Fitness-for-Service Fracture Assessment of Structures Containing Cracks A Workbook based on the European SINTAP/FITNET Procedure
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Fitness-for-Service Fracture Assessment of Structures Containing Cracks A Workbook based on the European SINTAP/FITNET Procedure Uwe Zerbst Manfred Schödel Stephen Webster Robert A. Ainsworth
Amsterdam • Boston • Heidelberg • London • New York • Oxford Paris • San Diego • San Francisco • Singapore • Sydney • Tokyo Academic Press is an imprint of Elsevier
Academic Press is an imprint of Elsevier 84 Theobald’s Road, London WC1X 8RR, UK Radarweg 29, PO Box 211, 1000 AE Amsterdam, The Netherlands Linacre House, Jordan Hill, Oxford OX2 8DP, UK 30 Corporate Drive, Suite 400, Burlington, MA 01803, USA 525 B Street, Suite 1900, San Diego, CA 92101-4495, USA First edition 2007 Copyright © 2007 Elsevier Ltd. All rights reserved No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means electronic, mechanical, photocopying, recording or otherwise without the prior written permission of the publisher Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone (+44) (0) 1865 843830; fax (+44) (0) 1865 853333; email:
[email protected]. Alternatively you can submit your request online by visiting the Elsevier web site at http://elsevier.com/locate/permissions, and selecting Obtaining permission to use Elsevier material Notice While the authors believe that the information and guidance given in this book are correct, all parties making use of it must rely on their own skill and judgement. The authors cannot assume any liability for loss or damage caused by any error or omission in the application of the SINTAP/FITNET procedure. Any and all such liability is disclaimed. The authors do not give any warranty or guarantee whatsoever that the information and guidance given in this book does not infringe the rights of any third party or can be used for any particular purpose at all. Any person intending to use the same should satisfy himself as to accuracy and the suitability for the purpose for which it is intended to be used. British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress ISBN: 978-0-08-044947-0 For information on all Elsevier Science publications visit our web site at books.elsevier.com Printed and bound in Great Britain 07 08 09 10 11
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Contents
Nomenclature
ix
1. Introduction 1.1. SINTAP 1.2. FITNET 1.3. The Topic of the Present Book
1 1 3 3
2. Brief Overview on the Development of Flaw Assessment 2.1. General Aspects 2.2. Ligament Yielding 2.3. The TWI (The Welding Institute) Design Curve Approach 2.4. The Early FAD Approach of CEGB (Central Electricity Generating Board) 2.5. The EPRI (Electric Power Research Institute) Approach 2.6. The Reference Stress Method of CEGB 2.7. The R6-Revision 3 Approach of CEGB 2.8. The Engineering Treatment Model (ETM) Approach of GKSS 2.9. The SINTAP Approach
7 7 8 9 10 11 12 13 15 16
3. Basic Features of SINTAP/FITNET 3.1. Fitness-for-Service 3.2. Potential Tasks of a SINTAP/FITNET Analysis 3.3. Multi-Optional Concept 3.4. FAD versus CDF Analyses 3.5. Integrated Concept
19 19 20 21 24 25
4. The Input Parameters 4.1. Loading Input Parameters 4.1.1. General Considerations 4.1.2. Primary and Secondary Stresses 4.1.3. Definition of the Stress Profile for Use with the K-Factor 4.1.4. Definition of the Stress Profile for Yield Load Determination 4.2. Flaw Characterisation 4.2.1. Planar and Volumetric Flaws 4.2.2. Basic Crack Types
29 29 29 29 30 35 36 36 36
vi
Contents
4.2.3. Crack Shape Idealisation 4.2.4. Interaction Effects of Multiple Cracks 4.2.5. Crack Re-characterisation 4.2.6. Crack Orientation and Projected Crack Depth 4.3. Deformation Characteristics of the Material 4.3.1. General Remarks 4.3.2. Engineering and True Stress-Strain Curve 4.3.3. Modulus of Elasticity (Young’s Modulus) and Poisson’s Ratio 4.3.4. Yield Strength and Tensile Strength 4.3.5. Flow Stress 4.3.6. Strain Hardening Coefficient 4.3.7. Yield Strain (Lüders’ Strain) 4.3.8. Tensile Data Relevant to Welds 4.3.9. Temperature and Strain Rate Dependency 4.4. Toughness Characteristics of the Material 4.4.1. General Remarks 4.4.2. The Fracture Toughness Transition Curve 4.4.3. Lower Shelf Fracture Toughness 4.4.4. Upper Shelf Fracture Toughness 4.4.5. Ductile-to-Brittle Transition 4.4.6. Constraint Dependency of Fracture Toughness 4.4.7. Reference Toughness Based on Charpy Data 5. The Model Parameters 5.1. The Stress Intensity Factor (K-Factor) 5.1.1. Sources for Analytical K-Factor Solutions 5.1.2. Types of Analytical K-Factor Solutions 5.1.3. Superposition of K-Factors 5.1.4. Treatment of Geometry Factor Solutions Available in Table Format 5.1.5. Individual Determination of K-Factors by Finite Element or Comparable Methods 5.2. Net Section Yield Load FY , Reference Stress ref and Ligament Yielding Lr 5.2.1. Methods for the Generation of Yield Load Solutions 5.2.2. Global vs. Local Yield Load 5.2.3. Conservatism in Yield Load Determination 5.2.4. Sources for Analytical Yield Load Solutions 5.2.5. Types of Analytical Yield Load Solutions for Homogenous Components 5.2.6. Equivalent Yield Load Solutions for Strength Mismatch Components
36 38 40 41 43 43 44 44 45 46 46 47 48 50 50 50 51 53 54 56 74 85 89 89 89 90 110 110 111 112 112 113 114 115 116 129
Contents vii
6. Structural Assessment 6.1. Acceptable or Critical Conditions of a Component 6.2. Assessment Based on the FAD Philosophy 6.2.1. The FAD 6.2.2. The Assessment Point (or Path) 6.2.3. Types of FAD Analysis 6.2.4. Non-unique Solutions 6.3. Assessment Based on the CDF Philosophy 6.3.1. The CDF Functions 6.3.2. The Determination of the Critical Condition 6.4. The f(Lr ) Function According to the Different Analysis Levels 6.4.1. General Remarks 6.4.2. Option 0 (“Basic Option”) 6.4.3. Option 1 (“Standard Option”) 6.4.4. Option 3 (“Stress–Strain Defined Option”) 6.5. Examples for SINTAP/FITNET Analysis 6.5.1. Determination of the Critical Load 6.5.2. Determination of the Critical Crack Size 6.5.3. Determination of the Required Minimum Toughness 6.5.4. Determination of the Instability Load (R-Curve Analysis) 6.6. Combined Primary and Secondary Stresses 6.6.1. General Remarks 6.6.2. The Correction Term V in the FAD and CDF Approaches 6.6.3. The Determination of V 6.6.4. Welding Residual Stress Profiles 6.6.5. Examples of Analysis for Combined Primary and Secondary Stresses 6.6.6. Further Remarks on Welding Residual Stress Profiles 6.7. Constraint Effects 6.7.1. Consideration in the FAD and CDF Approaches 6.7.2. Example of an Assessment Including Constraint Effects 6.8. Mixed Mode Loading 6.8.1. General Aspects 6.8.2. FAD Analysis 6.8.3. CDF Analysis 6.9. Rapid Loading and Crack Arrest 6.9.1. General Aspects 6.9.2. Quasi-Static vs. Dynamic Analysis 6.9.3. Crack Arrest 6.10. Thin Wall Structures 6.10.1. General Aspects
137 137 139 139 139 141 142 143 143 144 147 147 147 148 151 153 153 157 158 159 162 162 163 164 165 172 185 188 188 188 191 191 193 194 194 194 196 197 198 198
viii Contents
6.11.
6.12.
6.13.
6.14.
6.10.2. Thin Wall Assessment Module 6.10.3. Examples of Thin Wall Assessments Strength Mismatch 6.11.1. The Strength Mismatch Phenomenon 6.11.2. The Strength Mismatch Options 6.11.3. Examples of Option 2 Analysis 6.11.4. The Option 3 Mismatch Module 6.11.5. Further Aspects of Strength Mismatch Weld Shape Imperfections 6.12.1. Weldment Specific K-Solutions 6.12.2. Misalignment 6.12.3. Example of an Assessment Taking into Account Misalignment Reliability Aspects and Significance of the Results 6.13.1. General Aspects 6.13.2. Reserve Factors and Sensitivity Analysis 6.13.3. Reliability Analysis 6.13.4. Example of a Simplified Reliability Analysis 6.13.5. Partial Safety Factors Potential Benefit of Applying Advanced Options and Modules
200 202 208 208 210 214 219 220 224 224 225 226 229 229 230 230 231 235 238
7. Validation Examples 7.1. Introduction 7.2. Pipelines and Pressurised Tubes 7.3. Thin Wall Structures 7.4. Strength Mismatched Configurations 7.5. Failure Investigation
241 241 241 244 247 248
References
251
Appendix: “Fracture Toughness Test Standards”
265
Glossary
269
Index
289
Nomenclature
(Remark: Not every symbol used in this book could be included in this nomenclature list. Symbols that are only used in specific contexts will be explained in the corresponding sections of the text.)
(a) Frequently used symbols a
amax ao A b B B c cgsY co CV CVus E E’ F fc Fc Fpc
Crack depth (embedded or surface cracks), sometimes also crack length in surface direction of through wall cracks (2a for tension geometries) Deepest of a population of adjacent coplanar cracks (in the context of combination criteria, Section 4.2.4.2) Original or initial crack depth before extension Cross section area of the section containing the crack Ligament length (W-a) Specimen thickness (refers to t in components) Normalised T stress (constraint correction of fracture toughness, Section 4.4.6.2) Crack length in surface direction (through wall, embedded or surface cracks), for tension geometries 2c Half maximum length of an idealised surface crack under plastic collapse conditions in a tension plate (Section 4.2.4.2) Original or initial crack length before extension Charpy energy Upper shelf Charpy energy Modulus of elasticity (Young’s modulus) Effective Young’s modulus (= E for plane-stress; = E/(1-2 for plane-strain) Load (general term), also tensile force Probability density function (bi-modal Master Curve approach, Section 4.4.5.3.4) Critical load Plastic collapse load
x
Nomenclature
fs FY FYB FYW FYM f(Lr fi , Fi h h H H J JBL Jc , Ju , Juc Jcen JLB Jmat Jmed Jp Jssy Jo J02/BL , J02 , Ji K Kcen Keq
Dynamic stress enhancement factor for crack arrest events (Section 6.9) Net-section yield load (general term, also tensile yield force) Yield load of base plate material (weldments) Yield load of weld metal (weldments) Equivalent mismatch corrected yield load (strength mismatch joints) Correction function for ligament yielding given for different assessment options Tabulated geometry functions of K-factor solutions Fit function to finite element results in the EPRI approach (Section 2.5) Stress triaxiality parameter (= h /e , sometimes also h /Y , Section 4.4.6.2) Spacing between cracks for alignment criteria (Section 4.2.4.1) Height of the weld strip (Sections 5.2.6 and 6.11), for tension geometries 2H J-integral (for its different interpretations see the Glossary “J-integral”) J-integral referring to the blunting range before crack initiation Fracture toughness in J-integral terms determined at the point of instability (for definition see Section 4.4.2) Census criterion of the alternative two-parameter Weibull distribution according to Eqn 4.40 (Section 4.4.5.2.2) Engineering lower-bound toughness of the alternative two-parameter Weibull distribution according to Eqn 4.41 (Section 4.4.5.2.2) General term of fracture toughness expressed in terms of the J-integral Average toughness value of the alternative two-parameter Weibull distribution according to Eqn 4.41 (Section 4.4.5.2.2) Plastic component of the J-integral Elastic or small-scale yielding component of the J-integral Scale parameter of the two-parameter Weibull distribution (Section 4.4.5.2.2) Resistance against stable crack initiation in J-integral terms (for definition see Section 4.4.2) Stress intensity factor (K-factor) Census criterion of the Master Curve approach (Section 4.4.5.2.2) Equivalent K-factor for mixed mode loading (Section 6.8)
Nomenclature
KJ K km Kmat c Kmat
Kmed Kmin Kr KIp KIs Kps Ko KI , KII , KIII KIA KIB KIA KIB KIC KIc
L Lr Lmax r m
xi
K-factor formally determined from the J-integral (Eqn 6.4) K-factor formally determined from the CTOD (Eqn 6.5) Misalignment magnification factor for K-factor determination (Section 6.12) General term of fracture toughness expressed in terms of the K-factor Constraint corrected fracture toughness (Sections 4.4.6.3 and 6.7) Mean toughness value of the three-parameter Weibull distribution (Master Curve approach, Section 4.4.5.2.2) Shift parameter of the three-parameter Weibull distribution (Master Curve approach (= 20 MPa · m1/2 for ferritic steels with yield strengths between 275 and 825 MPa) Ordinate of the Failure Assessment Diagram (FAD) (= K/Kmat Mode I stress intensity factor for primary loading (Section 6.6) Mode I stress intensity factor for secondary loading (Section 6.6) Plasticity corrected mode I stress intensity factor for secondary loading (Section 6.6) Scale parameter of the three-parameter Weibull distribution (Section 4.4.5.2) Mode I, II and III stress intensity factors (Section 6.8) K-factor at the deepest point of a semi-elliptical surface crack K-factor at the surface points of a semi-elliptical surface crack K-factor at the surface points and the centre point of a quarter-elliptical surface crack (Example 5.5) Plane strain fracture toughness (small-scale yielding) Crack front length in the component (for statistical size correction of the scale parameter of the Weibull distribution in the ductile-to-brittle transition, Section 4.4.5.2) Likelihood (bi-modal Master Curve approach, Section 4.4.5.3.4) Ligament yielding parameter (= F/FY = ref /Y , abscissa of the FAD and CDF diagrams Plastic collapse limit Lr value Constraint factors in various equations (for specific meanings consult the text)
xii Nomenclature
m
M Mb Mk n N
N NB NM NW p p P P Pf pY Po Q r r o , yo ReL ReH
Shape parameter of the Weibull distribution (= 4 in the Master Curve approach; = 2 in the alternative two parameter Weibull distribution of Eqn 4.39) Mismatch factor (= YW /YB Global bending moment Magnification factor for K-factor determination of weldments (Section 6.12) Ramberg-Osgood strain hardening exponent (see the Glossary “crack tip opening displacement”, Eqn G3) Strain hardening exponent used in SINTAP/FITNET; slope of the plastic branch of the true stress-strain curve in double-logarithmic scales. SINTAP/FITNET uses, however, an empirical lower bound to experimental data that were obtained by the general definition. In the ETM approach (Section 2.8) N refers to the engineering stress-strain curve Number of tests of a data set for statistical analysis in the ductile-to-brittle transition (Section 4.4.5.2) Strain hardening exponent of the base plate material (weldments) Equivalent strain hardening exponent of the strength mismatched configurations Strain hardening exponent of the weld metal (weldments) Number of invalid or censored data in the statistical analysis of ductile-to-brittle transition (Section 4.4.5.2) Internal pressure (pressurised components) Load (general) in EPRI terminology (Section 2.5) Failure probability of test specimens (e.g. in the Master Curve approach; Section 4.4.5.2) Failure probability of the component (Section 6.13) Yield internal pressure Reference load (quantitatively close to the yield load FY in the EPRI approach (Section 2.5) Q-stress (constraint parameter, Section 4.4.6.2) Number of valid or uncensored data in the statistical analysis of ductile-to-brittle transition (Section 4.4.5.2) Parameters for describing surface welding residual stress profiles (Section 6.6) Lower yield strength of a material displaying a yield plateau (Lüders’ plateau) Upper yield strength of a material displaying a yield plateau (Lüders’ plateau)
Nomenclature
Rm Ri , Ro , R s Sc Sr t,T tr T T To
T27J , T28J u
V
W Wb x
Y
xiii
Uniaxial tensile strength Inner, outer and mean radius of hollow cylinders Spacing between cracks for combination criteria (Section 4.2.4.2) Survival function (bi-modal Master Curve approach, Section 4.4.5.3.4) Abscissa of the original R6 Rev. 1-FAD (= F/Fpc Wall thickness of the component Rise time (rapid loading, Section 6.9) T stress (constraint parameter, Section 4.4.6.2) Temperature (in C) Transition temperature of the Master Curve approach (the temperature at which the mean value of the fracture toughness, Kmed , equals 100 MPa · m1/2 for 1T specimens) (of thickness of 25 mm) Transition temperature based on Charpy test data Distance from one surface (usually at the crack location) through the thickness. (in some applications also designated by x; see also the comments for x. If the distance is required in two directions this can be described by u and v coordinates (e.g. Example 5.5)) Correction factor for primary and secondary stresses interaction (replacement for the -factor in older versions of FAD procedures) Specimen thickness, for tension geometries 2W Section modulus of the component Distance from one surface (usually at the crack location) through the thickness of the component (in some applications also designated by u); Care has to be exercised since x (or u) refers in some applications to the wall thickness and in others to the crack depth (for specific applications consult the text). Geometry function of K-factor solutions (General term) Angle of inclination with respect to the applied stress direction(s) (Section 4.2.6), also designated by in Section 6.10 Component constraint factor (Section 4.4.6.3) Partial safety factor (Section 6.13) Crack tip opening displacement (CTOD) (for its different specifications see the Glossary “crack tip opening displacement”)
xiv
Nomenclature
c , u , uc i mat 02/BL 02 I 5
a, c
aBL
Tss p ref t Y pl r
Fracture toughness in CTOD terms determined at the point of instability (for definition see Section 4.4.2) Census factor of the Master Curve approach (= 1 for uncensored and = 0 for censored data) General term of fracture toughness expressed in terms of the CTOD Resistance against stable crack initiation in CTOD terms (for definition see Section 4.4.2) Definition of the CTOD measured at the surfaces of the specimen or component, used in SINTAP/FITNET for the Thin Wall Module (Section 6.10) Stable crack extension in depth and surface directions Crack extension due to blunting Lüders’ strain, refers to the portion of the stress-strain curve beyond yield and before strain hardening where the stress stays nearly constant with increasing strain (length of the yield plateau) Shift in the Charpy data based transition temperature due to the use of sub-sized Charpy specimens (Section 4.4.7.5) Strain Plastic strain Reference strain (Reference stress method, Section 2.6, also used in SINTAP/FITNET; obtained for any ref = Lr · Y as the corresponding strain on the true stress-strain curve) True strain (up to curve maximum of the engineering stress-strain curve = ln (1+ with being the engineering strain) Yield strain Shape function for determining the J-integral in test specimens Coordinate for describing points at the front of semi-elliptical cracks Polar coordinates used to describe the stress-strain field ahead of a crack Biaxial tension loading parameter (= x /y Parameter used in conjunction with Option 1 and 2 analyses for modelling the yield plateau effect at Lr = 1 for materials displaying such a yield plateau (Section 6.4) Mean value of a given statistical distribution (general term)
