539, which has now been incorporated into BS 7910, procedures. BS 7910 contains the most recent update of the approach. It can be applied to creep and creep-fatigue loading conditions. The background to the high temperature defect assessment procedure in BS 7910 will be presented. The steps in the procedure will be outlined and a worked example of an axial defect in a pressurized cylinder considered. A further example of an elliptical defect in a plate will also be included to illustrate the sensitivity of the predictions to the choice of formula for calculating reference stress.

Code Application - Within the Creep Range

7.2

129

Description of creep crack growth

Creep crack growth occurs by the accumulation of damage in a process zone local to a crack tip, as shown in Fig. 7.1. Initially on loading, in the absence of plasticity, an elastic stress distribution is generated ahead of the crack tip which is described by the elastic stress intensity factor K. At elevated temperatures, with time, stress redistribution will take place because of creep until, when creep dominates, a steady state is reached which is described by the creep fracture mechanics parameter C*. In this situation, when creep rates is expressed in terms of stress a by the law

where C, e0, and <JD are material parameters and n is the stress sensitivity of creep, the steadystate stress and strain rate distributions at distance (r,ff) ahead of the crack tip become

respectively. In these equations /„, cfr (6,n), and ey (0, n) are normalizing factors (5). The transition time tT for the stress redistribution to be complete is given by

where G is the elastic strain energy release rate. During this transition period the stress distribution is given by a term C(t) which gradually relaxes to the value of C* as steady-state conditions are approached (5). The codes are based on the assumption that creep crack growth rate a is governed by the stress distribution generated ahead of a crack tip. It is postulated (6, 7) that crack advance takes place when the creep ductility £•*/ appropriate to the state of stress at the crack tip is exhausted in successive elements ahead of the crack tip, as illustrated in Fig. 7.2. With this approach, once steady-state conditions have been attained, creep crack propagation rate is given by an expression of the form

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Reference Stress Methods - Analysing Safety and Design

where A and q are material properties. Their values can be obtained from models of the cracking process or from experiments conducted according to the specifications of ASTM E1457 (8) on fracture mechanics specimens. With this standard, creep crack growth rate is measured as a function of C*. For guidance, some values of A and q are included in BS 7910 for a selection of materials. When no values are available, BS 7910 allows estimates to be obtained from models of the cracking process (5-7) using the uniaxial creep deformation and fracture characteristics of a material. In this case

when &. (6,n) and £ti(d,n) are taken to have their maximum values of unity. Typically q is a fraction close to unity and A is mainly governed by the ductility term £•*/. For plane stress conditions s*f is taken to be the material uniaxial creep ductility Sf and for plane strain conditions to be about Sf/30. Generally closest correspondence is achieved with plane stress predictions for ductile situations and with plane strain estimates when brittle behaviour is observed. For many instances equation (7.5) can be simplified to

where a is in m/h, £*/is a fraction and C* is in MJ/m2h. Equations (7.5) and (7.8) are relevant when a steady-state distribution of damage has developed ahead of a crack tip. This situation will not exist initially as time is required for the damage to form and often an incubation period ti is observed, (as shown in Fig. 7.3), prior to the onset of cracking. The magnitude of ti will depend to some extent on the sensitivity of the equipment used to detect crack growth. If the minimum extension that can be detected is rc, then

where a is the initial cracking rate prior to the steady-state distribution of damage being achieved at the crack tip. A lower bound to ti, can be obtained by substituting the steady-state value of a from equations (7.5) or (7.8) into equation (7.9) since the initial cracking rate will be lower than the steady-state value.

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131

Usually rc can be taken to be a few grain diameters in size. BS 7910 allows an incubation period to be calculated from equation (7.9) and subsequent crack extension from equation (7.5) or (7.8). A check is made for stress redistribution to be complete at the crack tip. When it is not crack growth rate is doubled to allow for the fact that C(t) is greater than C*. However in order to make assessments it is necessary to have a method for calculating C* in components.

7.3

Calculations of C*

In most cases, computer finite element analysis is required to make accurate determinations of C* for cracks in components. Solutions for C* are only available for a restricted range of crack geometries and loading conditions. In general these can be written in the form

where h is a non-dimensional function of crack size a, component dimensions, state of stress, and the stress dependence of creep for the material. When computer solutions for C* are not available, it has been found that satisfactory approximate estimates can be produced using limit analysis techniques and reference stress concepts (9). The reference stress crref is defined in terms of the applied load P, yield stress oy and the plastic collapse load of a cracked body PLC as

Use of this value for stress with its corresponding strain rate eref causes h to be relatively insensitive to the value of n in the creep law. This means that h can be obtained from elastic calculations from

where E is the elastic modulus, so that substitution for h a in equation (7.10) gives

Equation (7.12) allows C* to be determined when creep strain rate data are available. When only creep rupture data are known, e ref can be approximated to Sf /%•<$ where tR(ref) is the rupture life at the reference stress so that

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Reference Stress Methods - Analysing Safety and Design

Since solutions for K and aref are available for a wide range of cracked components (1-4, 9) equations (7.12) and (7.13) are convenient expressions for estimating C* when other solutions are not available. Also incorporation of equation (7.13) into equation (7.8) for e*f- =0.03, to represent relatively brittle behaviour gives

for a in m/h. Similarly insertion of equation (7.13) into equation (7.9) for rc » 50 urn and assuming approximate plane strain conditions in conjunction with the initial cracking rate produces a lower bound estimate of the incubation period of (3, 4)

with ti in hours. In BS 7910 equations (7.1), (7.4), (7.5), (7.8), (7.12), (7.14), and (7.15) are used for estimating incubation periods and creep crack extension. Also the reference stress concept is employed for determining the proximity of the uncracked ligament ahead of the crack to failure by creep rupture. This is achieved by updating tR(ref) as crack advance occurs and calculating the fraction of creep damage Dc accumulated. When Dc approaches unity, failure by net section rupture is deemed to have taken place.

7.4

BS 7910 worked example

The basic procedure for assessing creep failure in components in BS 7910 is contained in its Section 9. The step-by-step details are given in Annex T and a worked example, showing how secondary stresses and combined creep and fatigue are dealt with, in Annex U. The worked example concerns a postulated axial elliptical defect in a pressurized cylinder subject to transient thermal stresses during start-up operations. The material of the vessel is type 316 stainless steel. It is required to establish whether the plant can continue to be operated for a further 15 years. Further details of the vessel and operating conditions are given in the Appendix A of this Chapter. The calculation procedure follows a similar format to those used for making fast fracture and fatigue crack growth assessments. It is first necessary to establish the plant operating conditions and material properties. The material properties needed over the appropriate temperature range are: • • • •

yield strength or 0.2 per cent proof strength; ultimate tensile strength; creep strain and creep strain rate data; stress rupture data;

Code Application - Within the Creep Range

133

• creep crack growth rate data; • fatigue crack growth data; • fracture toughness properties. No specific material properties data are provided in the problem. Consequently the creep data and tensile properties are taken from A16 (2) and the other data from guidelines included in the Standard. It is then required to characterize the defect, calculate K, make a check for fatigue and estimate the proximity to fast fracture as indicated in the appendix. If this is satisfactory, the creep damage in the uncracked ligament is determined using a ductility exhaustion criterion in conjunction with the reference stress. Also the crack extension is calculated for the particular operating conditions. Finally the whole procedure is repeated for successive time steps until failure or the desired lifetime is achieved. A sensitivity study is performed to give added confidence in the predictions. Protection against creep rupture is achieved by limiting the ductility exhaustion in the uncracked ligament to a suitable fraction and prevention of fracture by restricting the amount of cracking allowed. In the example it is found that a further 15 years of life can be justified as indicated in Figs. A7.3, A7.4, and A7.5 in the appendix.

