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 + 0. The objective of shakedown analysis is to define a value of A = /l s , the shakedown value, so that for any & 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