Nomenclature
a b e f gb h Kmat m p p R RL RT ref s s t t x (xx
xv
Parameter used in conjunction with Option 1 and 2 analyses for correcting the f(Lr function at, and slightly below, Lr = 1 for materials without yield plateau (Section 6.4) Poisson’s ratio Function for determining the correction factor V (effect of secondary stresses on the crack driving force) (Section 6.6) Standard deviation of a given statistical distribution (general term) Proof test load (Section 6.6.6.2) Bending stress component (linear stress distribution across the wall). Note that different definitions have to be applied for use in K-factor and yield load determination. Equivalent stress (Section 4.4.6.2) Flow stress, equivalent yield strength taking into account strain hardening in a simplified way, in SINTAP/FITNET = 0.5(Y + Rm Global bending stress (= Mb /Wb Hydrostatic stress (Section 4.4.6.2) Standard deviation of fracture toughness in terms of the K-factor Membrane stress component (uniform stress distribution across the wall). Note that different definitions have to be applied for use in K-factor and yield load determination. Stress due to internal pressure Primary stresses Residual stresses (general term) Residual stresses in longitudinal direction (in weldments, parallel to the fusion line) Residual stresses in transverse direction (in weldments, normal to the fusion line) Net section reference stress (= Lr · Y Maximum induced bending stress due to misalignment (Section 6.12) Secondary stresses True stess (up to curve maximum of the engineering stress-strain curve = 1 + with being the engineering stress and the engineering strain) Stress due to thermal loading (Example 5.3) Normal stress in x direction (Fig. 4.23; refers to the in-plane or ligament direction)
xvi
Nomenclature
y (yy z (zz Y YB YW YM w 1 2 1 2 i
ij
Normal stress in y direction (Figure 4.23) Normal stress in z direction (Figure 4.23; refers to the out-of-plane direction) Yield strength (general term; for its different meanings see Section 4.3.4) Yield strength of the base plate material (weldments, general term) Yield strength of the weld metal (weldments, general term) Equivalent mismatch corrected yield strength of mismatched joints (Mismatch module of Option 3, Section 6.11.4) Weibull stress (Beremin model, Section 4.4.6.3) Principal stresses Stresses from a linear stress profile extrapolated to the front and back surface (definition for K-factor determination, Section 4.1.3) Stress coefficients (1 , 2 , …, n for modelling stress profiles across the wall of the un-cracked section (n = order of the polynomial, Section 4.1). Note that in some applications re-calculations of i are necessary before being applied to K-factor determination. Fundamental period of a component subjected to rapid loading (referring to its period of oscillation, Section 6.9) Shear stresses (Fig. 4.23; i and j = x, y or z) Angle of inclination with respect to the applied stress direction(s) (Section 6.10, in Section 4.2.6 also designated by ß) Angle to the rolling direction in anisotropic sheets Geometry parameter for describing mismatched configurations (= (W-a)/H)
(b) Frequently used subscripts A, B
ax i i; o i L mat
Deepest point and surfaces points of a semi-elliptical surface crack. In the case of quarter-elliptical corner cracks A and C mark the surface points and B the inner point Axial Original, initial Inner, internal and outer Number (counting variable) Longitudinal Material (referring to toughness)
Nomenclature
max B c T W us Y
Maximum Base plate material (weldments) Critical Transverse Weld metal Upper shelf Yield
(c) Frequently used abbreviations CDF CMOD COV CTOA CTOD C(T) FAD HRR HS LS M(T) OM PSF R-curve RT SSC UM 1T
Crack driving force Crack mouth opening displacement Coefficient of variation (ratio of standard deviation go mean value of a given statistical distribution) Crack tip opening angle Crack tip opening displacement ( Compact tension specimen Failure assessment diagram Hutchinson-Rice-Rosengreen field (stress-strain field the magnitude of which is defined by J or Higher strength (material of bi-material joints) Lower strength (material of bi-material joints) Middle cracked tension specimen Overmatching Partial safety factor Crack (extension) resistance curve Room temperature Small scale yielding Undermatching Specimen with a thickness of one inch (∼25 mm)
xvii
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Chapter 1
Introduction This chapter briefly describes the European SINTAP and FITNET projects and the aim of this book.
1.1. SINTAP SINTAP (Structural INTegrity Assessment Procedure) was a multidisciplinary collaborative project, part-funded by the European Union, with the aim of devising a unified procedure for the assessment of fracture behaviour, offering a range of assessment routes with maximum applicability from the smallest SME (small and medium-sized enterprise) to major industrial users. The project commenced in April 1996 and was completed in April 1999. Although many fracture assessment methods were in existence when the SINTAP project commenced, as will be outlined in Chapter 2, there were conflicting approaches and unspecified levels of empiricism. These approaches did not fully reflect either the performance of modern materials or the current state of knowledge. SINTAP covered both modelling and experimental work, and a large part of the project was concerned with the transfer of knowledge and data between industries and scientific organisations together with its compilation and interpretation to provide the required solutions. The culmination of this project was a procedure that is applicable to a wide cross-section of users because of its ability to offer routes of varying complexity, reflecting data quality and the scope for a final interpretation reflecting the preference of the user. In view of the comprehensive nature of the project and the necessity to consider carefully the requirements of the end user, a wide-ranging consortium was established for the project. This comprised a material supplier (British Steel, now part of the Corus Group), an electricity generator (British Energy), an oil and gas supplier (Shell), a chemical processor (EXXON), safety assessors (Health & Safety Executive and SAQ Inspection Ltd., now part of DnV), research institutes (GKSS, Fraunhoffer IWM, lnstitut de Soudure, TWI, VTT, JRC (IAM), two universities (Cantabria and Gent), a software developer (Marine Computation Services) and a consultancy (Integrity Management Services). The project was co-ordinated by Swinden Technology Centre, Corus (UK), with the five principal areas of study being led by Task Leaders as summarised below.
2
Fitness-for-Service Fracture Assessment of Structures Containing Cracks
Task 1: Mismatch – Leader: GKSS To quantify the behaviour of strength mismatched welded joints and to provide recommendations for their treatment in a procedure. Task 2: Failure of Cracked Components – Leader: British Energy To extend the understanding of the behaviour of cracked components in the specific areas of constraint, influence of yield strength (YS) to tensile strength (UTS) ratio, prior overload, leak-before-break, and stress intensity factor (SIF) and yield load (YL) solutions. Task 3: Optimised Treatment of Data – Leader: VTT To provide an industriallyapplicable method for a reliability-based defect assessment procedure. Task 4: Secondary Stress. – Leader: IdS The determination and validation of the most appropriate method of accounting for residual stress, including a compendium of residual stress profiles. Task 5: Procedure Development – Leader: Corus Development and validation of the procedure. Each task comprised a number of sub-tasks, (see Table 1.1), and was structured into three basic steps: As a first step a comprehensive collation of existing data,
Table 1.1: Sub-tasks in project Task 1: Mismatch
Task 2: Cracked Components
1.1 Review 2.1 Review 1.2 Bi-Materials 2.2 Constraint 1.3 Multipass Weld 1.4 Modelling 1.5 Procedure
Task 3: Optimised Data Treatment 3.1 Review 3.2 Toughness
Task 4: Secondary Stress
4.1 Review 4.2 Collate Profiles 2.3 YS/UTS 3.3 Charpy 4.3 Experiment Ratio Correlations & Modelling 2.4 Prior 3.4 NDI 4.4 Profiles Overload Guidance Library 2.5 Leak-before- 3.5 Probabilistics 4.5 Procedure break 2.6 SIF & YL 3.6 Procedure Solutions 2.7 Procedure
Task 5: Procedure Development 5.1 Review 5.2 Procedure 5.3 Software 5.4 Validation 5.5 Documentation
Introduction
3
procedures and codes was completed and then experimental work was carried out to cover omissions and to validate the approaches and assumptions relevant to the particular study area. The best practice approach within each task was finally collated within Task 5, with consideration of the needs of the practising engineer in order to formulate the procedure itself.
1.2. FITNET FITNET (Fitness-for-Service Network) was a European thematic network, which ran for four years from February 2002 to May 2006. The overall objective of the group was to develop and extend the use of fitness-for-service (FFS) procedures for welded and non-welded metallic structures throughout Europe. It was partly funded by the European Commission within the fifth framework programme and comprised about 50 organisations from 17 European countries but also included contributions from organisations in the USA, Japan and Korea. The FITNET FFS procedure is built up in four major analysis modules, namely: Fracture, Fatigue, Creep and Corrosion and the full procedures, together with background information on this is given in [1.1]. The procedures were developed in parallel as a CEN (Comit´e Europ´een de Normalisation) Workshop Agreement (a CEN document). The overall structure of the FITNET-FFS procedure is given in Fig. 1.1. The Fracture Module of the FITNET FFS procedure was mainly based on the previous developments carried out within the SINTAP project as well as later advances in other documents such as the Revision 4 of the British Energy R6 procedure [1.2] and amendments to the British Standard BS 7910 [1.3]. In addition, the results from other European Union projects were used to extend the treatment of several problem areas, such as the effects of constraint and the treatment of thin walled structures.
1.3. The Topic of the Present Book The emphasis of the present book is on the practical application of FFS tools for predicting the fracture behaviour of structures with real or potential cracks. The content is mainly based on the SINTAP procedure, which is basically the same as the fracture module of FITNET in many aspects, but complemented by additional topics such as mixed mode loading, the treatment of thin-walled structures and some further items. Since the name “SINTAP” has now been in use for more than seven years and has become quite widespread, the name is retained here but modified to “SINTAP/FITNET” to reflect the inclusion of elements of FITNET,
4
Fitness-for-Service Fracture Assessment of Structures Containing Cracks
Application Areas of FITNET FFS Procedure Design of new structures
Fabrication support
Failure analysis
In-service assessment
Information required for Assessment (Inputs) Flaw Information
Material properties
Stresses
Assessment Modules Fracture
Fatigue
Creep
Corrosion
Assessment and Reporting of Results
Alternative Approaches and Specific Applications (Leak-before-break, Local approach, Crack arrest, Mixed Mode . . .)
Additional Information Compendia (K solutions etc.)
Validation and case studies
Figure 1.1: Overall flowchart of the FITNET FSS procedure.
which did not belong to the original SINTAP document. On the other hand, the concentration on fracture, i.e., only the fracture module of FITNET, implies a restriction to the end-of-life conditions of a damaged component whereas FITNET actually covers a wider range; containing further modules for fatigue life, fatigue crack extension, creep and corrosion, as illustrated in Fig. 1.1. The intention of the present book is not to provide the reader with the SINTAP or FITNET procedure itself. The latter is available as a CEN document, as mentioned above. Instead the intention is to provide the potential user with detailed guidance on how to use the procedure in practical applications, complemented with background information that needs to be considered to avoid mistakes. The book is intended to be useful for training by containing a wider range of practical information than the procedural documents. For this purpose, additional details
Introduction
5
on the scientific background, alternative approaches in the FFS literature and practical tips are given in the text and in the Glossary. A large range of worked examples are included as tutorials. However, it should be recognised that the book does not contain all elements of the SINTAP/FITNET documents, such as the extended annexes of K-factor solutions, yield load solutions etc. which by themselves constitute a complete volume of FITNET. It is therefore intended to be used in conjunction with, rather than instead of, the procedural documents and in particular with the CEN-FITNET document.
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Chapter 2
Brief Overview on the Development of Flaw Assessment 2.1. General Aspects Engineering design codes traditionally work on the philosophy of comparing the applied stress with some limit stress level, such as the yield strength of the material and, as long as the latter is greater than the former, the component is regarded as safe. This approach assumes two things; the material is homogeneous and is free from defects. If a structural discontinuity (e.g., a crack) is present then the principle may not apply and comparisons need to be carried out on the basis of crack tip parameters (i.e., fracture mechanics). The behaviour of the component can then be analysed in terms of the critical applied load or a critical crack size. As long as the deformation behaviour of the structural component is linear elastic then the relevant parameter is K and comprehensive compendia of K-factor solutions exist in handbooks and computer programs. If the component behaves in an elastic–plastic manner then the situation is more complex as the crack tip loading is also influenced by the deformation behaviour of the material, as dictated by the appropriate stress–strain curve. This makes handbook solutions a very difficult task although they are available for some limited configurations [2.1]. Hence analytical methods are required to answer the vast majority of structural integrity questions. These approaches have been under development for more than 30 years and comprehensive reviews of the various methods available have been provided in a special edition of the International Journal of Pressure Vessels and Piping [2.2] and in a volume of the Elsevier encyclopaedia Comprehensive Structural Integrity (CSI) [2.3]. The principles behind the various methods are discussed in state-of-the-art papers in [2.4–2.6] and developments across the world are summarised in other publications. One principle common to all the methods is that they aim to produce conservative results and, hence, if an analysis leads to an “unsafe” result this does not mean that the component will fail, just that it is necessary to examine it more closely. The drive in recent years has been to reduce this degree of uncertainty using staged levels of increasing sophistication in analysis to provide scope for simple to complex evaluations.
8
Fitness-for-Service Fracture Assessment of Structures Containing Cracks
2.2. Ligament Yielding Note that, as a general rule, the crack driving force is correlated with the local strain rather than with stress. To use a simple comparison: in the elastic range of a stress–strain curve the stress and strain are simply related by Young’s modulus and it does not matter which parameter is given to reconstruct the whole curve. However, above the yield strength Y , the deformation behaviour becomes nonlinear and strain increases non-proportionally with further increase in stress. If we compare the determination of crack driving force by the linear elastic factor K with the determination of strain from a stress value using Young’s modulus we get the same effect. The crack driving force is underestimated for loads larger than approximately 60% or 70% of the net section yield load FY of the cracked component, and the greater the underestimation the lower the strain hardening exponent of the material, that is, the flatter the plastic branch of the stress–strain curve. The common flaw assessment methods are not restricted to a linear elastic deformation pattern, but cover the whole range up to elastic–plastic. Hence the yield load relevant to a cracked component plays an important role in almost all models and this marks the change from contained to net-section yielding (for a definition of these terms see the Glossary “crack tip plasticity”). In the literature the term “limit load” is frequently used instead of the term “yield load”. The reason for this is that FY is frequently determined for ideally plastic, that is, non-hardening material. In such a case the terms “yield load”, “limit load” or “collapse load” describe the same load level that is associated with the failure of the component. However, if the material work hardens beyond yield, the yield load does not correspond to failure, which may not occur until the plastic collapse load is reached. With the exception of the Design Curve approaches the concepts discussed below, independent of their different individual backgrounds, all have in common that they are based on the K-factor but “correcting” this for plasticity effects. Even if they, as in the case of the FAD (Failure Assessment Diagram) approaches, are still written in terms of K this is not in line with the original linear elastic stress intensity factor concept. Rather they describe the crack driving force in terms of a formal elastic–plastic K-factor or, alternatively, in terms of the J-integral or the crack tip opening displacement (CTOD). The underlying principle is schematically illustrated in Fig. 2.1. The advantage of this approach is that it can make use of K-factor solutions that are nowadays relatively simple to determine or are even available in compendia, thus avoiding the need for individual finite element determination of the crack driving force.
Brief Overview on the Development of Flaw Assessment
Formal elastic-plastic K (real crack driving force CDF)
Crack driving force Ligament yielding factor f(F/FY)
9
linear elastic K
CDF = K · f (F/FY)
1 0.6 – 0.7
F/FY
Figure 2.1: Schematic illustration of the effect of ligament yielding on the crack driving force.