7.5

Influence of predictions on reference stress

Not all codes use the same formula for calculating reference stress. The example of an austenitic stainless steel plate subject to combined tension and bending loading and containing a semi-elliptical surface defect, as shown in Fig. 7.4, is considered. Further details are included in Webster et al. (10). Formulae for reference stress are taken from A16 (2), BS 7910 (4), and R6 (11). A significant variation is found as given in Table 7.1. Table 7.1 Predictions of incubation periods based on different definitions of reference stress Reference stress (MPa)

Aa (fjm)

50 200

A16

R6

BS 7910

93.3

153.4

148.2

Incubation period (hours) Experiment

A16

R6

BS7910

35 140

214 857

12.0 48

14.5 58

It is found that predictions of incubation periods and crack extension (Fig. 7.5) of up to an order in magnitude difference can be obtained. Care needs to be exercised therefore in choice of equation for reference stress. Those used in BS 7910 and R6 are selected to be conservative. Recent work (12,13) is leading to the view that reduced estimates can be justified.

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Reference Stress Methods - Analysing Safety and Design

7.6 Conclusions Methods of estimating failure by creep using reference stress and fracture mechanics concepts have been presented. The steps involved in making an assessment have been outlined. A worked example has been considered to illustrate how the procedure is carried out using BS 7910. A further example is included to demonstrate that significant variation in predictions can be obtained depending on the formula employed for calculating reference stress.

References (1) (2) (3)

(4) (5) (6) (7)

(8) (9) (10)

(11) (12) (13)

R5. Assessment procedure for the high temperature response of structures containing defects, Issue 2, British Energy Generation, 1999. AFCEN, Design and construction rules for mechanical components of FBR nuclear islands. RCC-MR. Appendix A16. AFCEN, Paris, 1985. BSPD 6539, Methods for the assessment of the influence of crack growth on the significance of defects in components operating at high temperatures. BSI, London 1994. BS 7910, Guide on methods for assessing the acceptability of flaws in fusion welded structures. BSI, London, 1999. Webster, G. A. and Ainsworth, R. A., High temperature component life assessment. Chapman and Hall, London, 1994. Nikbin, K. M., Smith, D. J., and Webster, G. A., Prediction of creep crack growth from uniaxial creep data. Proc. Roy. Soc., A 396, 1984 183-197. Nikbin, K. M., Smith, D. J., and Webster, G. A., An engineering approach to the prediction of creep crack growth. J. Eng. Mat. & Tech., Trans ASME, 108, 1986, 186-191. ASTM E 1457, Standard test method for measurement of creep crack growth rates in metals. ASTM, 03.01, 1992, 1031-1043. Miller, A. G., Review of limit loads of structures containing defects. Int. J. Pres. Ves. & Piping, 32, 1988, 197-327. Webster, G. A., Nikbin, K. M., Chorlton, M. R., Celard, N. J. C., and Ober, M., A comparison of high temperature defect assessment methods. Materials at High Temp, Vol. 15, 1998,337-346. R6, Assessment of the integrity of structures containing defects, Revision 3. British Energy Generation, 2000. Goodall, I. W. and Webster, G. A., Theoretical determination of reference stress for partially penetrating flaws in plates. Int. J. Pres. Ves. & Piping, 78, 2001, 687-695. Lei, Y., J-integral and limit load analysis of a semi-elliptical surface crack in a plate under combined tensile and bending load. British Energy Generation Limited, E/REP/ATEC/0015/GEN/01, 2001.

Code Application -Within the Creep Range

Fig. 7.1 Creep process zone ahead of crack tip showing the elastic and steady-state creep stress distributions

Fig. 7.2 Accumulation of damage in successive elements ahead of an advancing crack

135

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Reference Stress Methods - Analysing Safety and Design

Fig. 7.3 Example of an incubation period prior to the onset of creep crack growth in a SENT specimen of 21/4 per cent CrMo steel weld metal at 565 °C

Fig. 7.4 Austenitic stainless steel plate subjected to mixed axial and bending loading

Code Application - Within the Creep Range

Fig. 7.5 Comparison of experimental crack growth at deepest point in plate with predictions based on different definitions of reference stress

137

138

Reference Stress Methods - Analysing Safety and Design

Appendix A A7.1 BS 7910 calculation procedure The worked example in Annex U is provided to indicate how the procedure can be applied. The problem investigated is that of a plant which began operation in April 1985. It is assumed that a crack was detected during an inspection in July 1990 and that it is required to know whether the plant can continue to be operated until June 2005. The operation schedule is shown in Table Ul (A7.1) of Annex U. Similarly the defect dimensions and transient thermal stress distributions are presented in corresponding Figs A7.1 and A7.2.

Table A7.1 Operating conditions Month

Pressure (bar)

Temperature (°C)

April, May, June, August, September, October, January, February, March November, December July

40

575

60 Shut down

550

The plant operated for 28 days (672 hours) each month. The fracture toughness, fatigue, and creep crack growth properties of the stainless steel are listed below. Fracture toughness KK Lower bound = 105 MPa m'/2 Fatigue crack growth, upper bound (da/dN)f = C' (AK)m with

C' = 8x 10-11 and m = 3 for crack propagation in m/cycle Creep crack growth

a = A(C*)9 A = 0.023 upper bound A = 0.003 mean q = 0.81

for a in m/h, C* in MPa m/h

Code Application - Within the Creep Range

139

An estimate of fatigue crack growth is given in Annex U.3.7. For high-pressure start-up, which corresponds to the worst loading conditions, the stress intensity factor ranges &Kl and AKa at the deepest, and surface points of the defect are, respectively,

so that substitution in the fatigue law gives

This crack growth during start-up is negligible and can be ignored. Determination of the margin against fast fracture is dealt with in Annex U.3.8. In making the estimation, a level 2 failure assessment diagram is adopted, as shown in Fig. A7.3. It is required to calculate Kr and Lr. In obtaining Kr, it is necessary to determine an interaction term p for combining the plasticity effects of the primary and secondary (thermal) components of stress as shown below. At the time of the inspection (1990) it is apparent from Fig. A7.3 that there is an adequate margin against fast fracture to justify proceeding with the calculations.

At surface Deepest point

K(p+s) (MPa^lm)

p

13.60 19.29

0.0212 0.0184

Kr

0.151 0.202

Lr 0.96 0.96

The steps for calculating the proximity to creep rupture are given in Annex U.3.8.3. It is conservative to assume that the crack was present but not detected in April 1985. The creep damage Dc up to July 1990 is then given by

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Reference Stress Methods - Analysing Safety and Design

for 10 months operation at high pressure and 48 months at low pressure.

Since Dc«1 this gives adequate margin against creep rupture. It is now necessary to determine performance from August 1990. The steps for doing this are identified in Annexes U.3.8.4-6. It is required to calculate the increment of damage ADC and creep strain Aec for each successive month from updated values of crref, K, and C* for each new crack length. For example, for August 1990

CTref

ADC

AEC

Kap p

K, C*a C*i

72.3

MPa

1.53x10 -4

9.17 x 10-6

9.05 MPa m1/2 5.96 MPa m1/2 1.54x lO-8MPam/h 6.69 x 10-9 MPa m/h

so that for this month,

and

These calculations are repeated every month and the crack extension and damage accumulated plotted as shown in Figs. A7.4 and A7.5. The high crack growth rate initially in Fig. A7.4 is due to stress redistribution not being complete during this period. Note: Several units are used for calculating C*. The conversion factor is as follows Units of C*

.

1 MJm2h = 1 MPa m/h = 1000 N/mmh

Code Application - Within the Creep Range

To convert crack growth law

with a in mm/h and C* in MJ/m2h to

with a in mm/h and C* in N/mmh

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Fig. A7.1 Defect in type 316 stainless steel cylindrical vessel

Fig. A7.2 Transient thermal stress distribution through vessel wall (this stress distribution is superimposed on pressure stresses for a period of 15 minutes after start up)

Code Application - Within the Creep Range

Fig. A7.3 Margin against fast fracture for high pressure start up (U.3.8)

Fig. A7.4 Creep crack growth for period August 1990 to July 2005

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Fig. A7.5 Increase in creep damage from start of operation in April 1985 to July 2005 G A Webster Department of Mechanical Engineering, Imperial College, London, UK

8 Fracture Assessment of Reeled Pipelines C Arbuthnot and T Hodgson

Abstract Pipelines installed from reel barges undergo significant plastic deformation as the pipeline is reeled on to a large diameter drum before being transported to site and then unreeled on to the seabed. The plastic deformation occurring during reeling is described in detail and discussed with reference to a recent failure. Differences in pipe dimensions and strength levels are shown to have a significant effect on the plastic collapse and fracture of pipeline girth welds during reeling. Further work is recommended to supplement existing design codes and help prevent further failures.

8.1

Introduction

Reel barges provide a rapid and economical method of installing rigid subsea pipelines. Unlike traditional pipe-laying methods, which require girth welds between successive lengths of pipe to be completed offshore, reel barge installation allows long lengths of pipeline to be fabricated and coated onshore under shop conditions. The completed pipeline is then reeled around a large diameter drum on board a purpose-built vessel, transported to site and simply installed by unreeling the pre-fabricated pipeline, thereby avoiding the costly offshore welding required with traditional pipe-laying methods. One of the first recorded uses of this technique occurred during the Second World War and involved the installation of a 3 in diameter pipeline, codenamed PLUTO, to provide fuel for the invasion of Normandy (1). Installation of commercial pipelines by reel barge began in earnest in the 1960's in the Gulf of Mexico and is now widely used for small to medium diameter pipelines throughout the world. Pipeline dimensions typically range from 4-16 in diameter with wall thickness up to approximately 25 mm. A wide range of materials has been used including normalized and quench-and-tempered carbon manganese steels (API 5L

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Grades X42 to X65) and corrosion resistant alloys such as duplex stainless steels and 13Cr steels. More recent developments involve the possible use of titanium alloys for catenary risers, pipe-in-pipe thermally insulated pipelines, and polyethylene coated lines. Pipeline installation by reel barge has the principal advantage that welding and coating operations can be conducted onshore without impeding other aspects of field development. This has both technical benefits in terms of weld and coating quality and economic advantages for short lengths of pipeline (10-20 km), which can be installed in a single operation. Pipeline diameters are limited by the dimensions of the reel and the reeling process imposes obvious limitations on non-flexible or fragile coatings.

8.2

Pipeline fabrication and installation procedures

Pipelines are fabricated in a number of specialized facilities, which have deep-water access for the reel barge (Fig. 8.1). Individual lengths of pre-coated seamless pipe, approximately 12 m long, are welded together to form 'stalks', which can be over 500 m in length (Fig. 8.2). Completed welds are normally inspected by radiography, which may be supplemented by magnetic particle inspection (MPI) or ultrasonic examination (UT), prior to application of coatings to weld preparation areas. Following completion of pipeline 'stalk' pre-fabrication, individual 'stalks' are welded together as they are spooled on to the reel barge in turn. The first 'stalk' is winched over a radiused ramp, passed through a tensioner and secured to the reel. Torque is then applied and the pipeline is progressively wound around the reel against a back tension supplied by the tensioner mechanism, as shown in Fig. 8.3. On arrival at site, the pipeline is unreeled under tension and passed through aligning and straightening mechanisms to remove residual curvature. Final operations, such as field joint welding for longer pipelines or attachment of sacrificial anodes, are carried out before the pipeline is passed through the stern roller and laid on the seabed.

8.3

Pipe deformation during reeling

The theoretical outer fibre strain induced in a uniform pipe during reeling is dependent on the diameter of the reel (D) and the diameter of the pipe (d):

Outer fibre strains in the pipe following reeling typically range between one and three per cent. For high strength X65 pipe, which has a high yield to tensile stress ratio, most of the cross-section of the pipe is plastically deformed during reeling, apart for a narrow elastically deformed region close to the neutral axis. Figure 8.4 shows that the stress throughout the pipe is approximately constant and close to the ultimate tensile stress after reeling with a maximum outer fibre strain of 2.5 per cent. Under such conditions, the bending moment is equal to over 90 per cent of the ultimate moment capacity for ductile rupture of the pipe, (i.e. the moment corresponding to the point when the surface strain is equal to the strain at the UTS), as shown in Fig. 8.5.

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Full-scale tests are frequently carried out to simulate the reeling process and verify the integrity of the pipeline and coating system. Testing typically involves bending the pipe against a former with the same radius as the reeling drum, followed by straightening against another radiused former with reversed curvature, as shown in Fig. 8.6.

8.4

Recent failures

A recent failure has highlighted the problem of excessive plastic deformation occurring in weaker sections of the pipe during reeling. The failure occurred in an Inconel lined connector between two 500 m long pipeline 'stalks', which had been internally lined with polyethylene to prevent corrosion. The connector was welded to the pipeline 'stalks' using an Inconel welding consumable to provide a corrosion resistant joint between the sections of polythene lined pipeline. The Inconel lining in the connector had been recessed to provide a grooved compression fitting for the polyethylene liner, resulting in a 2 mm reduction in wall thickness in the connector. The pipeline was successfully reeled on to the drum but, on unreeling, each connector failed by ductile rupture. Finite element analysis and consideration of the moment-curvature relationships for the pipe and the connector showed that the bending moment required to induce an outer fibre strain of 2.5 per cent in the thicker pipe section had exceeded the ultimate moment capacity of the connector. Unrestrained bending about a plastic hinge had then resulted in excessive straining and ductile rupture of the connector. Significantly, there was no indication of failure during full-scale reel simulation tests carried out prior to installation in which the pipe was first bent and then straightened against radiused formers. During testing, the steep bending moment gradient in the pipe restricts plastic deformation to a small zone close to the point of contact of the pipe with the former. Excessive straining in a weakened section of the pipe is limited by local rotation of the pipe and contact with the former. A similar situation exists when the pipe is deformed around the reel; however, the bending moment gradient is more gradual and additional plasticity can take place in weaker sections before contact between the pipe and the reel prevents further rotation. Reverse bending during unreeling tends to be less restrained and additional straining may again occur in weaker sections. As a result of this failure, attention was given to the effect of variations in wall thickness and tensile strength on plastic deformation during reeling. On subsequent projects, the tensile strength and wall thickness of the connectors has been matched to that of the adjacent pipe to ensure that failure did not occur. In addition, fracture mechanics assessments have been carried out to assess the risk of plastic collapse or fracture from defects present in welds or adjacent parent material.