2.3. The TWI (The Welding Institute) Design Curve Approach Burdekin and Dawes [2.7] formulated the earliest analytical methods in 1971 as the Design Curve approach, based on original ideas from Wells [2.8]. This basically relates the applied strain to the CTOD () or, latterly, the J-Integral via a quadratic equation in the contained yield region and linearly in the net-section yielding regime. Comparison of the material toughness with that necessary to withstand the applied strains gives the failure prediction. This approach was modified several times, finally, for deeper cracks, by Dawes [2.9] who also rewrote the equations in terms of stress to fit into the FAD approach that had then been adopted for the British PD 6493 procedure [2.10, 2.11]. It is still part of the current BS 7910 [2.12] and, with some modifications, is contained in US and Canadian guidelines for welded pipes, API 1104 [2.13] and CSA Z662 [2.14], and in the Chinese pressure vessel code CVDA-1984 [2.15]. A preferential application field is shallow cracks originating in notches.
10
Fitness-for-Service Fracture Assessment of Structures Containing Cracks
The original TWI-CTOD Design Curve was based on a semi-empirical correlation of the crack tip opening displacement, CTOD, with the local strain at the surface of a tension plate. Today, the method has been extended for tension and bending, for including strain hardening and for the use of the J-integral as crack driving force parameter in addition to the CTOD (for an overview see [2.4] and [2.6]).
2.4. The Early FAD Approach of CEGB (Central Electricity Generating Board) In parallel with the activities at TWI, a flaw assessment concept was developed by the British nuclear power industry in the 1970s. The early R6 routine of British Energy (formerly CEGB) [2.16] introduced the concept of the so-called two-criteria approach and the Failure Assessment Diagram (for a discussion of the early development see [2.5]). Failure was assumed either when the stress intensity factor in the component exceeds the fracture toughness in terms of the linear elastic KIc or when the applied load exceeds the plastic collapse load of the net section of the component containing the crack. The advantage of this approach was that both criteria were sufficiently well understood at a time when the theories of post-yield fracture mechanics were still under development. In between the extreme cases of KIc based fracture and plastic collapse, the failure mode of elastic–plastic fracture is modelled by an interpolation line based on empirical data and a strip yield approach [2.17]. A further part of R6-Revision 1 was that the applied K-factor is normalised by the fracture toughness Kmat and the applied load F by the plastic collapse load Fpc : Kr = K/Kmat
(2.1)
Sr = F/Fpc
(2.2)
and
In this way, the interpolation line K r = Sr
−1/2 8/ 2 ln sec Sr /2
(2.3)
becomes a failure line designated as a FAD (Fig. 2.2). In the FAD approach, the assessment of the component is based on the relative location of a geometrydependent assessment point (Kr Sr ) with respect to the FAD line which is assumed
Brief Overview on the Development of Flaw Assessment
11
potentially unsafe
C
1.0
B increasing load
Kr = K/Kmat
0.8 0.6
A 0.4
C FAD line B increasing crack size
A
0.2
safe 0
0
0.2
0.4
0.6
0.8
1.0
Sr = F/Fpc
Figure 2.2: Failure Assessment Diagram approach according to R6-Revision 1.
to be a universal curve roughly independent of the component geometry and the material. In the simplest application the component is regarded as safe as long as the assessment point lies within the area below the failure line. It is regarded as potentially unsafe if it is located on the line or outside the shaded area in Fig. 2.2. An increased load or larger crack size would move the assessment point along the loading path towards the failure line, as indicated by the locus A → B → C.
2.5. The EPRI (Electric Power Research Institute) Approach At about the same time that the R6 routine was first published, the American EPRI approach was developed [2.18]. The following explanations refer to the J-integral as the crack driving force parameter. Note, however, that solutions for CTOD in terms of the 45 definition (see the Glossary “crack tip opening displacement”) are available as well. In the EPRI method the J-integral is split into small scale yielding (ssy) and widespread yielding (p) components J = Jssy + Jp
(2.4)
The small scale yielding part, Jssy , is obtained from the elastic stress intensity factor with minor adjustments based on plastic zone corrections to the crack size (see the Glossary “crack tip plasticity”). For the plastic part, the HRR (Hutchinson-Rice-Rosengreen) field equation (see the Glossary “crack tip opening
12
Fitness-for-Service Fracture Assessment of Structures Containing Cracks
displacement” and “J-integral”) is solved for J. Replacing the HRR stresses by the applied load P and a reference load Po (EPRI terminology) gives an expression Jp = o o hL
P Po
n+1
(2.5)
In the EPRI approach, as in the R6 routine, the load is normalised by a reference load Po which, in principle, may be freely chosen but is usually identified with the yield load of the cracked component. The quantities o , o , and n are the fitting parameters of the Ramberg–Osgood formulation of the stress–strain curve (see the Glossary “crack tip opening displacement”, Eqn G3). L is a characteristic dimension that is commonly identified with the ligament length or crack size. In order to specify the function h for specific geometries, large sets of 2D finite element calculations were performed varying the component and crack dimensions as well as the Ramberg–Osgood strain hardening exponent n of the material. Tables of h for a range of plate and cylindrical configurations form the main items of the EPRI handbooks [2.1]. The EPRI approach was primarily developed in what in SINTAP/FITNET is called a CDF format, an acronym standing for “Crack Driving Force”. In contrast to the FAD philosophy used in R6, the CDF philosophy strictly separates the applied and material sides. The determination of the crack driving force in the component and its comparison with the fracture resistance of the material are two separate steps. It was, however, also implemented in a FAD format, see for example, [2.19]. In contrast to the CEGB-FAD, the EPRI-FAD is based on the yield load FY instead of the collapse load Fpc of the cracked component and the FAD line becomes a function of the strain hardening of the material. The advantage of the EPRI approach is that it is based on finite element analyses although the solutions in the EPRI handbook are restricted to 2D calculations. On the other hand its application is limited to only a few configurations, namely those for which tables of h factors are available. Another critical point is its use of the Ramberg–Osgood fit to the stress–strain curve since most materials do not follow this fit satisfactorily and, in particular, the important region near the yield strength is usually poorly described.
2.6. The Reference Stress Method of CEGB These limitations were overcome in the so-called Reference Stress method developed in the early 1980s at CEGB in the context of steady state creep fracture mechanics [2.20]. It was subsequently shown that the method could be interpreted as a generalisation of the EPRI approach [2.21]. In the Reference Stress approach,
Brief Overview on the Development of Flaw Assessment
13
the deformation behaviour is considered by a piece-wise introduction of the true stress–strain curve allowing an exact description of any material. In addition, it was shown in [2.21] that the dependency of h on the strain hardening coefficient n in Eqn (2.5) could be minimised by redefining the reference load Po . In this way it becomes possible to set h for any strain hardening approximately equal to the value for n = 1, the corresponding value for linear elastic material behaviour and related to the linear elastic K-factor. This gives the basic equation of the reference stress method K 2 E ref −1 (2.6) Jp = E ref with being equal to 0.75 for plane strain and 1 for plane stress. Since the modified reference load Po is found to be close to the yield load FY in many cases, the reference stress ref can be defined as ref =
F FY Y
(2.7)
The reference strain ref can be determined as the strain at the true stress–strain curve referring to ref . The term Y denotes the yield strength as a general term being ReL for materials with, and Rp02 for materials without, a Lüders’ strain. In Eqn (2.6) the second term can be interpreted as a ligament yielding correction to the linear elastic K-factor. For a more detailed discussion see [2.5] and [2.6].
2.7. The R6-Revision 3 Approach of CEGB Although it could also be applied in a CDF format the Reference Stress method was primarily developed for use in the R6-FAD. Eqn (2.3) is replaced by
E ef L3 + r Y Kr = K/Kmat = Lr Y 2E ref
−1/2 (2.8)
or in more general terms by Kr = f2 Lr
(2.9)
where f2 Lr =
E ef L3 + r Y Lr Y 2E ref
−1/2 (2.10)
14
Fitness-for-Service Fracture Assessment of Structures Containing Cracks
and Lr = ref /Y = F/FY
(2.11)
in R6-Revision 3 [2.22]. The first term (E ref /Lr Y ) in Eqn (2.8) describes both the elastic and fully plastic behaviour of the component but not the intermediate region between both limiting conditions. This is modelled by the second term (Lr3 Y /2E ref ) which was originally based on EPRI considerations but modified by adapting to additional finite element results on various geometries (see [2.23]). A maximum value for Lr , Lrmax , is introduced to cover failure by plastic collapse. This is determined by Y + Rm 2 max (2.12) Lr = Y Two examples of Eqn (2.8) for different materials are shown in Fig. 2.3. The curves reflect the different shapes of the stress–strain curves of the materials. The fLr function in Eqns (2.9) and (2.10) is designated as f2 Lr with the index “2” referring to what is called “Option 2” in R6-Revision 3. In addition, the procedure contains an “Option 1” which is still applicable when only the yield and tensile strength of the material under consideration, but not its complete stress–strain curve, are available. The “Option 1” fLr , designated as f1 Lr , is derived as a fit to “Option 2” curves for a variety of materials but biased towards a lower bound. It is given by (2.13) f1 Lr = 1 − 014Lr2 · 03 + 07 exp −065Lr6 with Lrmax as defined by Eqn (2.12).
Kr = K/Kmat
1.0
C-Mn Steel
0.8 0.6
Option 1 Curve
Austenitic Steel
0.4 0.2 0
0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
Lr = σref /σY = F/FY
Figure 2.3: Failure Assessment Diagrams for various materials according to R6-Revision 3 (according to [2.23]).
Brief Overview on the Development of Flaw Assessment
15
The two options in R6-Revision 3 follow a principle of stepwise graded conservatism. The lower “Option 1” allows less sophisticated analyses which, nevertheless, will give satisfying results in many cases. The higher “Option 2” requires more effort, but the user is “rewarded” by less conservative results. That also means that an unacceptable “Option 1” result may provide a motivation for repeating the analysis at “Option 2” rather than claiming that the component is proven to be unsafe. The principle of stepwise graded conservatism is developed further in the SINTAP/FITNET procedure. In the 1990s the R6 routine was adopted by a number of flaw assessment procedures such as the British Standards BS 7910 [2.24], the American API document API 579 [2.25], the Swedish SAQ procedure [2.26] and others (for overviews see [2.4, 2.27, 2.28]). Note that in R6-Revision 4 [2.29], as well as in the 2005 revised BS 7910 document [2.12], elements of the SINTAP procedure have been used.
2.8. The Engineering Treatment Model (ETM) Approach of GKSS During the early 1980s, at about the same time that the EPRI approach and the Reference Stress method were under development in the US and Britain, an independent method was developed at GKSS in Germany. The basic concept of the ETM is illustrated in Fig. 2.4. The deformation behaviour of the ligament ahead of the crack is assumed to be satisfactorily described by a piece-wise power law E·
for < Y (2.14) = Y / Y N for ≥ Y fitted to tensile test data. Replacing by F, Y by FY , by 5 and Y by 5Y gives an expression 5 /5Y = F/FY N
for
F ≥ FY
The contained yielding branch, F < FY , is described by
2 1 Keff F 5 = K + E mEY FY
(2.15)
(2.16)
with 1 and m being constraint and material dependent constants, for example, for plane stress in steel 1 = 241 and m = 1. Keff is the Irwin-type
Fitness-for-Service Fracture Assessment of Structures Containing Cracks
log σ/σY
16
1
1
ε/εY
log F/FY
1
1
1
1
N
N
δ5/δ5Y
Figure 2.4: Engineering Treatment Model (ETM): Basic principle.
plastic-zone-corrected stress intensity factor (see the Glossary “crack tip plasticity”). A special aspect of the ETM concept is its use of 5 , a special definition of the CTOD, that is, measured at two gauge points at the surface of the component (see the Glossary “crack tip opening displacement”). This makes the concept particularly suited to thin-walled structures. This method also includes a J-integral based approach. The ETM concept has been worked out in handbook format [2.30] and consists of a number of modules (basic module, notch module, strain-based module etc., see also [2.6, 2.31, 2.32]). Special emphasis is put on strength mismatch problems for which a specific methodology was developed and complemented by a compendium of modified yield load solutions [2.33].
2.9. The SINTAP Approach The aim of the SINTAP project was to develop a unified European procedure to combine the strengths of the methods available at the end of the 1990s. In a preinvestigation it was found that the results of these methods were very similar (see Fig. 2.5) so that the existing range of approaches was more of a handicap than an advantage to the user. In [2.34] the authors list a number of improvements and
Brief Overview on the Development of Flaw Assessment
(a)
17
(b)
1,0
f(Lr) 0.6 0.4
ETM R6-Opt.2 FEM [M(T), a/W = 0.5] FEM [SE(B), a/W = 0.25]
0.2 0
0
0.5
1
ETM R6-Opt.2 FEM [M(T), a/W = 0.5] FEM [SE(B), a/W = 0.25] 1.5
0
0.5
Lr
1
1.5
Lr
(c)
(d)
1,0
f(Lr) 0.6 0.4
ETM-MM R6-Opt.2 FEM
0.2 0
0
ETM-MM R6-Opt.2 FEM
0.5
1
Lr
1.5
0
0.5
1
1.5
2
Lr
Figure 2.5: Comparison between ETM, R6-Revision 3 Option 2 and finite element results for tension [M(T)] and bending [SE(B)] specimens ((a) and (b) [2.37], (c) and (d) [2.38]). (a) Homogeneous, material with yield plateau (e.g., ferritic steel); (b) Homogeneous, material without yield plateau (e.g., austenitic steel); (c) Weldment, strength mismatch base plate and weld metal with yield plateau (e.g., medium strength ferritic steels); (d) Weldment, strength mismatch neither base plate nor weld metal with yield plateau (e.g., austenitic steels).
novel features of the SINTAP procedure compared to the approaches available then. The most important changes are: • Extension and more detailed outline of the principle of stepwise graded conservatism • Redefined fLr functions for materials with and without yield plateau • Guidance on statistical aspects, among others with respect to the fracture toughness • A modified approach for treating combined primary and secondary stresses.
18
Fitness-for-Service Fracture Assessment of Structures Containing Cracks
Note that the SINTAP procedure of 1999 [2.35] has become the major part within a module of the European FITNET method that has recently been completed [2.36]. Within this project some minor changes were made on SINTAP and the procedure was extended to include further aspects. The present book refers to the most recent development and makes this recognisable by choosing the term SINTAP/FITNET instead of SINTAP.
Chapter 3
Basic Features of SINTAP/FITNET 3.1. Fitness-for-Service SINTAP/FITNET provides a fitness-for-service (sometimes referred to as a “fitness-for-purpose” or “engineering critical assessment”) procedure. In general, a structure is fit for service when it is capable of withstanding all the loads imposed by service or, in other words, when the conditions to cause failure are not reached. It should be noted that a wide range of failure modes including fracture, corrosion damage, cavity, erosion and creep damage are encountered in industrial practice. Only the final fracture of cracked components, caused by cleavage and micro-ductile fracture mechanisms and fracture modes such as stable crack initiation and extension, unstable crack initiation and plastic collapse, is considered in this book. It should be noted that the comprehensive FITNET method also includes modules which allow fatigue, high temperature creep and corrosion damage to be considered in analyses. Frequently the term “fitness-for-service” is used in contrast to what is called a “good workmanship” or “quality control” philosophy, which also deals with critical defect sizes but on the basis of experience and the potential (and limitations) of non-destructive inspection (NDI). Defects that are less severe than permitted by a certain quality-control specification are usually regarded as acceptable without any further consideration. Obviously, good workmanship arguments – to a certain extent – are arbitrary and usually conservative, but they are, nevertheless, of irreplaceable value in monitoring and maintaining high standards during production and fitness-for-service arguments should not be used for justifying poor quality standards. However, since fitness-for-service provides individual assessment, this is much more precise than good workmanship arguments. There will be cases where a component that fails to satisfy quality-control criteria is shown to be nonetheless safe, but there might also be cases where a component which meets these criteria in reality is not safe. Good workmanship and fitness-for-service should generally be regarded as complementary approaches with each having its own benefits. Today, the latter is frequently applied “after the fact”, that is, it is used to assess a component in which a conventionally unacceptable defect was found during manufacturing or in
20
Fitness-for-Service Fracture Assessment of Structures Containing Cracks
service. In addition to this important kind of application there is, however, an even more important future perspective in combining both assessment philosophies so that fitness-for-service results are used as input for specifying quality-control measures more precisely and, where required, more individually.
3.2. Potential Tasks of a SINTAP/FITNET Analysis The questions to be answered by SINTAP/FITNET follow the so-called “fracture mechanics triangle” (Fig. 3.1). The fracture behaviour of a given component is controlled by the three corners of the triangle: (a) The loading of the component. This includes the applied (or primary) loads and secondary loads such as residual stresses. (b) The crack dimensions and shape; through crack, embedded crack, surface crack, etc. (c) The fracture toughness of the material. Further parameters affecting the behaviour of a cracked component are the deformation properties of the material (its stress–strain curve) and constraint issues (geometry dependence of the toughness). All these parameters are covered by the SINTAP/FITNET procedure. The potential tasks of a SINTAP/FITNET analysis are based on the triangle. If two “corners” are known, the limiting value of the third can be determined. (a) A crack is detected or postulated (e.g., based on the NDI detection limit). With known loading, defect size and shape the question whether the component is safe or not can be answered. In conjunction with a fatigue crack extension analysis – which itself is not part of the fracture module of FITNET – the residual lifetime up to final failure can be determined as, for example, input information for establishing inspection intervals. (b) If the applied load as well as the material properties, including the fracture toughness, are known, the desired information is the critical size of a postulated crack, or – again combined with a fatigue crack extension analysis – the size of the crack that could grow to its critical size within a certain time, for example, the time up to the next inspection. This information is then used as input for the NDI, that is, what crack size has to be detectable with high confidence.
Basic Features of SINTAP/FITNET
21
Loading (primary and secondary)
Crack size and shape
Fracture toughness
Figure 3.1: The so-called “fracture mechanics triangle”.
(c) The third “edge” marks the fracture toughness. A minimum required fracture toughness can be specified for a postulated crack large enough to be reliably detected in quality control or in-service inspection. The rest of this chapter introduces some basic features of the procedure, these will be described in detail in subsequent chapters.
3.3. Multi-Optional Concept The SINTAP/FITNET flaw assessment module permits analyses at multiple levels of complexity and accuracy. Higher options are much more complex than a lower one and need improved input information; however, the user is “rewarded” by less conservative results. The various analysis options are mainly defined by the quality and completeness of the input information on the deformation and fracture behaviour of the material. SINTAP/FITNET comprises the following options (Table 3.1). (a) Option 0 – “Basic Option” This option is not recommended for general use but can be an alternative for cases of very limited knowledge of the material properties. It requires the information on yield strength and Charpy data only. The results of an Option 0 analysis can be unduly conservative. (b) Option 1 – “Standard Option” This is the minimum recommended option. It requires knowledge of the yield strength and the ultimate tensile strength of the material and of the toughness, which ideally should be available from at least three toughness tests. It can also be applied to strength mismatch components, for example, weldments, with yield-strength mismatch ratios less than 10%. In most such applications, the lower of the base and weld tensile properties have to be used.
22
Fitness-for-Service Fracture Assessment of Structures Containing Cracks
Table 3.1: SINTAP/FITNET assessment options Option
FITNET designation
Original SINTAP designation
Data needed
Application range remarks
Basic Option 0
Basic Option
Default Level
Yield strength, Charpy data
When no other tensile data available
Standard Options 1
Standard Option
Basic Level
2
Mismatch Option Stress-Strain Defined Option
Mismatch Level
3
4
J-Integral Analysis
5
Constraint Option
Stress-Strain Level
Yield strength, Tensile strength, Fracture toughness Yield strength, Charpy data Complete stress-strain curve, Fracture toughness
Advanced Options J-Integral Level Complete stress-strain curve, Fracture toughness Constraint Level Geometry dependent fracture toughness
When no complete stress-strain curve available For yield strength mismatch >10% Assessment modules for homogenous & yield strength mismatched components Numerical determination of crack tip parameter, e.g., J Based on two-parameter concepts
(c) Option 2 – “Mismatch Option” This option refers to a modification of Option 1 for strength-mismatch components with yield strength mismatch ratios larger than 10% and ligament yielding Lr > 075. For the terminology see Section 5.2.
Basic Features of SINTAP/FITNET
23
(d) Option 3 – “Stress–Strain Defined Option” For this option the complete stress–strain curve of the material, as well as fracture toughness data, are required. Both homogeneous and strengthmismatch components can be assessed in special assessment modules. Note that, in some cases, a complete stress–strain curve will be available in conjunction with Charpy data only instead of with fracture toughness data, or vice-versa. In such cases the options may be “mixed” but the user should always be aware of the potential conservatism of his analysis varient. In addition to these options the procedure comprises further modules that can help to reduce conservatism. (e) Option 4 – “J-Integral Analysis” This option includes the finite element determination of the crack driving force in terms of J (or crack tip opening displacement CTOD) in the frame of a general SINTAP/FITNET analysis. Only limited guidance on numerical aspects is provided in the documents and this will not be discussed in this book. (f) Option 5 – “Constraint Option” This option provides rules for the estimation of geometry dependent toughness values, in particular in low constraint geometries such as thin sections or predominantly tension-loaded plates. In this book Option 5 will not be described as a separate option but is included in Sections 4.4.6 and 6.7. No discussion will be provided on the original SINTAP Level 6 (Leak-beforebreak). The philosophy of different analysis levels enables the user to perform uncomplicated and straightforward analyses in a number of cases. If a lower-option analysis indicates safety, no further analysis is necessary. On the other hand, the application of a higher option is advisable when a more accurate, that is, a less conservative, analysis is needed. This requires more complete input information and sufficient expertise to handle the additional complexity. As a consequence of the multi-optional concept, an unacceptable result at a lower analysis option does not necessarily mean the failure of the component. Instead, it rather provides a motivation for repeating the analysis at the next higher level such as illustrated in Fig. 3.2. Note that the effect of assessment at a higher option on the final conservatism of the SINTAP/FITNET analysis does depend on the ligament yielding parameter Lr (for definition see Section 5.2). This will be discussed in Section 6.14.
24
Fitness-for-Service Fracture Assessment of Structures Containing Cracks
Only σY (Re or Rp0.2) available
σY and Rm available
Option 0
yes
no
Safe service?
Strength mismatch?
no
yes
Option 1 (∗)
Option 2 (∗)
yes
Safe service? no Option 3 (∗∗)
(∗) σ and Y
Rm have to be available
(∗∗)
yes Safe service? no
Complete stress-strain curve has to be available
Complete FEM analysis yes Safe service?
Component is safe!
no
Component is not safe!
Figure 3.2: Multi-optional assessment concept of SINTAP/FITNET.
3.4. FAD Versus CDF Analyses The terms FAD and CDF analysis and their scientific and historical backgrounds have already been introduced in Sections 2.4 and 2.5. In the FAD approach, a roughly geometry-independent failure line is constructed by normalising the crack driving force by the material’s fracture resistance. The assessment of the component is then based on the relative location of a geometry-dependent assessment point with respect to this failure line. In contrast to this, in the CDF philosophy the applied and material sides are strictly separated. The determination of the
Basic Features of SINTAP/FITNET
25
crack driving force in the component, and its comparison with the fracture resistance of the material, are two separate steps. Note that both analysis types are harmonised with each other in the SINTAP/FITNET procedure. In other words, FAD and CDF lead to identical results. Whether to follow an FAD or a CDF philosophy is merely a matter of personal preference and not of importance for the correctness of the results. More detailed guidance on FAD and CDF analyses including worked examples will be provided in Chapter 6.
3.5. Integrated Concept As mentioned above, the fracture of a specimen or component can occur by different mechanisms such as • cleavage fracture, • micro-ductile fracture or • plastic collapse. While these possibilities have explicitly to be taken into account in the determination of the fracture toughness of the material, they are covered by the same set of equations f(Lr ) for the crack driving force in SINTAP/FITNET, see Fig. 3.3. There is no need to determine whether a cracked component operates in small-scale yielding, net-section yielding or in the plastic-collapse regime (for a definition of these terms see the Glossary “crack tip plasticity”). This is of great benefit since this distinction would be much more complicated for a component than for a test specimen. Another aspect of the integrated concept is that the FITNET document provides a number of extensive annexes with solutions for input and model parameters Small-scale yielding 1
Contained yielding Net-section yielding Plastic collapse
f (Lr)
Lr = F/FY = σref /σY
1
Lrmax
Figure 3.3: Ligament yielding ranges covered by SINTAP/FITNET.
26
Fitness-for-Service Fracture Assessment of Structures Containing Cracks
such as stress intensity factors, yield loads, welding residual stress profiles, stress magnifications due to misalignment and constraint parameters for a wide range of component geometries. These solutions make it possible, in many cases, to perform SINTAP/FITNET analyses without consulting additional sources. In Fig. 3.4 only those annexes are included which are of importance for the topic of this book. Annex A: Stress intensity factor (SIF) solutions Flat plates with and without hole Round bars and hollow cylinders Nozzles Welded joints Through thickness cracks Embedded cracks Extended and semi-elliptical surface cracks Corner cracks Annex B: Yield load solutions Flat plates with and without hole Round bars and hollow cylinders Welded joints (with and without strength mismatch) Tubular joints Through thickness cracks Embedded cracks Extended and semi-elliptical surface cracks Annex C: Welding residual stress profiles Plate and pipe butt welds Plate and pipe T-butt welds Set-in nozzles Repair welds Surface profiles Through thickness profiles Longitudinal and transverse profiles Annex I: Bending stresses due to misalignment Axial and angular misalignment Ovalisation Flat plates and hollow cylinders Butt and fillet welds Annex K: Input for constraint analysis T stress and β solutions Flat plates and hollow cylinders
Figure 3.4: FITNET annexes containing compendia of solutions needed for fracture analysis.
Basic Features of SINTAP/FITNET
27
A further annex (Annex L) provides the user with literature sources in which information on material properties can be found. Tables of specific values that are required for use in conjunction with the SINTAP module, as sometimes given in industry-specific documents, are avoided. Instead, the user is referred to conservative estimates based, for example, on Charpy information. At this point a warning is advisable with respect to literature data on fracture toughness. The user should always be aware that toughness values are much more sensitive to even minor changes in the microstructure than strength properties or the Paris constants of a da/dN-K curve. An example is that structural steels of identical yield or tensile strengths, which comply with the same manufacturer’s specification, can show quite different fracture resistance and this is even the case for materials that undergo heat treatment by the final producer. Data compendia do, however, make sense in specific industries where a limited variability of materials is used and where the materials are subject to strict input quality control.
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Chapter 4
The Input Parameters 4.1. Loading Input Parameters 4.1.1. General Considerations All relevant loads have to be taken into account in a SINTAP/FITNET analysis. These may include: • • • •
Applied service loads such as external forces, moments or pressure, Dead weight and inertia loads, Thermal stresses Residual stresses.
The loads can be available as forces, moments or pressure, or as stress profiles across the section of the component. The latter may be obtained by numerical analysis techniques or handbook solutions. The stress profiles used in SINTAP/FITNET analysis Options 0 to 3 refer to the uncracked geometry and are commonly available from conventional strength analyses. Note, however, that caution has to be exercised for structures with multiple load paths, as the failure of one load path will affect the stress distribution on the remaining uncracked path(s).
4.1.2. Primary and Secondary Stresses In the analyses, the stresses are categorised as primary or secondary. As a rule, primary stresses arise from the applied mechanical load, including any dead weight or inertia effects, whilst secondary stresses result from suppressed local distortions, for example during the welding process, or are due to thermal gradients. Secondary stresses are self-equilibrating across the structure, so that net force and bending moment are zero. This separation is necessary because primary stresses contribute to plastic collapse but secondary stresses do not. As a consequence, the K-factor determination is based on both primary and secondary stresses, whereas only the primary stresses are taken into account for the determination of the yield load, FY (or the reference stress ref or ligament yielding factor Lr respectively, see Section 6.6).
30
Fitness-for-Service Fracture Assessment of Structures Containing Cracks
Note, however, that categorisation is not always straightforward because secondary stresses, which are self-equilibrating in the whole structure, will not always be self-equilibrating on the section containing the crack. Caution has to be exercised when the crack is small compared to the spatial extent of the secondary stress distribution, or in cases with significant elastic follow-up from the rest of the structure. If it is uncertain whether a stress is primary or secondary, its treatment as primary is conservative.
4.1.3. Definition of the Stress Profile for Use with the K-Factor The stress profile can be approximated by a polynomial expression or it can be linearized. If possible, the former method is to be preferred. However the solution that will be finally applied depends on the available K-factor expression for the case under consideration. (a) If the uncracked structure stress distribution x in the thickness direction is available, from, for example, a finite element analysis, this can be represented by a polynomial x/t =
n
i x/ ti = o + 1 x/ t + 2 x/ t2 + · · · + n x/ tn
(4.1)
i=0
with n being the order of the polynomial, t the wall thickness and x the distance from the surface through the thickness of the component (Fig. 4.1). Polynomial expressions of the stress distribution can be used, for example, in conjunction with weight function solutions to determine the K-factor. K solutions of that type are usually based on third- to sixth-order polynomials. Note that the quality of a polynomial approximation should always be tested (e.g., by eye). Example 4.1: A stress profile across the wall of a component is provided in Table 4.1. The approximation by Eqn (4.1) gives polynomials, for example, Third order:
x t
in MPa = 22883 − 68758 · + 90513 ·
x 2 t
x t
− 41958 ·
x 3 t
The Input Parameters
Sixth order:
x t
in MPa = 23009 − 59856 · + 637755 ·
x t
x 3
− 58350 ·
31
x 2
t x 4
− 12 89693 · t t x 5 x 6 + 10 86727 · − 3 36601 · t t The eye check of Fig. 4.1 shows that both expressions describe the empirical stress distribution well. In this case it is more convenient to use the lower-order polynomial for the further analysis. 300
Sixth order polynomial Third order polynomial
Stress in MPa
250
200 x t
150
100
50
0
0
0.2
0.4
0.6
0.8
1
x /t
Figure 4.1: Example 4.1: Approximation of the empirical stress profile of Table 4.1 by third and sixth order polynomials. Table 4.1: Example 4.1: Empirical stress profile across the wall of a component, for example as the result of a finite element analysis x/t
Stress perpendicular to the assumed crack plane in MPa
x/t
Stress perpendicular to the assumed crack plane in MPa
x/t
Stress perpendicular to the assumed crack plane in MPa
0.0 0.1 0.2 0.3
230 170 120 90
04 05 06 07
73 63 55 46
08 09 10
40 34 30
32
Fitness-for-Service Fracture Assessment of Structures Containing Cracks
For hollow cylinders it is common to present the stress distribution as a combination of a local non-linear stress distribution and a global bending stress bg obtained as the quotient of bending moment, Mb , and section modulus, Wb x =
n
i x/ti + bg =
i=0
n
i x/ti + Mb Wb
(4.2)
i=0
An example for this type of approximation is provided in Fig. 5.3. First, the global bending stress bg has to be subtracted from the finite element distribution and then the remaining profile is fitted by the polynomial. An example will be provided in Section 5.1.2.2. If the stress profile across the wall is available from a finite element analysis, but only a K solution is available for linear stress profiles in terms of membrane and bending stress components, m and b , the stress profile has to be approximated by a straight line over the position of the crack. The principle is illustrated for surface and embedded cracks in Fig. 4.2.
Stress
Empirical stress distribution Fitted stress distribution σ1 Stress
σ1
σ2
σ2
Potential crack
Potential crack
t
t
Figure 4.2: Stress linearisation for the determination of membrane and bending stress components.
The Input Parameters
33
From the stresses at both surfaces, 1 and 2 , the membrane and bending stress components, m and b , can then be determined by m = 05 · 1 + 2
(4.3)
b = 05 · 1 − 2
(4.4)
and Example 4.2: The empirical stress profile of Example 4.1 is used to obtain the membrane and bending stress components for a relative crack size of a/t = 03. The procedure is illustrated in Fig. 4.3. With 1 = 230 MPa and 2 = −23667 MPa the stress components are determined as m = −334 MPa and b = 23334 MPa 300 250 σ1 = 230 MPa 200 x t
150
Stress in MPa
100 50
Potential crack
0 – 50 – 100 – 150 – 200 σ2 = – 236.67 MPa – 250
0
0.2
0.4
0.6
0.8
1
x /t
Figure 4.3: Example 4.2: Stress linearization. Determination of the 1 and 2 stresses.
34
Fitness-for-Service Fracture Assessment of Structures Containing Cracks
Note that polynomial and linearized stress profiles will frequently be used in conjunction with substitute geometries for K-factor determination. This means that the stress profile will be determined – usually by finite element analysis – for the component section to be assessed. It can then be approximated as described above and finally used as input to a K solution for a substitute geometry, such as a plate or a cylinder (Fig. 4.4). The component to be assessed and the substitute geometry should not be too different with respect to their stiffness. If the component is stiffer than the substitute geometry the resulting K-factor will be over-estimated and, thus, conservative. According to Fig. 4.2 and Eqns (4.3) and (4.4), a special feature of the membrane and bending stress components is their dependency on the crack length. In Example 4.2, the bending stress will become smaller and the membrane stress larger for a crack size greater than the assumed a/t = 03. Note that this is in contrast to the conventional definition of the stress components m = F/A
and
b = Mb /Wb
(4.5)
Component A A t
x
t
x A
A Stress distribution in section A-A
x
x=0
x=t
x
x=0
x=t
Approximation of the stress field by a polynomial or linearly Determination of the K-factor for a substitute geometry t
t
Figure 4.4: Determination of K-factors using substitute geometries.
The Input Parameters
35
with F being the tensile force, A the cross section area, Mb the bending moment and Wb the section modulus. A number of K solutions available in the literature and in compendia are also based on the conventional definition. The type of specific K solution to be applied defines which m and b are relevant for an individual application. For a linear stress distribution across the section of a component without a geometric discontinuity, the use of the polynomial approximation (Eqn 4.1), the straight-line approximation (Eqns 4.3, 4.4) and the definition according to Eqn (4.5) will yield identical results. In this case the 0 and 1 terms of the polynomial refer to the membrane and bending stress components, m and b . For a non-linear stress distribution the linearization technique will usually tend to be more conservative than the polynomial approach, slightly overestimating the resulting K values. An example for this effect is provided in Section 5.1.2.4.