8.5

Defect tolerance

8.5.1 Plastic collapse Typical results are presented below for plastic collapse assessment of defects in X65 pipe subject to a nominal reeling strain of 2.5 per cent. Under these conditions, the reference stress for the defect-free pipe, as defined by BS 7910 (2), exceeds the flow stress of the material

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(average of the yield stress and ultimate tensile stress). A strict interpretation of BS 7910 would indicate that plastic collapse could occur during reeling without any defects being present. However, the ultimate collapse moment, corresponding to an outer fibre strain equal to the strain at the UTS, comfortably exceeds the collapse moment based on the flow stress suggesting that the reference stress approach is unnecessarily conservative in these circumstances. Use of the ultimate collapse moment, which can be readily calculated from the stress-strain curve for the material, is less restrictive and enables the size of defects required to cause plastic collapse to be estimated taking account of the reduced cross-section containing the defect. The results of typical calculations based on the limit load expressions given by Miller (3) are shown in Fig. 8.7 for defects located in joints between pipes of the same thickness and for joints between pipes, which differ in thickness by 5 per cent and 10 per cent. Figure 8.7 shows that gross surface-breaking defects over 6 mm deep are required to cause plastic collapse in a joint between pipes of the same thickness at typical reeling strains between one and three per cent. Surface-breaking defects in excess of 6 mm and equivalent buried defects should be easily detected by standard non-destructive examination methods. However, defect tolerance is reduced significantly for joints between pipes of different thickness because the increased bending moment required to deform the thicker section effectively reduces the size of defects that may be tolerated in the thinner section. Figure 8.7 indicates that a 10 per cent difference in wall thickness is sufficient to cause plastic collapse on its own during reeling without any defects being present at all. Similar results are obtained for defects located in joints between pipes with the same dimensions but different strengths. The tolerable defect sizes given in Fig. 8.7 are based on the requirement for the bending moment to be continuous along the pipe until it conies in contact with the reel or other support in the system. If the length of the weaker section is short relative to the stronger sections and the bending radius, then the bending moment will be approximately equal to that in the stronger section, resulting in plastic collapse (as occurred in the failure described earlier). Conversely, short sections of stronger pipe will experience the bending moment in the adjoining weaker sections, although some local strain concentration will occur to accommodate the change in curvature across the interface. Where both weak and strong sections have comparable length, the bending moment at the interface will represent a compromise between the nominal bending moments in the two sections. Consequently, the tolerable defect sizes given in Fig. 8.7 represent the worst-case (i.e. for a short weak section of pipe) and will be conservative for longer pipe lengths. 8.5.2 Fracture Fracture mechanics assessment of defects in reeled pipelines are complicated by the high levels of plasticity and possible changes in residual stress and mechanical properties resulting from plasticity during the reeling process. The accuracy of current reference stress-based assessment procedures may be in doubt at high strains and strain-based J estimation procedures, such as those described in (4), may be more appropriate. The effect of welding residual stresses might be expected to be minimal, since plastic deformation during reeling sets up compressive residual stresses in the outer fibre of the pipe and tensile residual stresses on the inside, which try to uncoil the pipe. On unreeling, the stresses are reversed before being removed entirely when the pipe is passed through the

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straightening mechanism. BS 7910 allows the residual stress used in fracture assessments to be reduced to a minimum of 40 per cent of the yield stress if the reference stress exceeds the flow stress, which is likely to be conservative for defects located in parent material given the strains involved. Interestingly, measurements (5) indicate that residual stresses before and after reeling are of comparable magnitude. The tensile strength of Grade X52 parent metal and weld metal appears to increase slightly after reeling (6), although this effect appears to be less significant in higher strength X65 material (7). The crack growth resistance of X52 parent material has been found to improve after reeling but a modest deterioration in crack growth resistance has been reported for HAZ and weld metal (6). Again no detectable differences were reported between crack growth resistance curves for reeled and undeformed HAZ and weld metal in Grade X65 pipe (7). Fracture mechanics calculations in accordance with BS 7910 typically result in small tolerable defect sizes, less than 2 mm deep by 20 mm long, for X65 pipe based on nominal reeling strains (7). Probability of detection data for radiographic inspection (8), which is shown in Fig. 8.8, indicates defects of this size should be detected with a probability of 50-60 per cent at best. If effects due to thickness and tensile strength tolerances are included, tolerable defect sizes are reduced significantly. Lower detection probabilities for smaller defects imply that an appreciable number of 'critical' defects could remain undetected. The absence of significant numbers of weld failures during pipe reeling is almost certainly due to the fact that the tensile properties of the weld metal and HAZ regions normally exceed that of the parent material. Fracture assessments based on parent material properties are known to become increasingly conservative for overmatched welds when the applied loads are greater than 80 per cent of the limit load (9). While assessment of overmatching effects in the HAZ is less straightforward, the higher strength of the HAZ and the additional strengthening provided by the weld cap are likely to make an important contribution to the integrity of pipeline welds during reeling.

8.6

Conclusions

The failure described in this Chapter has demonstrated that apparently minor differences in pipe wall thickness or tensile properties can significantly reduce the defect tolerance of pipe girth welds and can result in plastic collapse during reeling. Although this effect is particularly marked where the length of the weaker component is small compared to the bend radius, reduced but still significant effects, would also be expected for longer pipe-to-pipe connections. Further work is required to investigate the phenomenon in detail and establish acceptable tolerances for pipe wall thickness and strength levels. The higher strength of over-matched weld metal and the additional strengthening from the weld cap appear to make a significant practical contribution to the defect tolerance of pipeline girth welds and may help prevent plastic collapse or fracture during reeling. Neither of these factors are currently taken into account by existing pipeline codes (10). The introduction of more challenging applications for reeled pipelines, such as catenary risers, and the possible introduction of new welding methods and exotic materials, will put greater demands on the integrity of the reeled pipelines. A better understanding of tolerance and over-

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matching effects is clearly desirable and further research in this area may help to prevent further failures.

Acknowledgements The authors wish to thank Stolt Comex Limited, DSND Limited, and Stena Offshore Limited for permission to use photographs. Discussions with Phil Deboys (AEC Limited) and Graham Stewart (LRS) are gratefully acknowledged.

References (1) (2)

A.M.E., Reeling of Small Diameter Pipelines, 1988. British Standards Institution, Guide on methods for assessing the acceptability of flaws in fusion welded structures, BS 7910: 1999. (3) Miller, A. G. Review of limit loads of structures containing defects, International Journal of Pressure Vessels and Piping, Vol. 32, pp 197-327, 1988. (4) Linkens, D. C., Formby, L., and Ainsworth, R. A. A strain-based approach to fracture assessment - example applications, 5th International Conference on Structural Integrity Assessment, Cambridge, September 2000. (5) Bell, M. Fatigue Performance of Steel Catenary Risers Installed by Reel Ship, Deepwater Pipeline and Riser Technology Conference, Houston, 2000. (6) Pisarski, H. G., Phaal, R., Hadley, L, and Francis, R. Integrity of Steel Pipe during Reeling, Offshore Mechanics and Arctic Engineering Conference, Houston, 1994. (7) Harland, M. O. Reeled Pipe for Dynamic Riser Applications, OTC 10978, Offshore Technology Conference, Houston, 1999. (8) Forli, O. Reliability of Radiography and Ultrasonic Testing, Proceedings of the European-American Workshop - Determination of Reliability and Validation Methods forNDE, Berlin, 1997. (9) Lei, Y. and Ainsworth, R. A. A J integral estimation method for cracks in welds with mismatched mechanical properties, International Journal of Pressure Vessels and Piping, Vol. 70, pp 237-245, 1997. (10) DnV, Rules for Submarine Pipeline Systems, 2000.