4.1.4. Definition of the Stress Profile for Yield Load Determination Yield load (FY ), reference stress (ref ), or ligament yielding parameter (Lr ) solutions are sometimes based on bending and membrane stress components. Note, however that the m and b used for determining the yield load are different from those used for determining K as described in Section 4.1.3. In general they are obtained by m =
1 t dx t 0
(4.6)
and 6 t t b = 2 − x dx t 0 2
(4.7)
If the stress profile is available as a sixth-order polynomial (Eqn 4.1), m and b , can be solved as 1 1 1 1 1 1 m = 0 + 1 + 2 + 3 + 4 + 5 + 6 2 3 4 5 6 7
(4.8)
1 1 9 2 5 9 b = − 1 − 2 − 3 − 4 − 5 − 6 2 2 20 5 14 28
(4.9)
If the polynomial is lower than sixth order, the superflous i coefficients are simply set to zero, for example, for a third degree polynomial 4 = 5 = 6 = 0.
36
Fitness-for-Service Fracture Assessment of Structures Containing Cracks
Example 4.3: The empirical stress profile of Example 4.1 is used to determine m and b for yield load determination. With the polynomial coefficients of Example 4.1 the results are Third-order polynomial: m = 8185 MPa and b = 8003 MPa, Sixth-order polynomial: m = 8166 MPa and b = 8066 MPa. Note that the m and b stress components used for yield load determination deviate significantly from those used for the K-factor determination (m = −334 MPa and b = 23334 MPa) obtained in Example 4.2.
4.2. Flaw Characterisation 4.2.1. Planar and Volumetric Flaws Flaws can be planar or non-planar/volumetric. Planar flaws are cracks, laminations, lack of fusion in welds, undercuts, sharp groove-like localised corrosion, branch-type cracks due to environmental effects, etc. Volumetric flaws are, for example, cavities, aligned porosity, solid inclusions and local thinning as a consequence of corrosion. Planar flaws are generally treated as cracks, but the decision as to when a volumetric flaw should be treated as a crack is rather complicated and has to include the user’s experience of the structure under consideration. It should be taken into account that non-destructive examination might not be sensitive enough to find out whether or not microcracks have initiated from a volumetric flaw. The following discussion refers to cracks and crack-like flaws.
4.2.2. Basic Crack Types Flaw characterisation means that an existing or postulated crack is modelled by a simpler geometry such as a through crack with a straight crack front, an embedded crack with elliptical shape or a surface crack with a semi-elliptical shape. The basic types of idealised planar flaws are given in Fig. 4.5. Note that for throughwall crack configurations the nomenclature a or 2a is sometimes used instead of 2c, for example, in conjunction with strength mismatched configurations as in Sections 5.2.6 and 6.11.
4.2.3. Crack Shape Idealisation Crack shape idealisation becomes necessary when real cracks have been detected during an inspection. Idealisation means that a flaw or crack with a complex
The Input Parameters
37
shape is modelled as one of the basic crack types shown in Fig. 4.5, but with conservative dimensions, so that the idealised crack geometry should be more severe that the actual one. The principle is shown in Fig. 4.6 in which the idealised cracks are characterised by the containment rectangles of the actual cracks.
Surface crack
Through wall crack
Edge crack
a 2c
2c
c
Corner crack
Embedded crack 2a
a c
2c
Figure 4.5: Basic types of planar flaws.
a
a 2c
2c
2c
2c
c
c
a
a c
c
2a
2a 2c
2c
Figure 4.6: Idealisation of detected planar flaws for a SINTAP/FITNET analysis.
38
Fitness-for-Service Fracture Assessment of Structures Containing Cracks
4.2.4. Interaction Effects of Multiple Cracks 4.2.4.1. Alignment Criteria Multiple cracks can be coplanar or non-coplanar. Non-coplanar cracks under mode I loading tend to shield one another, their K factors being reduced relative to those of the single cracks. In most cases, therefore, the interaction of noncoplanar cracks is of little concern for an assessment that has the aim of being conservative. Exceptions are closely spaced non-coplanar cracks that are aligned to the same cross section. The SINTAP/FITNET criterion for this is based on spacing between the cracks H ≤ min 2a1 2a2 for embedded cracks (Fig. 4.7a) and H ≤ min 2a1 a2 for surface and embedded cracks (Fig. 4.7b) Note that the flaw alignment criteria are based on the maximum principal stress plane. 4.2.4.2. Combination Criteria If coplanar cracks, that is, multiple cracks on the same cross section, are in close proximity to one another, they may show some interaction effects that make them more serious than the individual cracks. In such cases, rules are given (a)
(b)
Plane normal to the maximum principal stress
Plane normal to the maximum principal stress
2a1
2a1
Crack 1
Crack 1 Crack 2
Crack 2
2a2 H
a2 H
Figure 4.7: SINTAP/FITNET flaw alignment criteria – (a) Embedded cracks; (b) Surface and embedded cracks.
The Input Parameters
39
in SINTAP/FITNET to decide when they have to be combined and treated as one single crack whose dimensions encompass the individual cracks. After the correction is done, usually no further interaction between the obtained effective cracks has to be considered. If the spacing between coplanar cracks is sufficient to avoid interaction, they are treated individually. Only the worst-case crack needs to be considered in a fracture analysis. No complete overview will be given here on the SINTAP/FITNET interaction rules. Fig. 4.8 shows some examples for illustrating the principle. With the terminology of Fig. 4.8, interaction has to be assumed for s ≤ min 2c1 2c2
s ≤ max 05a1 05a2
or
for a1 /c1 or a2 /c2 > 1
otherwise s ≤ max 05a1 05a2 in Fig. 4.8 (a) and for s ≤ a1 + a2 in Fig. 4.8 (b). The reason why interaction effects have to be taken into account is that the local stress field at the tip of a crack controls the crack driving force and this can be significantly affected by adjacent cracks. In the case of coplanar cracks, the interaction criteria are defined for a maximum tolerable increase in the crack driving force. Since “maximum tolerable” is a term open to discussion, slightly different interaction criteria are adopted by various documents. An overview on the criteria of ASME, BS 7910, API 579 and SINTAP is provided in [4.1]. The interaction rules discussed so far consider only elastic behaviour. It has, however, to be expected that in the net-section and gross-section yielding regimes
(a)
a2
a1 2c1 s (b)
2c2
max(a1, a2 ) 2c1 + 2c2 + s
2c1
2a1
s
2a2 2c2
2a1 + 2a2 + s
max(2c1, 2c2 )
Figure 4.8: Interaction rules for co-planar flaws – (a) Surface cracks; (b) Embedded cracks.
40
Fitness-for-Service Fracture Assessment of Structures Containing Cracks
the magnitude of the interaction effect will also be affected by the plastic deformation pattern. Therefore, SINTAP/FITNET provides a separate rule for situations where failure by brittle fracture can be excluded. A maximum crack length 2cgsY is determined for a rectangular surface crack of a depth amax which refers to the crack depth of the deepest single crack: 2cgsY =
1 − Y /Rm W · t · 1 + Y /Rm amax
(4.10)
Equation (4.10) is obtained for a flat plate of width W and thickness t and gross-section yielding conditions; the plastic deformation is not confined to the crack plane but also includes the remote cross section. For large plates (W > 300 mm), an effective W has to be specified based on experience or experiments. A correction for interaction has to be carried out for cracks the spacing of which is s≤
1 + Y /Rm 2ci − 2cgsY 2 · Y /Rm
(4.11)
where i is a counter variable for multiple cracks. For plates showing strength mismatch, the mismatch ratio M plays an additional role (for strength mismatch see Section 6.11.1). Equation (4.10) is then modified by 2Y /Rm B W·t 2cgsY = 1 − · (4.12) M · 1 + Y /Rm
amax with M being the yield strength mismatch ratio YW /YB (W = weld metal; B = base plate).
4.2.5. Crack Re-characterisation Crack re-characterisation means that an embedded or a surface crack is recharacterised as a surface or through-wall crack respectively, as shown in Fig. 4.9. There are cases where local failure is calculated in the ligament(s) ahead of the embedded or surface crack but, when the crack is re-characterised as a (larger) surface or through-wall crack, the component is predicted to be globally safe. If this condition is given, the failure analysis can be based on the re-characterised surface or through-wall crack. Allowance has, however, to be made for dynamic effects combined with a certain amount of additional crack growth at the surface for the break-through event. This is covered by adding a crack portion 2a + p in Fig. 4.9 (a) and t in Fig. 4.9 (b). If the ligament failure is predicted to be brittle,
The Input Parameters
41
(a)
2a
2a + p
2c
p
2c + 2a + p
(b) t
2c
2c + t
Figure 4.9: Crack re-characterisation in general application – (a) embedded crack; (b) semi-elliptical surface crack.
the fracture resistance used in the assessment of the re-characterised crack should be the dynamic or crack arrest toughness. Special care should be exercised for cases of large difference between static and dynamic toughness [4.2]. Note that, when it is used for assessing leakage rates, crack re-characterisation is also required in conjunction with a leak-before-break analysis. This information may be necessary to specify the time between the break-though of the crack and the detection of the leak. In this context, conservative assessment means an overestimation of this time span, which refers to an underestimation of the leakage area defined by the crack dimensions at break through. As a consequence, the crack size immediately after leakage should be modelled as realistically as possible and not potentially overestimated, as in Fig. 4.9. The SINTAP/FITNET recommendations for crack re-characterisation for leak-before-break are shown in Fig. 4.10.
4.2.6. Crack Orientation and Projected Crack Depth Frequently, cracks are oriented along a plane normal to the maximum principal stress direction – mode I crack opening. Even if they initially grow in another orientation, they will turn to mode I during further extension. Therefore, as a rule, a postulated crack will be aligned with the principal stress axes in cases where there is no clear indication of mixed mode loading, as described below. If the orientation of a real crack deviates from the mode I plane it can be projected onto the principal plane referring to the maximum principal stress.
42
Fitness-for-Service Fracture Assessment of Structures Containing Cracks
after break-through c
at break-through (local instability) tensile loading t 2c
2c + t t
through-wall bending t 2c
2c + t
Figure 4.10: Crack re-characterisation in leak-before-break application. σ1
2c0 σ2
2c2
β
σ2
2c1
σ1
σ1
a
σ1
Figure 4.11: Simplified projection method for cracks not oriented on a principal plane (according to [4.1]).
A simple method for this is shown in Fig. 4.11. The projected crack depth, a, is given in the figure and the projected crack length, c, can be determined by √ √ c for 1 c1 ≥ 2 c2 √ √ c= 1 (4.13) c2 for 1 c1 < 2 c2 This simple solution can, however, yield non-conservative estimates in some cases. As an alternative, an approach based on energy release rate considerations is presented in [4.1] although it will not be described here in detail. The author gives an approximate solution for biaxial loading as c/co = cos2 + 05 1 − B sin cos + B2 sin2
(4.14)
The Input Parameters
for cracks projected to the maximum principal stress (1 plane, and
c/co = 1/B2 · cos2 + 05 1 − B sin cos + sin2
43
(4.15)
for cracks on the 2 principal stress plane, with B being the biaxiality ratio defined as B = 2 /1
for 2 > 1
(4.16)
In Eqns (4.14) and (4.15) the crack length c corresponds to half flaw length as shown in Fig. 4.11, or to the total length in the case of edge or corner cracks. Note that Eqns (4.14) and (4.15) are valid only if both principal stresses are positive. If 2 is negative (compressive), B should be set to zero and Eqn (4.14) should be applied [4.3, 4.4]. A conservative option is to set the equivalent crack size c equal to the measured crack size co irrespective of the crack orientation. The projection of a crack to the mode I plane is not admissible [4.5] when • the angle between the principal plane and the plane of the actual flaw exceeds a value of 20 , • there is only a small difference between the stress intensity factors on the planes of projection, • the maximum stress intensity factor and the maximum limit load belong to two different projection planes, • one of the principal stresses is significantly compressive, where significant means of an order similar to the maximum-principal stress. In such a case, a mixed-mode analysis should be performed taking into account the shear stresses as well. Guidance on this is given in Section 6.8.
4.3. Deformation Characteristics of the Material 4.3.1. General Remarks As with other material properties, tensile data should be obtained for the specific material and batch, in the correct product form and, if possible, at the relevant temperature and loading rate. If there is any indication of a possible change or degradation in properties as a result of ageing over the lifetime of the component, or as a consequence of a fabrication process, then this has to be taken into account. In some cases, such as rolled material, the specimen orientation can affect the tensile data. Usually, lower bound values of the yield strength and tensile strength
44
Fitness-for-Service Fracture Assessment of Structures Containing Cracks
have to be used for an assessment. There are, however, exceptions that will be explained in the corresponding sections below. It has already been mentioned in Section 3.3 that the SINTAP/FITNET procedure is organised in different analysis options mainly with respect to the quality and completeness of the tensile data.
4.3.2. Engineering and True Stress-Strain Curve Both engineering and true stress-strain curves are used in a SINTAP/FITNET analysis. Based on the engineering curves, the yield strength Y and the ultimate tensile strength Rm are required for Option 1 and Option 2 analysis whereas the complete true stress–strain curve is used for Option 3 analysis. Finite element analyses in fracture mechanics contexts are usually based on true stress–strain curves. The true stress and strain data, t and t , can be determined from the engineering data, and , using: t = 1 +
(4.17)
t = ln1 +
(4.18)
and
Because Eqns (4.17) and (4.18) are based on the assumption of a homogeneous strain distribution along the gauge length of the tensile specimen, these equations are applicable only up to the onset of necking. Beyond this, the true stress can be determined from measurements of the actual cross section diameter in the necking region. In addition, since the neck – which by its nature is a notch – introduces a complex triaxial stress state, a further correction, such as the “Bridgman correction”, is needed to provide an estimate of the uniaxial stress that would exist if no necking took place. The relation between the engineering, the true, and the Bridgman corrected true stress–strain curves is schematically illustrated in Fig. 4.12.
4.3.3. Modulus of Elasticity (Young’s Modulus) and Poisson’s Ratio Young’s modulus and Poisson’s ratio values are also required for the temperature relevant to the component analysis to be carried out. Compendia can be used if no individual values are available; some approximate values are provided in Table 4.2 [4.6].
The Input Parameters
45
Stress
Ultimate tensile strength
Yield strength
True curve (correction for necking) True curve (no correction for necking) Engineering curve
Strain
Figure 4.12: Schematic comparison of engineering and true stress–strain curves. Table 4.2: Approximate values of Young’s modulus and Poisson’s ratio for various classes of materials (taken from [4.6]) Material Temperature Ferritic steels Steels with c. 12% Cr Austenitic steels Aluminium alloys Titanium alloys
Modulus of Elasticity, GPa
Poisson’s ratio
20 C
200 C
400 C
600 C
20 C
211 216 196 60−80 112−130
196 200 186 54−72 99−113
177 179 174 − 88−93
127 127 157 − 77−80
c. 0.30 c. 0.30 c. 0.30 c. 0.33 0.32−0.38
4.3.4. Yield Strength and Tensile Strength Note that, in SINTAP/FITNET, the yield strength of the material is designated by the general term Y . However, with respect to specific applications, it has different meanings varying from case to case. For materials with continuous yielding, that is, without yield strain (Lüders’ strain), Y refers to the proof strength of the material, Rp 02 , which is defined for a plastic strain of 0.2%. For materials showing a yield plateau, it refers to the lower yield strength ReL . For Option 0, 1 and 2 analyses, instead of the individually determined yield strength, specified minimum values given by manufacturer’s standards are acceptable. However, care should be taken as either the lower or upper yield plateau, ReL and ReH , can be quoted on test certificates, although it is usually the latter value. Therefore, in order to avoid non-conservatism, the value provided should
46
Fitness-for-Service Fracture Assessment of Structures Containing Cracks
be factored by 0.95 when applied to the determination of the yield load FY (or ref or Lr respectively), unless it is certain that the data refer to the lower yield plateau. Note that, in the SINTAP/FITNET procedure, conservatism of the yield strength – and ultimate tensile strength – is defined quite differently depending on its use in the analysis. For determining FY , ref or Lr , conservatism is given by lower bound values. However, for estimating welding residual stresses (Section 6.6.4), conservatism refers to upper-bound values. Special care should be exercised for leak-before-break analyses where the conditions of conservatism can be very complex. Another case where upper-bound values are required is the estimation of a maximum critical crack size after an overload or proof test. In one special case – for estimating welding residual stress profiles – Y is defined as the room temperature proof strength for a strain of 1% for austenitic steels or, if this is not available, as 1.5 Rp02 . For the definition of mismatch ratio, M, the mean values of Y are used for both weld and base materials. In cases in which more than one measurement of Y is available, the lowest or highest value has to be chosen depending on what is conservative for that specific application.
4.3.5. Flow Stress The so-called flow stress, f , can be understood as an effective yield strength taking into account strain hardening. In the SINTAP/FITNET procedure it is defined as f = 05Y + Rm
(4.19)
Because the definition of f is somewhat arbitrary, variations of this definition are used in the literature and some other flaw assessment procedures. In the context of SINTAP/FITNET, only the specification according to Eqn (4.19) is permitted. For definition of the cut-off, Lr max , mean values of f and Y are used.
4.3.6. Strain Hardening Coefficient In general, the strain hardening coefficient N (0 > N > 1) is defined as the slope of the plastic branch of the stress–strain curve, when plotted in double-logarithmic co-ordinates. Based on this, the SINTAP procedure uses a conservative estimate, given by
N = 03 1 − Y (4.20) Rm
The Input Parameters
47
Strain hardening exponent N
0.4
Lower bound curve 0.3
0.2 0
0.1
0 0.6
0.7
0.8
0.9
1
σY/Rm
Figure 4.13: Derivation of the SINTAP/FITNET lower bound equation for the strain hardening coefficient N.
Equation (4.20) was obtained as a lower-bound curve (Fig. 4.13) to a large data set of individually determined N versus Y /Rm pairs. This set was based on a wide range of steels with yield strengths between 300 and 1,000 MPa and Y /Rm ratios between 0.65 and 0.95, although data from some aluminium alloys, brasses and nickel alloys were also included. Nevertheless, the general application to nonferrous materials should be treated with caution. Note, however, that if structures made of materials for which Eqn (4.20) is potentially not applicable have to be assessed, the general definition for determining N as the slope of the doublelogarithmic plotted plastic branch of the true stress–strain curve can still be used. The same type of approach is recommended by the authors of the present book for cases where an upper-bound N is needed for reasons of conservatism, as in the context of a proof test, although using a factor of 0.5 instead of 0.3 in Eqn (4.20) provides a mean fit to the data. Alternatively, Fig. 4.13 could be used to derive an upper-bound curve. Note that a number of further definitions of the strain hardening exponent are used in the literature. Although the parameter is commonly also designated as N or n, the values may strongly diverge so no use should be made of N values provided in external sources unless it can be demonstrated that the basic definition is compatible with SINTAP/FITNET or, at least, conservative with respect to this.
4.3.7. Yield Strain (Lüders’ Strain) In Option 2 and 3 analyses materials with and without yield strain must be identified. In many situations the data will be complete enough to establish the
48
Fitness-for-Service Fracture Assessment of Structures Containing Cracks
yielding characteristics, but there are other cases where this may not be so. For structural steels, some indication can be obtained from the yield strength, the composition of the material and the process route. Guidance for the decision as to whether a yield plateau is expected or not is given in Table 4.3, in which the factors mentioned have been grouped according to standard specifications. It should, however, be recognised that the presence of a yield plateau is affected not only by the material and its test temperature, but also by the test method, the loading rate and the specimen design. Table 4.3 applies only to the steels listed. For other materials, the yielding behaviour should be individually established. If there is some doubt regarding whether a material shows a yield plateau or not, it is usually conservative to assume its presence. Again, exceptions are proof test and (partly) leak-beforebreak applications. If no individual measurements of the Lüders’ strain are available, a conservative estimate can be made using: 003751 − 0001 ReH for ReH ≤ 1000 MPa (4.21)
= 0 for ReH > 1000 MPa As in the case of the strain hardening coefficient in Eqn (4.20) the general solution for the Lüders’ strain was obtained empirically, but as a best fit to the data. Nevertheless, the solution is assumed to be conservative in many cases, since the Lüders’ strain is known to be smaller or even to disappear in large-scale tests, in particular in the presence of bending stress components.
4.3.8. Tensile Data Relevant to Welds In the case of weldments where the difference in yield strength between the base material (denoted B) and the weld metal (denoted W) is greater than 10%, strength mismatch effects have to be taken into account in the SINTAP/FITNET procedure. Guidance on how to do this is given in Section 6.11. Complete stress– strain curves for both materials have to be available for Option 3 analyses. For Option 2 analyses only the yield and ultimate tensile strength values of these materials are required. Local stress-strain curves can be obtained by mini-tensile tests [4.7, 4.8]. No detailed discussion on these types of test is provided here. Note, however, that care should be taken to avoid any affect of the machining procedure on the measured load-deformation curves. Care should also be exercised when mini specimens of rectangular cross sections are used, since the yielding pattern across the section differs from that of a circular cross section and this may affect
The Input Parameters
49
Table 4.3: Guidance for determining whether a yield plateau should be assumed Yield strength range (MPa) ≤350
Process route As–rolled
Normalised
>350 ≤500
NA
Yes
Mo, Cr, V, Nb, Al or Ti present
NA
No1
EN 10025 type compositions without microalloy additions
Conventional normalising
Yes
EN 10113 type compositions with microalloy additions
Conventional normalising
Yes
EN 10113 compositions
Controlled Rolled
EN 10113 compositions
Quenched & Tempered
As-Quenched
Assume yield plateau?
Heat treatment aspects
Conventional steels (e.g. EN 10025 grades) without microalloy additions
Controlled Rolled
Quenched & Tempered
>500 ≤1050
Composition aspects
—
Yes
Light TMCR schedules2 Heavy TMCR schedules3
Yes
Mo or B present with microalloy additions Cr, V, Nb or Ti
Heavy tempering favours plateau
Yes
Light tempering favours no plateau
Yes1
Mo or B not present but microalloying additions Cr, V, Nb or Ti are (V particularly strong effect)
Heavy tempering
Yes1 No1
Mo or B present with microalloy additions Cr, V, Nb or Ti
Tempering4
Light tempering
5
Tempering
Yes1
No1 No
Mo or B not present but microalloy additions Cr, V, Nb or Ti are
Tempering4 Tempering5
Yes No1
All compositions
NA
No
1) uncertain; 2) Y < 400 MPa; 3) Y > 400 MPa; 4) to Y < ∼690 MPa; 5) to Y ≥ ∼690 MPa.
50
Fitness-for-Service Fracture Assessment of Structures Containing Cracks
the result at higher strains. A hybrid method consisting of measurements and finite element simulations is provided in [4.9]. Usually, lower-bound material tensile properties should be used for both base material and weld metal. It should, however, be kept in mind that the real strength of the material may deviate significantly from the specified minimum for the grade of material, so that the real yield strength of the material is frequently higher than required. Note that for the definition of mismatch ratio, M, the mean values of Y are used for both weld and base materials as stated before. It should also be kept in mind that weld tensile properties may vary through the thickness of a component and may be dependent on the specimen orientation. The tensile properties used should be the lowest (or highest) within the weld, irrespective of orientation and position, in order to provide conservative results.
4.3.9. Temperature and Strain Rate Dependency For situations in which the operating temperature is below room temperature but only the room temperature yield strength is available, the yield strength may be estimated by YT = YRT +
105 − 189 MPa 491 + 18 T
(4.22)
with Y given in MPa, T being the temperature of interest in C and RT being the room temperature. The strength parameters of ferritic or bainitic steels tend to increase with increasing loading rate and hence, in general, the use of quasi-static tensile properties will yield conservative results. Care should, however, be exercised for materials showing dynamic strain aging effects.
4.4. Toughness Characteristics of the Material 4.4.1. General Remarks Instructions on the determination of the fracture toughness are not part of the SINTAP/FITNET procedure. For these the user is referred to existing test procedures and standards, a list of which is given in the Appendix “Fracture Toughness Test Standards”. The information provided in this section is restricted to statistical and transferability aspects of fracture toughness, which, within SINTAP/FITNET, are generally designated by an index “mat”, although it refers to different meanings in
The Input Parameters
51
different applications, for example, resistance against cleavage, resistance against ductile crack initiation or resistance against ductile tearing (R-curve). The toughness data should, wherever possible, be obtained for the specific material and batch, in the correct product form, and at the relevant temperature and loading rate. Additionally, the orientation of the test specimens should correspond to the flaw orientation in the structure unless it can be shown that the specimen orientation does not affect the toughness. Particular attention should be paid to the size and number of specimens. In cases in which it is not possible to meet the component conditions, the data should be conservative with respect to the SINTAP/FITNET analysis. In most cases “conservative” means either lower-bound values, or a relatively small percentile value if a complete statistical analysis is performed. An upper-bound toughness value has to be used to specify the maximum crack size after a proof test. Note that, although in principle possible, the use of fracture toughness data from the open literature or from compendia should be exercised with care because toughness data may be much more sensitive than strength data to even minor changes in the manufacturing process, or to a particular fabrication route for a component.
4.4.2. The Fracture Toughness Transition Curve Materials with body-centred cubic and hexagonal lattices, such as ferritic and bainitic steels, show a transition behaviour from cleavage fracture at low temperatures to micro-ductile fracture at high temperatures. A schematic drawing is provided in Fig. 4.14. It is characterised by a very wide scatter band in the
Fracture Resistance
micro-ductile tearing + final cleavage
ductile-brittle transition
upper shelf
stable ductile crack initiation
cleavage
lower shelf Temperature
Figure 4.14: Ductile-to-brittle transition in ferritic or bainitic materials.
52
Fitness-for-Service Fracture Assessment of Structures Containing Cracks
ductile-to-brittle transition range, as compared to rather moderate scatter bands in both the lower and the upper shelf. Most other materials, including austenitic steels, with face-centred cubic lattice will not fail by cleavage and, therefore, do not show such a transition behaviour. Note that the designation “cleavage” and “microductile” refers to microscopic fracture mechanisms and should not be mixed up with macroscopic features such as “brittle” and “ductile” or “flat” and “slant”. Although brittle fracture characterised by a flat fracture surface will frequently refer to cleavage, it can also occur in materials that do not fail by cleavage but are micro-ductile. When, following the common designation, the term “ductile-to-brittle transition” is used in this section, it should be kept in mind that “microductile-to-cleavage transition” is meant. In Fig. 4.15 the most frequently used fracture toughness parameters are summarised with respect to the toughness-temperature transition curve (of ferritic materials) and the force-displacement diagram of the test record. The parameters describe stable crack extension as well as unstable fracture. “Stable” means that the specimen fails only when the applied load is increased. In contrast, “unstable” means specimen failure will occur in any case, even if the load is kept constant or decreased. The failure of a specimen or component in a stable or unstable manner depends on a number of factors, including the material, the temperature and loading rate and whether loading is applied using a load controlled or displacement controlled regime. Information on whether a component will potentially fail in an unstable manner or not cannot, therefore, simply be taken from the record of a laboratory test.
lower shelf
ductile-brittle transition
upper shelf
δu Ju δ-Δa curve J-Δa curve δc KIc
Jc
δuc Juc
Temperature
δ0.2 /BL
δu Ju δuc Juc Force
Fracture Toughness
(a) Fracture parameters for unstable crack extension – As long as the forcedisplacement record is essentially linear up to unstable fracture, a plane strain fracture toughness KIc is determined. If it is non-linear prior to unstable fracture,
δc Jc
J0.2 /BL δ0.2 J0.2
KIc
δ0.2 /BL δ0.2 J0.2 /BL J0.2
Displacement
Figure 4.15: Fracture parameters relevant to temperature effects of ferritic and bainitic steels and to a force displacement diagram [4.10].
The Input Parameters
53
but the stable crack extension including blunting is less than 0.2 mm, the parameters c and Jc are determined at the point of instability. This type of test record is usually obtained in the ductile-to-brittle transition range. If stable crack extension exceeds 0.2 mm prior to unstable fracture the parameters will be designated as u and Ju . This is usually associated with a material condition close to the upper shelf. If the amount of stable crack extension prior to unstable fracture is unknown the fracture parameters are designated as uc and Juc . (b) Fracture parameters for stable crack extension – The parameters 02/BL and J02/BL characterise the resistance against stable crack initiation. They are, however, based on the pragmatic definition of the material resistance at a value of 0.2 mm stable crack extension, excluding the crack extension due to blunting. This definition, on the one hand, allows easy and accurate measurements of crack extension and, on the other hand, is still close enough to the “real” crack initiation. In contrast to this, the parameters 02 and J02 represent the material resistance at a value of 0.2 mm of total crack extension including the blunting component. Further alternative parameters such as i and Ji , which are based on stretch zone measurements in a scanning electron microscope, are not shown in the figure. If the resistance against unstable ductile crack extension has to be determined, crack resistance (R) curves based on or J, - a or J- a are used.
4.4.3. Lower Shelf Fracture Toughness Lower shelf fracture toughness is usually characterised by the plane strain fracture toughness KIc . Note that the use of KQ values, which refer to a geometrydependent fracture toughness in cases where the size criterion of linear elastic fracture mechanics is not fulfilled, is not recommended here. Instead it is recommended that the tests are repeated with larger specimens or, better still, that the evaluation of the tests is based on the concepts of elastic-plastic fracture mechanics such as the crack tip opening displacement or the J-integral. If no more than three replicate tests are available for a material condition, which commonly refers to the minimum requirement of the test standards, a simplified statistical analysis based on the lowest value should be performed. According to BS 7910 [4.5], an equivalent of the minimum of three is recommended if more than three tests are available. This equivalent refers to • the lowest value for 3 to 5 test data, • the second lowest for 6 to 10 test data, and • the third lowest for 11 to 15 test data.
54
Fitness-for-Service Fracture Assessment of Structures Containing Cracks
Caution should, however, be exercised when the minimum toughness in terms of CTOD, mat , is less than half the average of three specimens, or the maximum toughness is more than twice this average, and the same should be assumed for the J-integral. In terms of Kmat this refers to a minimum toughness less than 70% or a maximum toughness more than 140% compared to the average of three specimens. Such pronounced variations point to more excessive scatter with the consequence that more test data are needed. If enough data are available, the Master Curve approach, as described in detail in Sections 4.4.5.2 and 4.4.5.3, should be applied as a maximum likelihood method.
4.4.4. Upper Shelf Fracture Toughness At the upper shelf the fracture behaviour is characterised by the so-called R or fracture resistance curve. The increase in fracture toughness in terms of the J-integral, J, or the crack tip opening displacement, , is determined as a function of stable crack extension a. Based on this curve an initiation fracture toughness is defined for the onset of stable ductile-crack extension (ductile tearing) where different definitions (02/BL , J02/BL , 02 , J02 , i , Ji ) are used in the common test procedures and standards as described in Section 4.4.2 (b). Although the slope of the R-curve is, in principle, a geometry dependent measure, the resistance against stable crack extension can be regarded as geometry independent. Usually, R-curves obtained from centre-cracked tension-loaded panels show a significantly higher slope than R-curves obtained from bend specimens. Other parameters affecting the slope of the R-curve are the crack length, the ligament length-to-thickness ratio and the strain hardening of the material. A more detailed discussion of the problem is provided in Section 4.4.6. As the scatter in data representing the onset of ductile tearing is rather small, it is usually sufficient to base the characteristic initiation toughness on the minimum value of three valid test results. However, if the lowest value in the data set is more than 10% below the highest value in terms of Kmat or more than 20% in terms of mat or Jmat , this indicates inhomogeneous behaviour. In this case, metallographic sectioning should be undertaken to ensure that the pre-fatigued crack tip is located in the microstructure of interest, and consideration should be given to performing more tests. If more than three specimens have been tested, a complete statistical analysis can be performed. The characteristic value should then be taken as a mean-minusone standard deviation or a 20% percentile of the fitted R-curve. According to [4.11] the relative scatter band in upper-shelf toughness has been found to be broadly independent of crack extension and to be typically less than 10% of the mean value corresponding to a crack extension of 1 mm. Although this
The Input Parameters
55
practical approach is not recommended in the final documents of SINTAP and FITNET (however, see R6 [4.3]) it is used in the following example to determine a lower-bound R-curve from an experimental data set. Example 4.4: The J- a curve is given by the data points of Table 4.4. The regression analysis according to ESIS P3 [4.12] J = A + C · aD
(4.23)
provides the coefficients A = 0, C = 800 and D = 069 (J in N/mm, a in mm). Based on these coefficients a lower bound curve reduced by 10% is obtained as J = 720 · a069 (Fig. 4.16). With the blunting line determined by JBL = 375 · Rm · aBL
(4.24)
and a tensile strength of Rm = 640 MPa, a J02/BL lower-bound design value is obtained as 283 N/mm. Note that for ferritic or bainitic steels, it is important to ensure that the temperature of interest is high enough to avoid any risk of brittle fracture occurring from proximity to the ductile-to-brittle transition. This is, however, not a trivial task because it refers to component conditions rather than test specimen conditions. Nevertheless, in order to avoid the risk of cleavage, it is recommended that the specimens are tested at just below the temperature of interest and that they are investigated for cleavage events, using fracture surface and metallographic sectioning as appropriate, following the test. Useful information can also be obtained from Charpy data. If there is some indication that the component could fail in the ductile-to-brittle transition regime, a Master Curve analysis according to Section 4.4.5.2 should be carried out. Table 4.4: Example 4.4: Experimental R curve data J-integral in N/mm 160 195 215 250 280
a in mm
J-integral in N/mm
a in mm
J-integral in N/mm
a in mm
0.09 0.13 0.15 0.16 0.25
345 395 425 455 545
035 035 040 045 055
660 675 680 725 1095
080 065 095 080 160
56
Fitness-for-Service Fracture Assessment of Structures Containing Cracks
1200
Blunting line: J = 3.75 Rm Δa
J-integral in N/mm
1000
Fit + 10%
800
Fit – 10% (Lower bound)
0.2 mm 600
400
J0.2,BL
Fit: J (N/mm) = 800 ⋅ Δa (mm)0.69
200
Rm = 640 MPa 0
0
0.4
0.8
1.2
1.6
Stable crack extension Δa in mm
Figure 4.16: Example 4.4: Lower bound R-curve pragmatically derived by lowering the mean curve according to Eqn (4.23) by 10%.