Fracture Assessment of Reeled Pipelines

Fig. 8.1 Spoolbase pipeline fabrication yard, Leith (Courtesy of DSND)

Fig. 8.2 Pipe stalks, Evanton (Courtesy of Coflexip Stena Offshore)

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Fig. 8.3 Pipeline installation on reel (Courtesy of Coflexip Stena Offshore)

Fig. 8.4 Stress distribution through X65 pipe following reeling

Fracture Assessment of Reeled Pipelines

Fig. 8.5 Change in bending moment with nominal outer fibre strain for X65 pipe

Fig. 8.6 Pipe bending rig (reproduced from Reference 7)

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Fig. 8.7 Effect of differences in pipe thickness on defect size required to cause plastic collapse (defect length = 20 x defect depth)

0

DEFECT DEPTH (mm)

10

Fig. 8.8 Probability of detection data for radiographic inspection (8) C Arbuthnot CA Associates T Hodgson Galbraith Consulting Limited formerly with Atkins Process, Visiting Professor at the Department of Naval Architecture, Universities of Glasgow and Strathclyde, UK

9 The Use of Reference Stresses in Buckling Calculations T Hodgson

Abstract Codes of practice normally assess buckling of individual members or stiffeners in a simplistic form, determining critical buckling loads or imperfection bending stresses assuming a basic, often uniform, variation of stress along a member. This approach is normally acceptable for framed structures, where dominant axial loads do not vary significantly, but is less accurate for stiffened plate structures where stresses can vary substantially along or across each component. Possible approaches for the derivation and use of reference stresses for the evaluation of buckling strength are demonstrated and discussed.

Notation C Cm E fa Fa fb Fb Fe' h i L s t tw x

buckling coefficient moment amplification reduction factor Young's modulus acting axial stress in a member allowable axial stress in a member acting bending stress in a member allowable bending stress in a member allowable Euler buckling stress height of web stress type (axial, lateral, shear, pressure) length of stiffener stiffener spacing plate thickness thickness of web location along stiffener

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1 H Hr v aa ab Cbi

reduced slenderness (A=V(
9.1

Introduction

The offshore oil and gas industry in the UK has traditionally focused on tubular steel space framed jacket structures that are piled into the sea floor (see Fig. 9.1). The steel jackets support topsides that are typically of truss or plate girder design. These structures have been designed using well established guidance documents such as API RP2A (1), which are typically based on equivalent rules for onshore framed structures [AISC (2)]. The industry has also made use of stiffened steel plate construction in the past, for components such as TLPs (tension leg platforms) and concrete GBS (gravity based structure) topsides. These structure types have typically used a combination of ship building and deep, plate girder bridge design rules to assess strength. In recent years, development has moved away from substantial fixed structures towards moored floating vessels of the FPSO (floating production, storage, and offloading) type (see Fig. 9.2). FPSOs are essentially stiffened steel plate ship hulls designed as floating structures using traditional ship building rules [e.g. (3, 4)] and recent guidance (5). These vessels have the following advantages over traditional fixed structures. • Their cost does not increase exponentially with water depth and, therefore, they offer a practical solution for the increasing number of deep water sites. • They are relocatable, permitting the economical development of marginal fields by use of a single vessel. • They provide oil storage facilities, which permits the use of shuttle tankers for offloading at remote fields.

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157

The fundamental difference in construction between steel space framed and stiffened steel plate structures requires alternative approaches to design and analysis. Loading on stiffened steel plate structures is typically more complex, involving many different directions, types, and variations of load. This Chapter investigates the rules that are available to determine the strength of stiffened steel plate construction and determines if these are adequate for practical details of FPSO and other current offshore structures. Linear elastic or non-linear finite element buckling analysis may be used to determine the factor of safety against collapse of individual components. However, such analysis is not usually practical for general design due to the large number of different combinations of loads, boundary conditions, and initial imperfections that must be considered for a wide range of details. The approaches described herein are, therefore, based on a simple reference stress methodology, determining the effective stress levels acting on the panel and using these in rule-based buckling calculations.

9.2

Framed structure buckling rules

The buckling of individual members within a space frame structure is generally based on semi-empirical methods, with equations such as the following (9.1) being used to relate the axial and bending stresses in the element to the allowable stresses and the allowable Euler buckling stress (F e ' )

This form of equation is relatively simple because there are only two significant stress components for the element. Of these, the axial stress typically does not vary significantly along the element, since there is normally little or no axial load applied between its ends. This allows an axial capacity to be determined, incorporating the anticipated build quality of the member and dependant only on the slenderness and degree of restraint on the element. Unlike axial stress, the bending moment diagram along the element can vary significantly from one end to another. It is for this purpose that the moment amplification reduction factor, Cm,, is introduced, to modify the contribution of bending to the above unity check and reflect the characteristic bending moment diagram, i.e.: • • • •

whether the element is primarily bent in single or double curvature; whether sideways is permitted; whether or not the element is laterally loaded along its length; whether or not the element is restrained at its ends.

9.3

Stiffened plate buckling rules

The buckling of stiffened steel plate components is more complex than space frame members, for the following reasons: • there are considerably more components and directions of loading to consider (along the stiffener, transverse to it, plate shear, stiffener bending, lateral pressure, etc.);

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• each of these components of stress will typically vary along the stiffener or across the plate, often in a complex fashion; • there are several different buckling modes for the panel (global stiffener buckling, plate buckling, lateral torsional buckling); • there are potentially complex fabrication imperfections (stiffener out-of-straightness, plate undulation, flange straightness, etc.). Considering firstly plate panel buckling, there are four principal load directions on each panel between stiffeners: in-plane stresses in each principle direction; in-plane shear; and out-ofplane pressure loading (see Fig. 9.3). Each of these stresses generally vary across the panel requiring codes and contemporary rules (6, 7, 8) to consider various simplified patterns of loading on the panel. Figure 9.4 shows formulae for the buckling coefficient, C (which varies depending on the relative magnitude and distribution of the acting stresses
The rules also present interaction formulae (such as those based on the von Mises equivalent stress) to allow the combined effect of simultaneous stress types to be assessed. The elastic buckling stress is not used directly in checks on structural strength, but is used to derive a critical stress, that allows for yield and plasticity by an equation of the form

Despite the reasonably length list of loads and load combinations presented in Fig. 9.4, this is by no means exhaustive. For example, local stresses that may result from concentrated loading on the plate panel are not represented. Stiffener buckling is even more complex, with the stiffener being potentially destabilized by the application of axial load, transverse plate stress, shear in the plate and stiffener bending (perhaps caused by lateral pressure, as shown in Fig. 9.5). The stiffener may globally buckle, the flange may trip (lateral torsional buckling), or the web may buckle. These effects are normally considered in the rules as follows: • transverse and shear stresses in the plate are converted to equivalent axial stresses and added to the existing axial stress to produce an overall buckling load [alternative approaches have been suggested (9,10), but these do not yet feature in the rules]; • the effect of shear lag and the resultant effectiveness of the plate as a flange for the stiffener is determined; • bending stresses due to external pressure and other loads are evaluated; • global buckling is considered using an interaction equation similar to that for space frame members; • lateral torsional buckling is prevented by restricting the direct stress in the flange of the stiffener; • web buckling is prevented by restricting the aspect ratio (h/tw) of the web, or else specifying stiffening.

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159

The most significant limitation in current offshore rules is that stresses for the above stiffener calculations are normally assumed to be uniform along the stiffener. More complex stress patterns such as those presented for plate panels are not consistently considered and locally concentrated loads are generally not covered. There is a need for simple guidance to derive buckling capacity for these more complex loadings.