In R6 [4.3] it is recommended that potential ductile-to-brittle transition data should only be assessed if the stable crack extension prior to fracture is shown experimentally not to exceed 0.2 mm. This means that the data analysed should be restricted to c or Jc rather than u or Ju as described in Section 4.4.2.
4.4.5. Ductile to Brittle Transition 4.4.5.1. The Weakest Link Model As a consequence of the usual large scatter band of fracture toughness data in the ductile-to-brittle transition range of ferritic or bainitic materials, which is explained by the so-called weakest link model, statistical treatment is indispensable. The scientific basis of the weakest link model, first proposed in [4.13], is the assumption of randomly distributed small regions of very low toughness, called “weak links”. When a critical stress is reached at the location of one of these “weak links” the whole specimen (or component) breaks, in a similar manner to a chain failing when one of its links breaks. The stresses ahead of a crack show a distinct peak which broadens and shifts into the ligament with increasing crack driving force in terms of or J. The distance of the first “weak link” from the
The Input Parameters
57
crack tip varies from specimen to specimen due to the irregular distribution of the “weak links” in the ligament. As a consequence, the shift of the stress peak necessary to trigger the “weak link” and the resulting crack driving force is also quite different and this explains the observed scatter of fracture toughness in the ductile-to-brittle transition. A further consequence of the weakest link mechanism is a statistically based “specimen thickness effect”. This has to be distinguished from the constraintbased geometry dependency which will be the subject of Section 4.4.6. Strictly speaking, the statistical “thickness effect” is not an effect of the specimen (or component) dimensions, but of the crack front length. The longer the crack front, the greater is the probability that find a “weak link” will be found near the crack tip. Therefore, from a statistical point of view, a larger crack front length must correspond with a smaller scatter band, but the lower bounds of the scatter bands of specimens with smaller and larger crack front lengths should be identical. 4.4.5.2. The Master Curve Approach (Stage 1) 4.4.5.2.1. Introduction In SINTAP/FITNET the scatter in fracture toughness is modelled by the so-called Master Curve concept of VTT (e.g., [4.14]), which also forms the basis of ASTM (American Society for Testing and Materials) test standard E 1921 [4.15] (latest version [4.16]). It is based on a three-parameter Weibull distribution of a number of replicate test results: m Kmat − Kmin (4.25) P = 1 − exp − Ko − Kmin with P being the failure probability (of the test specimens), Kmat the fracture toughness in terms of the K-factor, Ko the scale parameter, Kmin the shift parameter and m the shape parameter of the distribution. In general application, Ko , Kmin and m are fit parameters. However, in the Master Curve concept, two of these are fixed. For ferritic steels with yield strengths between Y = 275 and 825 Mpa, the shape parameter is given by m = 4 and the shift parameter by Kmin = 20 MPa ·m1/2 . In this section the Master Curve basic or “stage 1” approach is presented. The SINTAP/FITNET variation for inhomogeneous material states (stages 2 and 3) will be introduced in Section 4.4.5.2. The steps in the Master Curve approach are set out in detail for both homogeneous and inhomogeneous materials. This makes the presentation quite long compared to the presentations for fracture toughness determination in other regimes or for other materials, where there is less scatter and where fracture toughness determination is more straightforward.
58
Fitness-for-Service Fracture Assessment of Structures Containing Cracks
4.4.5.2.2. Determination of the Scale Parameter Ko The steps for determining the scale parameter Ko are summarised in Fig. 4.17. Step 1 – The fracture toughness data in terms of Jmat or mat are transferred into Kmat values by Kmat = m · mat · Y · E 1 − 2 (4.26) or Kmat =
Jmat · E 1 − 2
(4.27)
Input toughness data set [N data Jmat or δmat]
Step 1
Transfer toughness data to Kmat in MPa · m1/2
Step 2
Sort toughness data
Step 3
Determine the according failure probability
Step 4
Apply the census criterion
Step 5
If necessary: adjust to 25 mm specimen thickness
Step 6
Determine the scale parameter Ko for the test specimens
Step 7
Transfer Ko to component conditions
Step 8
Determine fractile values for the fracture toughness
Figure 4.17: Master Curve (Stage 1) approach: The determination of the scale parameter Ko and its transfer to component conditions.
The Input Parameters
59
respectively. In terms of the fracture parameters, according to Section 4.4.2 the Jmat or mat data can be c , Jc , u , Ju , uc or Juc values. They are determined at the point of instability and no validity criterion is applied prior to the Master Curve analysis. Note that the Kmat values in Eqns (4.26) and (4.27) have to be expressed in MPa · m1/2 . The parameter m in Eqn (4.26) is a constraint parameter which in SINTAP/FITNET is conservatively chosen as m = 15 for steels with a strain hardening coefficient N ≥ 005. Note that for conditions different from small-scale yielding, Kmat is a formally deduced parameter representing mat or Jmat in terms of K instead of being a critical stress intensity factor which is not defined for contained or net-section yielding conditions. If a CDF approach which is based on the CTOD or J-integral is applied, the Kmat data have to be re-converted to mat or Jmat using Eqn (4.26) or (4.27) after completing the Master Curve analysis. If an FAD approach is carried out, Kmat is directly used as the input parameter (with respect to FAD and CDF see Sections 6.2 and 6.3). Step 2 – The adjusted Kmat data have to be sorted beginning with the smallest value so that Kmat i < Kmat i + 1 < · · · < Kmat N
(4.28)
In Eqn (4.28) N designates the number of specimens tested and i is a counter variable (i = 1 to N). Step 3 – For each data point (i) a failure probability Pi (of the test specimen) is determined as Pi = i − 03/N + 04
(4.29)
Step 4 – All Kmat i ≥ Kcen have to be excluded (censored) from the subsequent analysis. The censoring criterion Kcen is introduced in order to avoid significant influence of geometry from low- constraint conditions on the toughness values. 1 Kcen = E · bo · Y (4.30) 30 In Eqn (4.30) the yield strength Y refers to test temperature conditions. The quantity bo is the initial length (W-ao ) of the uncracked ligament at the beginning of the test. Specimens that did not fail by cleavage are also censored. All other
60
Fitness-for-Service Fracture Assessment of Structures Containing Cracks
data are treated as uncensored. In addition, a censoring parameter i is introduced; for censored data i = 0 and for uncensored data i = 1. Note that in the 2005 update of ASTM E 1921-05 [4.16] a further censoring criterion is introduced for tests with more than 0.05 (W-ao ) or 1 mm (whichever is smaller) stable crack extension prior to fracture. Step 5 – In cases in which the toughness has been determined by specimens with a thickness B other than 25 mm (the latter referring to so-called 1T bend specimens with “1” standing for B = 1 inch, approximately 25 mm, according to ASTM nomenclature) Kmat has to be adjusted to B = 25 mm. This is done by Kmat 25 = 20 + Kmat − 20 · B/251/ 4
(4.31)
with Kmat and Kmat 25 in MPa · m1/2 and the specimen thickness B used in the test in mm. In the following, the Kmat term is generally assumed as adjusted values. Step 6 – Using the information available so far, the scale parameter Ko is determined as
Ko = 20 +
⎧ ⎪ ⎪ ⎨
⎫1 / 4 ⎪ ⎪ ⎬
N 1 Kmat i − 204 N ⎪ ⎪ ⎪ ⎪ ⎩ i i=1 ⎭
(4.32)
i=1
This refers to a failure probability P of the specimen of 63.2%. If the number of uncensored values is designated by r, Eqn (4.32) can be re-written as
N 1 K i − 204 Ko = 20 + r i=1 mat
1/ 4
(4.33)
Step 7 – So far the scale parameter Ko is defined for bend specimens with a thickness B of 25 mm. In order to apply it to real structures with crack lengths deviating from 25 mm, Ko has to be adapted to the real crack front lengths . This is done by
1 4 = 25 mm / Ko = 20 + Ko = 25 mm − Kmin · with Ko in MPa · m1/2 and in mm.
(4.34)
The Input Parameters
61
Step 8 – With m = 4, Kmin = 20 MPa · m1/2 and the component corrected Ko value of Eqn (4.34), Eqn (4.35) can be used to obtain fracture toughness values for arbitrary percentile values of P, 5% or 20%, for example: Kmat P = 20 + Ko − 20 · − ln 1 − P 1/ 4
(4.35)
again with Kmat P in MPa · m1/2 and in mm. Example 4.5: A data set of 10 Jmat experiments comprises the following data (all in N/mm): 22.5 / 31.5 / 15.0 / 46.5 / 24.5 / 58.5 / 41.5 / 30.5 / 46.0 / 48.5 for a steel with E = 210 GPa and = 03. These refer to the following Kmat values (in MPa · m1/2 : 72.1 / 85.3 / 58.8 / 103.6 / 75.2 / 116.2 / 97.9 / 83.9 / 103.0 / 105.8 (Step 1). Sorting the data (Step 2) and determining the according failure probabilities P (Step 3) gives the pairs of values summarised in Table 4.5. With a specimen width of 50 mm and an initial crack size of ao = 2815 mm, which gives a ligament length bo = 50–2815 mm = 2185 mm, the censoring criterion is obtained as Kcen = 287 MPa · m1/2 from Eqn (4.30) (Step 4). This means that none of the Kmat values above has to be censored in this special case. The values have been obtained by C(T) specimens with a thickness B of 25 mm and so no thickness adjustment (Step 5) had to be carried out. Using Eqn (4.32), the scale parameter Ko is obtained as Ko = 956 MPa · m1/2 (Step 6). Assuming a crack front length in the component of = 25 mm, no transfer of Ko (Step 7) is necessary in the present example. Finally, the analysis provides percentile toughness values of Table 4.5: Example 4.5: Sorted toughness values in terms of Kmat (i) (in MPa · m1/2 ) and the corresponding specimen failure probabilities Pi after Step 3 i 1 2 3 4 5 6 7 8 9 10
Kmat (i) in MPa · m1/2
Pi
588 721 752 839 853 979 1030 1036 1058 1162
0.067 0.163 0.260 0.356 0.452 0.548 0.644 0.740 0.837 0.933
62
Fitness-for-Service Fracture Assessment of Structures Containing Cracks
100
P = 1 – exp –
Failure probability P in %
80
(Kmat – Kmin)
m
(Ko – Kmin)
Kmin = 20 MPa ⋅ m1/2 m=4
60
Ko = 95 . 6 MPa ⋅ m1/2 (P = 63.2%) Kmat (P = 20%) = 72 MPa ⋅ m1/2 Kmat (P = 5%) = 56 MPa ⋅ m1/2
40
P = 20% 20 P = 5% 0
0
60 80 100 20 40 Fracture toughness Kmat in MPa ⋅ m1/2
120
Figure 4.18: Example 4.5: Resulting toughness distribution and percentile values for P = 5% and 20%.
Kmat P = 5% = 56 MPa · m1/2 and Kmat P = 20% = 72 MPa · m1/2 by applying Eqn (4.35) (Step 8). The result is shown in Fig. 4.18. The user should decide on which percentile value to finally base the SINTAP/ FITNET analysis. It should be noted that the failure probability of the test specimens is not identical with the failure probability of the component under consideration. SINTAP/FITNET offers a complete reliability analysis (Section 6.13) as an alternative to choosing a lower percentile value of the fracture toughness. In addition to the SINTAP/FITNET variant of the Master Curve approach, ASTM E 1921-05 [4.16] adds a criterion for identifying “outliers” which deviate greatly from the rest of the data set. First, the percentile values of Kmat for 2% and 98% are determined by Kmat 2% = 0415 · Kmed + 1170 MPa · m1/ 2
(4.36)
Kmat 98% = 1547 · Kmed − 1094 MPa · m1/ 2
(4.37)
with Kmat and Kmed being provided in MPa · m1/2 . Mean value Kmed and scale parameter Ko are correlated by Kmed = Kmin + Ko − Kmin · ln 2 1/ 4
(4.38)
The Input Parameters
63
If a value is outside this limit it is regarded as an “outlier”, the influence of which can be reduced by testing additional specimens. The “outlier” should, however not be discarded from the data used for determining Ko and Kmed , since further “outliers” can indicate inhomogeneous material. Within SINTAP/FITNET an inhomogeneity analysis (stages 2 and/or 3; see Section 4.4.5.3) is recommended in any case. In [4.17, 4.18] the authors propose an alternative lower-bound method based on a modified two-parameter Weibull distribution of the type √ ln 2/Jo · Jmat − JLB for P ≤ 05 (4.39) P= 1 − exp −Jmat /Jo m
for P > 05 with the shape parameter being m = 2 and Jo being the scale parameter in terms of the J integral (or the crack tip opening respectively). Unlike the Master Curve approach, Eqn (4.39) is not part of SINTAP/FITNET. For probabilities P > 05 Eqn (4.39) follows the Weibull distribution. Below this value the P-Jmat function is modelled by a straight line with the slope at P = 05. The toughness data in terms of Jmat are censored by a criterion Jcen =
1 b · 30 o Y
(4.40)
Values of Jmat i ≥ Jcen have to be excluded from the subsequent analysis. If no Jmat value is censored, an engineering lower-bound value, JLB , referring to P = 0 in the model or to a probability lower than P = 25% in the Master Curve approach, can be determined by JLB = 023Jo = 026Jmed
(4.41)
In Eqn (4.41), Jmed is simply the average value of the data set Jmed =
N 1 J N i=1 mat
(4.42)
If a number of p values are censored, JLB is obtained as JLB = 026 · · Jmed
(4.43)
with Jmed =
N−p 1 J N − p i=1 mat
(4.44)
64
Fitness-for-Service Fracture Assessment of Structures Containing Cracks
and = 1 + 1286 · p
(4.45)
The standard deviation of JLB is estimated by JLB ≈ 013 · Jmed /r
(4.46)
where r designates the number of uncensored data, that is, r + p = N. When the toughness distributions described so far have to be used in the context of a reliability analysis (Section 6.13), they usually have to be rewritten in terms of coefficients of variation, for example as quotients of the standard deviations (Kmat or Jmat and the mean values (Kmed or Jmed . The transfer is realised by Eqns (4.47) to (4.50). 1 (4.47) Kmed = Kmin + Ko − Kmin · 1 + m 2 1 2 − 1+ (4.48) Kmat = Ko − Kmin · 1 + m m With Kmin = 20 MPa · m1/2 , m = 4 and the Gamma functions 125 = 09064 and 15 = 088623, Kmed and Kmat are obtained as Kmed = 09064 · Ko + 1872
(4.49)
Kmat = 02543 · Ko − 5086
(4.50)
and
Example 4.6: The results of Example 4.5 (Ko = 956MPa · m1/2 , Kmin = 20 MPa · m1/2 and m = 4) are rewritten in terms of mean value and standard deviation of the Weibull distribution. Using Eqns (4.49) and (4.50) the mean value is obtained as Kmed = 885 MPa · m1/2 and the standard deviation Kmat = 192 MPa · m1/2 . Based on these data the coefficient of variation is COV = 022. 4.4.5.2.3. Determination of the transition temperature To The Master Curve reference temperature, To , is defined as the temperature T at which the mean value Kmed of a set of 1T size specimens (B ≈ 25 mm) is 100 MPa · m1/2 . Once this measure is known the complete Kmed -temperature curve Kmed = 30 + 70 exp 0019 T − To in MPa · m1/ 2
(4.51)
The Input Parameters
65
for this specimen size can be reconstructed with T and To being in C. In conjunction with Ko = 11033 · Kmed − 20653
(4.52)
and Eqn (4.35) Kmat P = 20 + Ko − 20 · − ln 1 − P 1/ 4
(4.53)
(Kmat and Ko in MPa · m1/2 , the toughness values for a specific probability P, can be determined for any temperature. The Master Curve according to Eqn (4.51) is defined for a temperature range To ± 50 C [4.16]. While To could be determined from a test set of data points at one test temperature by solving Eqn (4.51) for To , the method recommended in SINTAP/FITNET uses data points Kmat (i) at different temperatures Ti in conjunction with an iterative approach: N i exp 0019 Ti − To
11 + 77 exp 0019 Ti − To
i=1 N Kmat i − 20 4 exp 0019 Ti − To
− = 0 (4.54) 11 + 77 exp 0019 Ti − To 5 i=1 The censoring parameter i is i = 0 for censored data that either did not fail by cleavage or showed a toughness equal or larger than Kcen in Eqn (4.31). The yield strength Y refers for each data point to the corresponding test temperature Ti . For uncensored data, the parameter i is chosen as i = 1. The toughness values Kmat (i) are adjusted to a specimen thickness B = 25 mm. Example 4.7: Toughness data sets of a pressure vessel steel have been determined by 1T size specimens at −154 C, −91 C, −60 C, −40 C, −20 C, and 0 C (Table 4.6), the data being taken from [4.19]. In addition to the toughness values Kmat and the yield strength Y at test temperature, the censoring criterion according to Eqn (4.31) are provided in Table 4.6 for an initial ligament size of bo = 22 mm and a Young’s modulus of 210 GPa. At T = −40 C one out of ten specimens fails the censoring criterion, at T = −20 C seven specimens fail and at 0 C all of the specimens fail. The Master Curve analysis provided the Kmed and Ko values as well as percentile values of the toughness as a function of temperature and these are summarised in Table 4.7 and Fig. 4.19.