9.4

Practical examples of stiffened plate loading

Global stresses in the hull of an FPSO structure result from hull bending, shear, and fluid pressures (see Fig. 9.6). The size of the structure and the large number of stiffened panels mean that stress levels across any individual panel are relatively uniform, implying that current rules are generally adequate for strength calculations on such panels. However, there exist many locations in FPSO hulls and other structures where the pattern of stress is not so uniform. Specific locations include: • the supports for heavy items of topsides equipment, which are relatively discrete in nature and cause significant variations in local stress levels as these reactions are dissipated into the surrounding structure; • connections of columns, brackets, and buttresses, where local stresses are again significant; • around discontinuities and penetrations in the vessel. Examples showing stress variations in typical FPSO structures are illustrated by contour plots of stress in Figs 9.7 and 9.8. The first is a region of bulkhead plate under the deck of an FPSO where the plate panel carries the reaction from a support stool above. High local compressive stresses occur immediately below the ends of the support stool, approximately central in the plate panel below. The linear elastic vertical stress patterns in the panel are seen to be far from uniform. The second example is a buttress structure that carries the loading from a horizontal member into the hull bottom. Forward of the bulkhead, a partial bracket is provided which creates high localized vertical compression stresses in the panel below. Even with more substantial plate, local stresses aft of the bulkhead are still high, with all stress components increasing significantly towards the aft end of the buttress. This latter example is particularly significant, because buckling of this plate panel occurred in practice (see Figs 9.9 and 9.10). The buckle was located at the aft end of the aft buttress plate, where compressive and shear stresses were highest. It is worth pointing out that neither of the above examples constitutes particularly good design. In the first instance, a vertical stiffener below the stool was incorrectly located and would have been more effective below the most highly stressed region, splitting the panel into two parts with relatively linear stress patterns. Note that such a stiffener would tend to have a varying axial stress, being greatest below the support location, but reducing with distance away from this point as load is distributed by shear into the adjacent plate. This is a common design condition.

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In the second of the above examples, high compressive stresses in the web plate could have been reduced by providing a suitable continuous flange at the aft edge of the buttress. Additionally, the bracket forward of the bulkhead should have extended fully to the forward end of the buttress, to avoid high localized stresses. More use of vertical rather than horizontal stiffeners would also have improved the ability of the buttress to carry the intended load.

9.5

Selection of reference stresses for buckling

Where stresses vary significantly along a stiffener or across a plate panel, as in the previous examples, it is necessary to select stress levels for buckling calculations that are representative of the loading pattern on the component. The aim is to derive a set of equivalent simplified loads that represent the true variation of stress along/across the structure. Such simplified stress levels are referred to by structural engineers as reference stresses. It would be possible to base these reference stresses on the maximum stresses along the component being considered. This is undoubtedly safe, but is often far too conservative where stress levels vary significantly and can result in uneconomical design. An alternative approach is needed that reflects the buckling modes of the structure and the effective stress level on the component. The selection of suitable reference stresses is thus very important in buckling calculations. Where buckling coefficients are available for various theoretical load patterns, such as for plate panels, coefficients may be determined for patterns that most closely match the applied stresses. When this is not the case, however, average or representative stresses over the component need to be determined. Several options for the selection of reference stresses are discussed in the following section, for plate panels and stiffeners, respectively. 9.5.1 Plate buckling Experience has shown that the range of load patterns for plate panels in guidance such as DNV (6) is generally sufficient to simulate all expected load and restraint conditions with sufficient accuracy. Local high concentrations of stress at the comers of panels will tend to 'yield out' and can generally be ignored. The approach adopted should be to integrate the stresses acting on the panel in each direction and derive equivalent linear variations of stress that give the same total load. These equivalent stresses may then be used to assess plate buckling. The exceptions to this are where stress variations are significantly non-linear, due to: • intermediate reactions on the plate panel, as illustrated by the first of the examples shown above, or; • where significant concentrations of stress occur towards one end of the panel, as in the second example. In the first case, where bearing stresses are typically high and the consequences of failure significant, it is generally recommended that a local FE analysis be used to check for possible buckling of the panel, otherwise conservative checks should be performed using peak local stresses (6). This condition is relatively rare, however, as it is normal practice to include a load bearing stiffener beneath the reaction point to assist the plate in supporting this load.

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Where high local stresses occur over part of a panel only, the possibility of local buckling of this region alone should be considered. A practical approach to this problem is to consider a reduced length or width sub-panel subject to high localized stress over the selected region only, rather than to average stresses over the entire length or width of the panel. Both the local sub-panel and the global panel should demonstrate adequate strength. 9.5.2 Stiffener buckling Somewhat less information is available in the codes and rules for stiffeners. In general, it is normally assumed that axial, transverse, and shear stresses, where considered, are uniform along the length of the stiffener. One exception to this is BS 5400 (7), which suggests that one-sixth of the peak value of the 'bending' (linearly varying) component of any transverse compression in the plate should be added to the average transverse stress when considering the destabilizing load on the stiffener. This 'one-sixth rule' for transverse stress is derived from the Interim Design and Workmanship Rules (8), the result of substantial experimental and theoretical work carried out on steel bridge structures as a direct result of post-war failures of such structures. It is also consistent with the increase in the buckling coefficient for plate panels from 4.0 for uniform stress to 24.0 for bending stress as shown in Fig. 9.4. This effective contribution from lateral bending stress is useful in determining equivalent loads from stress distributions that peak towards the ends of the stiffener. The suggested approach is to consider the transverse stress in the plate as varying from au - at, to au + ab. The stiffener can then be designed using a reference transverse stress of au + ab/6. Although less well supported in codes and rules and not such a common load type, a similar approach is used for shear stresses that vary along the length of the stiffener. A useful approach for other stress patterns, member, is derived from modal analysis distribution on the buckling modes of the normally the first mode of the member, but the component may cause buckling in higher

such as those that peak towards the centre of a of structures. The effect of the acting load stiffener is considered. The important mode is loads that vary significantly along the length of modes.

The suggested method involves determining the product of the load times the normalized mode shape along the length of the stiffener, and comparing this with a similar calculation for a uniform load distribution. Thus for the first mode of a simply supported stiffener [represented by the mode shape sin(nx/L)], the equivalent uniform reference stress would be calculated by

Care should be taken when using this approach where stresses change sign along a stiffener. In this case, the above product can tend to zero. The previous approach, using the 'one-sixth' rule, implies that this is non-conservative, due to the contribution of stresses to buckling in the second mode. It is suggested that the bending stress approach should be adopted in such cases. The effect of axial load variation along a member is normally treated in the same way as above. There is less theoretical justification for this approach for axial load, but the method is shown to be reasonably accurate for the relatively uniform distribution of axial load along typical members of FPSOs.

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One example of this approach for axial load is provided for bearing stiffeners. AISC rules (2) suggest that 75 per cent of the stiffener length should be used in buckling calculations to account for axial load dissipating into the plate. For slender stiffeners, this suggests that buckling will occur at approximately 752/100 = 56 per cent of the uniform axial load buckling capacity of the member. Where the axial stress reduces from a maximum value under the load to zero at the other end, the above 'one-sixth' rule suggests the use of 50 + 50/6 = 58 per cent of the maximum stress, which shows good agreement with the AISC code. The above approach will provide a reasonable estimate of the buckling capacity of a stiffener under the action of reference stresses. However, this may be non-conservative where local high stresses coincide with stresses due to buckling. It is recommended that peak local stresses be incorporated into the analysis to safeguard against tripping, yield, and inelastic buckling at the ends of a stiffener, for example. This is achieved by adding the difference between the peak local stress and the buckling reference stresses to the stress checks that result from the codes of practice. The above methods have been found to be acceptable, when calibrated against FE buckling analysis, for the determination of stiffener strength where loads vary along the length of a stiffener, as long as these loads are not too localized (effectively point loads). In such cases, consideration should be given to the use of bearing stiffeners, else finite element analysis should be used to justify the structure. The approach is summarized as follows: • for each load type, i, on the stiffener, determine the approximate linear variation of stress along the stiffener and classify this variation as a 'bending' component, ±06,; • subtract this component from the total load and use the remainder to derive an equivalent uniform stress,
• perform buckling checks in accordance with selected rules, based on the above reference stresses to derive a buckling stress utilization, ftr, based on the reference stresses and the corresponding critical buckling stress (may require equivalent stresses to be used where one stress type is not dominant); • increase the strength checks to allow for the difference between peak localized high-axial stresses and the reference stresses used in the above buckling checks, p = fir + (aa - x ora)/0y; • where appropriate, allowable stresses in the above checks should be limited to allow for possible lateral torsional buckling and web buckling. The addition of peak local stresses into the utilization equation as above is potentially conservative, since these may not necessarily occur at the same location as the maximum buckling stresses. Where the results are thought to be unrealistically conservative, FE analysis may be used. As for plates, consideration should again be given to conditions where stresses are high over a part of the stiffener only, such that buckling of a higher mode is possible. The second of the two examples illustrated in Section 9.4 shows the partial buckling of the stiffener, suggesting that part of this buckled shape occurred in the second mode. Where such effects are

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163

pronounced, conservative reference stress levels should be adopted or FE analysis should be considered to check for stiffener stability. The destabilizing effect of the high stress transverse to the upper horizontal stiffener in the forward panel would be treated in such a way.

9.6

Summary and conclusions

Buckling calculations for stiffened steel plate construction now being used in the offshore industry are considerably more complex than for members of traditional space frame structures. This complexity arises due to the number and variability of stress components that act on the stiffener and plate, together with the different buckling modes and imperfection levels that are possible. Current codes and rules present equations to evaluate the buckling resistance of plate panels due to a variety of different load types, to the extent that most expected load directions and conditions are covered. The exceptions to this are discrete support reactions on the edge of panels or other conditions where plate stresses are locally high and do not vary linearly along or across the component. Specific checks, perhaps using finite element analysis, should be provided to ensure local plate stability in such regions, else local strengthening is required. Rules for the buckling of stiffeners attached to the plate are less well developed and variation of stress acting along and across the element is not usually considered. At least one example has been shown of a buckling failure caused by stresses that are much higher at one end of a member than at the other. It is generally conservative to base buckling behaviour on the maximum loads on a stiffener. A less conservative but still supportable approach is to identify reference stress levels that are representative of the buckling behaviour of the stiffener and apply these in design, in combination with local peak stresses that do not contribute significantly to buckling. More guidance is required in the rules to support the selection of these reference stresses. However, in the absence of this information, a simplified approach has been suggested that appears to give reasonable results when compared with FE analysis. This approach is based on calculating the reference stress for each direction of load on the stiffener based on the equivalent uniform stress in that direction, plus one-sixth of the peak linearized varying stress along the length of the stiffener. This approach compares well with rule guidance, where available, and with analysis of typical structures under typical loads. Once again, particular attention should be given to cases where the stress variation along the stiffener is significantly non-linear or very localized. Conservative reference stresses, checks on part panels, or local finite element buckling analysis should be used for these conditions.

References (1)

American Petroleum Institute, 'Recommended Practice for Planning, Designing and Constructing Fixed Offshore Platforms - Working Stress Design', API RP2A - WSD, 20th Edition, July 1993.

164

(2)

Reference Stress Methods - Analysing Safety and Design

American Institute of Steel Construction, 'Specification for Structural Steel Buildings Allowable Stress Design', June 1989. (3) Lloyds Register, 'Rules and Regulations for the Classification of Ship Structures - Part 3, Ship Structures', July 1999. (4) Det Norske Veritas, 'Rules for Classification of Ships', July 1997 (5) Lloyds Register, 'Rules and Regulations for the Classification of a Floating Offshore Installation at a Fixed Location - Part 4, Steel Unit Structures', July 1999. (6) Det Norske Veritas Classification AS, 'Buckling Strength Analysis', Classification Note 30.1, July 1995. (7) British Standards Institute, 'Steel, Concrete and Composite Bridges - Part 3, Code of Practice for Design of Steel Bridges', BS5400, 1982. (8) Merrison, A. E. et al. 'Inquiry into the Basis of Design and Method of Erection of Steel Box Girder Bridges - Interim Design and Workmanship Rules', Parts II and III, 1973. (9) Rahal, K. N. and Harding, J. E. 'Transversely Stiffened Girders Subjected to Shear Loading - Part 1, Behaviour', Proceedings of the Institution of Civil Engineers, Paper 9483, 1990. (10) Rahal, K. N. and Harding, J. E. 'Transversely Stiffened Girders Subjected to Shear Loading - Part 2, Stiffener Design', Proceedings of the Institution of Civil Engineers, Paper 9484,1990.

The Use of Reference Stresses in Buckling Calculations

Fig. 9.1 Typical offshore jacket structure

Fig. 9.2 Typical FPSO vessel model

165

166

Reference Stress Methods - Analysing Safety and Design

Fig. 9.3 Plate panel loading

The Use of Reference Stresses in Buckling Calculations

Fig. 9.4 Buckling coefficients for plate panels (6)

167

168

Reference Stress Methods - Analysing Safety and Design

STIFFENER PLATE =£>

BEAM-COLUMN

Fig. 9.5 Typical stiffener loads

HULL GIRDER BENDING STRESSES

Fig. 9.6 FFSO global hull stresses

The Use of Reference Stresses in Buckling Calculations

Fig. 9.7 Example 1 - vertical stress contours

Fig. 9.8 Example 2 - equivalent stress contours

169

170

Reference Stress Methods - Analysing Safety and Design

Fig. 9.9 Example 2 - region of buckling

Fig. 9.10 Example 2 - extent of buckling T Hodgson Galbraith Consulting Limited, previously with Atkins Process, Aberdeen and Visiting Professor, Department of Naval Architecture and Marine Engineering, Universities of Glasgow and Strathclyde, UK

Index Analyses: creep 87 limit 12 limit, load versus displacement 35 modal 160 shakedown 12 Boundary cracking, Type IV and fusion 83 Branch junctions, limit loads for cracked, welded piping 35 Bree problem, classic 19 British Standards published document BSPD 6539 128 BS7910 128 worked example 132 Buckling: analysis, elastic or non-linear finite element 156 of a higher mode 162 of individual members 156 interaction formulae 157 plate panel 157 stiffener 157 Butt-welded pipe 84, 86 Carbide precipitates, coarsening of 79 Cavitation damage, constrained 82 Cavity initiation and growth, ageing 82 Codes and rules for stiffeners 160 Codes, design 76, 88 Conditions, multi-axial 80 Convergence proof 15, 17 Crack: fully circumferential, internal, part-penetrating 34 growth and net section rupture 128 initiation and growth, creep 8 opening displacement 114 Cracked body, plastic collapse load of 131 Cracked elbow 38 Creep: analyses 87 brittle materials 4 cavitation 79 combined 132 crack 79 crack initiation and growth 8 deformation: and rupture 77 multi-axial rupture 78 steady-state 59 ductile materials 4 ductility e* 129 failure mechanisms for weldments 78 fracture mechanics parameter C* 129 hyperbolic-sine law 80 of multi-material components 62 power law 59, 80 properties, weld materials 59, 65 rupture 7, 140

data, multi-axial 84 life 20, 85 of structures 4 stress 20 steady state 2 strain rates, minimum 85 test technique, impression 60, 65 Critical stresses within material zones 58 Cylinder, axial elliptical defect in 132 Cylinders, thin and thick 62 Damage: creep rupture 4 evolution 84 in a process zone 129 DAMAGE XX 77, 87, 98 Data: isochronous stress-strain 7 multi-axial creep rupture 84 Defect assessment procedures 114 Deformation rate, normalized 63 Design: codes 76, 88 equations 95 implications - materials selection 89 methods 3 Detection probabilities 149 Distribution of damage, steady-state 130 Ductility exhaustion criterion 133 Elastic solutions 3 Elastic-perfectly plastic solid 13 Elastic-plastic fracture 5 Elbows subjected to closing in-plane bending 37 Energy theorems 3 Equations: constitutive 12, 78, 79, 95, 98 design 95 Example, BS 7910 worked 132, 138 Experimental: Data on elbows 37,39 Failure: assessment diagram 34, 116, 139 mode of pipes 92 recent, of pipeline 147 Finite element code ABAQUS 15 Finite element, elastic or non-linear buckling analysis 156 FITNET 119 Fracture: elastic-plastic 5 mechanics, post yield 114 parameter, J 5, 7,116 parameter C*, 8, 129, 131 French appendix Al6 128 Fusion boundary and Type IV failure 89, 90

Hull bending, shear, and fluid pressures 158 Incubation period 130 Isochronous stress-strain data 7 Jacket structures, tubular steel space framed 155 Limit: analysis 12 load versus displacement 35 loads 22 cracked, welded piping branch junctions 35 global 36 local 36 of cracked branch junctions 39 tangent intersection method 36 twice-elastic-slope 35 moments 40 Linear: matching method 12 problems 14 Load, plastic collapse 3 Loading, combined, internal pressure and in-plane bending 37 history 18 Material: constants 83 inhomogeneity, the weld region 58 selection, design implications 89 creep brittle 4 creep ductile 4 Modal method 77 Non-linear programming method, convergent 22 PD6493 114,118 Pipe: wall thickness, differences in 149 weld, thick-walled 63 butt-welded 84 Pipeline fabrication 146 Piping elbows, short-radius 34 Plastic collapse 6 load 3 of a cracked body 131 Plastic solid, elastic-perfectly 13 Plate: panel buckling 157 structures, stiffened steel 156 plane stress, 19 Pressure and moment loads, interaction of 40 Pressures with, hull bending and shear, 158 Process zone, damage in 129 R5 6, 95, 128 R6 7,34 diagram 117

Reel barges 145 Reference: parameters 61 stress 2, 65, 133, 156, 159 techniques 117 Scaling factor 59 Shakedown: analysis 12 limit 19,22 theorem, upper bound 13 SINTAP 118, 119 Steel plate structures, stiffened 156 Steel space framed: jacket structures, tubular 155 structures 156 Stiffeners, codes and rales for 160 Stress: concentration features, sharp 7 rupture criterion, multi-axial 77, 91, 96 state of 129 design and allowable 114 effects of thermal and residual 119 global 158 high compressive 159 high local 160 secondary 132 and primary 117 welding residual 148 within material zones, critical 58 Tension, equi-biaxial 87 Thermal and residual stresses, effects of 119 Tolerance, defect 148 Transition time 129 Transverse stress, one-sixth rule 160 Two-criteria approach 34, 115 Two-material, two-bar structure 61 TypelV: region 79 fusion boundary: cracking 83 failure 89,90 Upper bound: error in 17 shakedown theorem 13 Weld materials 77 creep properties 59 Welded pipes: lifetime, effect of loading condition 87 pressurized 76 Welding residual stresses 148 Weldments, creep failure mechanisms 78 Welds, creep properties for 65 Yield condition, von Mises 13,17

Engineering Measurements Methods and Intrinsic Errors By TA Polak and C Pande This book provides a valuable insight into the equipment and methods generally used in taking measurements, and helps engineers avoid or minimize the inaccuracies that can arise even when using highly accurate instruments. Many of the commonly used measurement methods are described, together with their pitfalls and problems. The authors also incorporate plenty of useful, practical examples.

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Handbook of Mechanical Works Inspection By Clifford Matthews The Handbook of Mechanical Works Inspection provides the techniques, guidelines, and technical data needed to perform inspections on mechanical equipment found in power and process plant applications. The Handbook concentrates on the core fitness for purpose issues that arise during the witnessing of material and equipment tests in the manufacturers' works. This step-by-step guide provides a comprehensive and practical source of information for engineers involved in the inspection of plant and equipment. It will help the non-specialist to carry out works inspections on the common mechanical equipment found in power and process engineering. Designed to encourage an effective approach to works inspection, it will help inspection engineers to perform their work in a professional way.

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IMechE Engineers' Data Book - Second Edition By Clifford Matthews This new edition of the Data Book has been completely revised and expanded. It has now increased significantly in content (by approximately 30 percent compared to the first edition), and many sections have been extended. There is a more general emphasis on the design process, and technical standards and references have been both updated and extended to take into account the current harmonisation of standards. In addition, more and more engineering information is now available in electronic form. This new edition contains details of a wide range of engineering-related web sites to assist the many engineering students who now use the internet as a source of reference for technical information. The Engineers' Data Book - Second Edition is the most practical source of mechanical engineering information available to assist engineers in their day-to-day activity.

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THE STRESS ANALYSIS OF CRACKS HANDBOOK THIRD EDITION by Hiroshi Tada, Paul C Paris, and George R Irwin Nearly double the size of the previous edition, The Stress Analysis of Cracks Handbook- Third Edition provides a comprehensive, easy-toaccess collection of elastic stress solutions for crack configurations. For each configuration, this volume presents crack tip stress intensity formulas and other relevant information such as displacements, crack opening areas, basic stress functions, source references, accuracy of solutions, and more. Throughout, it stresses formulas for application to test configurations. The introductory section details the methods of developing the information. A series of appendices presents special methods and special applications. The current Handbook, co-published with The American Society of Mechanical Engineers, offers a number of new features as follows: • New stress solutions to cracked configurations • Plates with pinching loads • Dislocations and cracks solutions • Plastic zone instability (explaining a potentially interceding "elastic" failure mechanism) • Estimation methods for stress intensity formulas • J-integral methods • Pure shear plasticity solutions Contents include: Introductory Information Stress Analysis Results for Common Test Specimen Configurations Two-Dimensional Stress Solutions for Various Configurations with Cracks A. Cracks Along a Single Line B. Parallel Cracks C. Cracks and Holes or Notches D. Curved, Angled, Branched, or Radiating Cracks E. Cracks in Reinforced Plates Three-Dimensional Cracked Configurations Crack(s) in a Rod or a Plate by Energy Rate Analysis Strip Yield Model Solutions Crack(s) in a Shell Appendices A. Compliance Calibration Methods B. A Method for Computing Certain Displacements Relevant to Crack Problems C. The Weight Function Method for Determining Stress Intensity Factors D. Anistropic Linear-Elastic Crack-Tip Stress Fields E. Stress Intensity Factors for Cracks in a Plate Subjected to Pinching Loads F. Cracks in Residual Stress Fields G. Westergaard Stress Functions for Dislocations and Cracks H. The Plastic Zone Instability Concept Applied to Analysis of Pressure Vessel Failure I. Approximations and Engineering Estimates of Stress Intensity Factors J. Rice's J-lntegral as an Analytical Tool in Stress Analysis K. Elasto-Plastic Pure Shear Stress-Strain Analysis (Mode III) L. Table of Complete Elliptic Integrals M. Table and Properties of Gamma Function References: Software Guide The authors also provide 30 new solution pages, plus modifications of older solutions. This book is an excellent reference, as well as a text for in-house training courses in various industrial and academic settings.

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