66
Fitness-for-Service Fracture Assessment of Structures Containing Cracks
Table 4.6: Example 4.7: Kmat data set at test temperatures between −154 and 0 C. The original data are taken from [4.19] T C
Y MPa
T C
Y MPa
−154 −154 −154 −154 −154 −154 −154 −154 −154 −154 −154
677 677 677 677 677 677 677 677 677 677 677
435 443 524 356 438 483 463 384 304 556 413
322 9 322 9 322 9 322 9 322 9 322 9 322 9 322 9 322 9 322 9 322 9
1 1 1 1 1 1 1 1 1 1 1
−40 −40 −40 −40 −40 −40 −40 −40 −40 −40
504 504 504 504 504 504 504 504 504 504
207 7 157 5 237 8 165 7 268 8 217 7 223 8 266 9 251 5 324 1
278 6 278 6 278 6 278 6 278 6 278 6 278 6 278 6 278 6 278 6
1 1 1 1 1 1 1 1 1 0
−91 −91 −91 −91 −91 −91 −91 −91 −91 −91
538 538 538 538 538 538 538 538 538 538
719 855 586 1036 753 1164 980 838 1031 1060
287 8 287 8 287 8 287 8 287 8 287 8 287 8 287 8 287 8 287 8
1 1 1 1 1 1 1 1 1 1
−20 −20 −20 −20 −20 −20 −20 −20 −20 −20
475 475 475 475 475 475 475 475 475 475
212 2 204 1 275 5 197 0 289 1 274 5 389 0 386 9 484 5 335 2
270 5 270 5 270 5 270 5 270 5 270 5 270 5 270 5 270 5 270 5
1 1 0 1 0 0 0 0 0 0
−60 −60 −60 −60 −60 −60 −60 −60 −60 −60
506 506 506 506 506 506 506 506 506 506
1950 1591 1171 1508 1105 1614 1847 1383 2138 1496
279 1 279 1 279 1 279 1 279 1 279 1 279 1 279 1 279 1 279 1
1 1 1 1 1 1 1 1 1 1
0 0 0 0 0 0 0 0 0 0
470 470 470 470 470 470 470 470 470 470
343 5 714 3 731 9 734 7 735 3 739 8 742 6 743 8 752 0 759 9
269 0 269 0 269 0 269 0 269 0 269 0 269 0 269 0 269 0 269 0
0 0 0 0 0 0 0 0 0 0
Kmat Kcen i MPa · m1/2 MPa · m1/2
Kmat Kcen i MPa · m1/2 MPa · m1/2
4.4.5.2.4. Minimum Numbers of Specimens to be Tested No statement can be made about the overall number of specimens to be tested in the ductile-to-brittle transition range. There exist, however, clues to the minimum
The Input Parameters
67
Table 4.7: Example 4.7: Results of the Master Curve analysis T C
Kmed MPa · m1/2 (P = 50%)
Ko MPa · m1/2 (P = 632%)
Kmat MPa · m1/2 (P = 5%)
Kmat MPa · m1/2 (P = 20%)
−154 −91 −60 −40 −20 0
489 925 1426 1946 2707 3820
51 8 997 1548 2121 2958 4181
351 579 842 1114 1512 2095
435 748 1127 1520 2095 2936
Validity range?
outside inside inside inside outside outside
9 Specimens 500
Fracture toughness Kmat in MPa ⋅ m1/2
To = – 85°C Ko (P = 63.2%)
400 Validity range for To
Kmed (P = 50%)
300 Kmat (P = 20%) 200 Kmat (P = 5%) 100 ?? ? ? ? ? ? ?? ?
0 – 80 – 60 – 40 – 20 0 20 40 60 80 Temperature difference T-To in °C
100
Figure 4.19: Example 4.7: Results of the Master Curve analysis.
number of “valid” tests needed to fulfil the requirement in Eqn (4.30). When the mean value of Kmed for 1T size specimens is greater than 83 MPa·m1/2 , the required minimum number of valid tests is six. For smaller Kmed values the required number goes up to ten (Table 4.8) [4.16]. The definition for valid tests of weldments is extended by a further criterion – metallographic consistency – with the consequence that the number of tests is increased. According to [4.5] a minimum of 12 valid tests is necessary for HAZ
68
Fitness-for-Service Fracture Assessment of Structures Containing Cracks
Table 4.8: Number of valid Kmat values required for Master Curve analysis according to [4.16] Kmed range 1T equivalent in MPa·m1/2 100 to 84 83 to 66 65 to 58 57 to 53 52 to 50
Number of valid Kmat values required 6 7 8 9 10
cracks and ductile-to-brittle transition range, otherwise excessive scatter has to be expected. Small test pieces such as pre-cracked Charpy specimens can lead to a very large number of “invalid” tests with respect of Eqn (4.31) when tested close to the T0 temperature. In such cases the testing should be performed at temperatures below T0 where more data can be expected to fulfil the censoring criterion. In addition, it is required that the fatigue crack has to sample the type and portion of the specified microstructure to characterise the HAZ toughness. If the HAZ exhibits upper-shelf behaviour the scatter is limited and in that case three “valid” data points are regarded as sufficient. However, when the HAZ operates in the ductile-to-brittle transition range, or excessive scatter is expected, a minimum of 12 valid tests, more in some cases, is indicated [4.5]. 4.4.5.3. Statistical Aspects of Fracture Toughness in Inhomogeneous Materials 4.4.5.3.1. The Inhomogeneity Problem The Master Curve introduced in Section 4.4.5.2 –stage 1 analysis – is intended for homogeneous materials showing ductile-to-brittle transition behaviour. Note, however, that fully homogeneous materials are exceptional rather than usual. Forgings, for example, will frequently have a different toughness at the plate centre and the surface. The effect is even more pronounced in weldments, which can include ductile as well as brittle zones across the weld and the adjacent material. This means in statistical terminology that different samples – according to the different material states – are mixed together. The result is schematically illustrated in Fig. 4.20. Assume two materials of quite different toughness characteristics, with the first material being the more brittle. Even though five data points of the second material are censored its distribution still tends towards much
The Input Parameters
(a)
69
(b) 100 Material 1
Failure probability P
Material 2
Material 1
Toughness
80
Mixed
60
40
Material 2
20
0
Census criterion
Toughness
Figure 4.20: Schematic illustration of the effect of inhomogeneous materials such as forgings or weldments on the toughness distribution.
higher toughness values. When the distributions from both materials are randomly mixed together and censored, the resulting distribution will be in between the two original distributions. It will show higher toughness values than the distribution from the first material, even at its lower tail. What is needed, therefore, is a procedure for “demixing” the various subsets of potentially inhomogeneous materials. Two methods are provided in SINTAP/FITNET in order to solve this problem. The first comprises the so-called stage 2 and stage 3 Master Curve analyses that were developed by VTT in the SINTAP project. These are lower bound analyses, that is, they describe the statistics of the more brittle material but neglect the fracture behaviour of the more ductile zones. The second method is designated the bi-modal Master Curve analysis and is incorporated in the FITNET procedure as additional information. For detailed information on the derivation and verification of both methods see [4.20]. The bi-modal Master Curve differs from the basic (stage 1) and the stage 2/ stage 3 methods in that it describes the toughness distribution of inhomogeneous materials as a combination of two separate distributions. This makes it particularly efficient for extreme inhomogeneity, for example, in describing heat affected zone data. In the following sections, both methods will be introduced.
70
Fitness-for-Service Fracture Assessment of Structures Containing Cracks
4.4.5.3.2. Master Curve approach Stages 2 and 3: Correction of the scale parameter Ko (a) Stage 2 analysis – lower tail estimation The stage 2 analysis is an iterative procedure performing a lower tail estimation, checking and correcting any undue influence of toughness values in the upper tail of the distribution by stepwise reduction of the censoring criterion Step 1 – K0 and Kmed are determined following the basic concept (Eqns 4.32 and 4.47/4.49). Kmed is designated as Kmed (step 1). Step 2 – The censoring criterion of Eqn (4.30) is replaced by Kcen = Kmed (step 1), which reduces the number of “valid” data and increases the number of censored data. With this new information the analysis is repeated, again based on Eqns (4.32 and 4.47/4.49) but resulting in new values of K0 and Kmed , the new Kmed being designated as Kmed (step 2). If Kmed (step 2) ≥ Kmed (step 1) no inhomogeneity is detected, otherwise the analysis has to be continued with step 3. Step 3 – The censoring criterion of Eqn (4.30) is replaced by Kcen = Kmed (step 2) which again reduces the number of “valid” data and increases the number of censored data. The analysis is repeated as before and the new Kmed is designated as Kmed (step 3). If Kmed (step 3) ≥ Kmed (step 2) no inhomogeneity is stated. Otherwise the analysis has to be repeated again, and so on. Final Step – If the overall number of data in the data set, N, is 10 or more the analysis is terminated when Kmed (step n + 1) ≥ Kmed (step n). If the number of data in the data set is less than 10, a stage 3 (or minimum value) estimation has to be performed as described below. (b) Stage 3 analysis – minimum value estimation At stage 3, a minimum value estimation is performed to check, and make allowance for, extreme inhomogeneity in the material. An additional safety factor is incorporated for cases in which the number of tests is small. As in stage 2, a stage 3 analysis consists of different steps Step 1 – The smallest value Kmat of the test set has to be identified and designated as Kmatmin . For this, K0 has to be determined as K0min by √ K0min = N/ln 21/ 4 Kmin − 20 + 20 in MPa m (4.55)
The Input Parameters
71
Step 2 – K0min is compared with the last K0 , that is, K0 (step n), of the stage 2 analysis, K0min > 09 K0 (step n) indicating that the data set is homogeneous. The values K0 (Step n) and Kmed (Step n) obtained in “Stage 2” can therefore be taken as representative and used for determining the final value for K0 in step 3 below. However, K0min ≤ 09 K0 (step n) indicates significant inhomogeneity. In this case K0 (step n) has to be replaced by K0min before it is used for determining the final value for K0 in step 3 below. Step 3 – The final values of K0 and Kmed which will be used in the subsequent failure assessment include a small data set safety correction K0 =
1 −1/ 2
1 + 025 r
K0 Step n − 20 + 20 in MPa
√
m
(4.56)
or K0 =
1 −1/ 2
1 + 025 r
K0min − 20 + 20 in MPa
√ m
(4.57)
respectively where r is the number of uncensored data points. The mean value Kmed is obtained from K0 by Eqn (4.49). It is recommended that all three analysis stages are employed when the number of tests is between three and nine. With an increasing number of tests, the influence of the penalty for small data sets is gradually reduced. For ten and more tests, only stages 1 and 2 need to be used. However, stage 3 may still be employed for indicative purposes, especially where there is evidence of gross inhomogeneity in the material, as occurs in welds or heat affected zone material. In such cases, it may be concluded that the characteristic value is based upon the stage 3 result, or alternatively, such a result may be used as guidance in a sensitivity analysis. 4.4.5.3.3. Master Curve approach Stages 2 and 3: Correction of the transition temperature To (a) Stage 2 analysis – lower tail estimation Step 1 – The temperature T0 is determined by Eqn (4.54) as described above and designated as T0 – step 1. Step 2 – Subsequently all data Kmati > 30 + 70 exp 0019 T − T0 step 1 in MPa
√ m
(4.58)
72
Fitness-for-Service Fracture Assessment of Structures Containing Cracks
have to be censored and replaced by Kmati = 30 + 70 exp 0019 T − T0 step1 in MPa
√ m
(4.59)
Based on this modified input information. T0 is determined using Eqn (4.54) with i = 0 for the censored data. The newly obtained value of T0 is then designated as T0 (step 2). If T0 step 2 ≤ T0 step 1 no inhomogeneity is present. Otherwise the analysis has to be continued with step 3. Step 3 – The censoring criterion is replaced by Kmati > 30 + 70 exp 0019 T − T0 step 2 in MPa
√ m
(4.60)
√ m
(4.61)
the censored data are replaced by Kmati = 30 + 70 exp 0019 T − T0 step 2 in MPa
and the analysis is repeated. If T0 step 3 ≤ T0 step 2 no inhomogeneity is stated. Otherwise the analysis has to be repeated again, and so on. Final Step – If the overall number of data in the data set, N, is 10 or more, the analysis is terminated when T0 step n + 1 ≤ T0 step n. If the number of data in the data set is less than 10, a stage 3 estimation has to be performed as described below. (b) Stage 3 analysis – minimum value estimation Step 1 – For each “valid” Kmati with respect to Eqn (4.30), a T0 value is determined by 1 1 1/ 4 T0 = Ti − ln Kmati − 20 × n/ln 2 − 11 (4.62) 0019 77 Step 2 – The largest of these values, T0max , is then compared to T0 step n of the stage 2 analysis. If T0max −8 C < T0 step n, the data may be considered to be homogenous and T0 step n is taken as representative. If, however, T0max −8 C ≥ T0 step n, this indicates inhomogeneous data. In this case T0max is assumed to be representative. Step 3 – In all cases, a small data set safety correction has to be performed using √ (4.63) T0 = T0 step n + 14 r
The Input Parameters
73
or T0 = T0max + 14
√ r
(4.64)
respectively, with r being the number of uncensored data. From T0 , the Kmed temperature curve is determined by Eqn (4.51) and the K0 -temperature curve by K0 = 31 + 77 exp 0019 T − T0 in MPa
√ m
(4.65)
4.4.5.3.4. The Bi-modal Master Curve Approach The total cumulative probability is expressed as a bimodal distribution 4 4 Kmat − Kmin Kmat − Kmin P = 1 − Pa · exp − − 1 − Pa · exp − K01 − Kmin K02 − Kmin (4.66) with the toughness, Kmat , and the scale parameters, K01 and K02 , of the two constituents in MPa · m1/2 , and Pa being the probability that the toughness belongs to distribution 1. Note that in the case of multi temperature data the scale parameters K01 and K02 are expressed in terms of the corresponding transition temperatures, T01 and T02 . In contrast to the one parameter of the basic Master Curve analysis three parameters have to be determined. The analysis is based on a maximum likelihood procedure where the likelihood, L, is expressed as L=
n 1− 1 · f 2 · S1−2 · · f n · S1−n (4.67) fcii · Sci i = fc11 · S1− c1 c2 cn c2 cn i=1
In Eqn (4.67) fc is the probability density function 4 Kmat − Kmin 3 Kmat − Kmin fc = 4 · Pa · 4 · exp − K01 − Kmin K01 − Kmin 4 Kmat − Kmin 3 Kmat − Kmin − 4 · 1 − Pa · 4 · exp − K02 − Kmin K02 − Kmin
(4.68)
and Sc is the survival function 4 4 Kmat − Kmin Kmat − Kmin + 1 − Pa · exp − (4.69) Sc = Pa · exp − K01 − Kmin K02 − Kmin
74
Fitness-for-Service Fracture Assessment of Structures Containing Cracks
The parameter is the censoring parameter as in the basic approach. Equation (4.67) is solved so as to maximise L. The numerical procedure can be simplified by taking the logarithm of L so that a summation equation is obtained: ln L =
n
i · ln fci + 1 − i · ln Sci
i=1
= 1 · ln fc1 + 1 − 1 · ln Sc1 + 2 · ln fc2 + 1 − 2 · ln Sc2
+ · · · + n · ln fcn + 1 − n · ln Scn (4.70) In this case ln(L) has to be maximised. 4.4.5.3.5. Application Ranges of the Different Approaches The stage 2/stage 3 Master Curve analyses have been developed for the analysis of small data sets in order to provide representative lower-bound toughness values as input data to structural assessment. However, they should not be used for the determination of transition temperature shifts or for the determination of the average fracture toughness. In contrast, the bi-modal Master Curve approach needs data sets of sufficient size since the accuracy of its fit parameters depends on the number of data belonging to the material areas of interest.
4.4.6. Constraint Dependency of Fracture Toughness 4.4.6.1. Constraint and Stress Triaxiality The fracture toughness, Kmat , is commonly derived from deeply cracked bend specimens with almost square ligaments, using recommended testing standards and validity criteria. These are designed to ensure high constraint conditions near the crack tip that correspond to lower-bound toughness values independent of specimen size and geometry. However, there is evidence that the material resistance to fracture is increased for specimens – or components – with shallow flaws, or panels loaded in tension since these conditions lead to lower constraint around the crack. In general, the fracture resistance is dependent on the component size and geometry as well as on the loading type – bending versus tension. The geometry dependency of R-curves is schematically illustrated in Fig. 4.21. Usually, constraint is modelled by parameters describing the triaxiality of the stress state at the crack tip. An overview on the various parameters that affect this triaxiality and, as a consequence, the fracture resistance of the material, is
The Input Parameters
75
(a)
Blunti ng line
J-integral or CTOD-δ
Middle crack tension
Bending
ng line
Increasing B
Blunti
J-integral or CTOD-δ
(b)
B
ng line
Increasing b/B
Blunti
J-integral or CTOD-δ
(c)
B
b
Stable crack extension Δa
Figure 4.21: The geometry dependency of R-curves – (a) Effect of loading geometry; (b) Effect of specimen thickness (out-of-plane constraint); (c) Effect of ligament slimness (in-plane constraint).
provided in Fig. 4.22. In addition, strength mismatch, for example, in weldments, can also have an effect on the constraint – see Section 6.11.5.3. A consequence of the constraint, or traxiality, effect is that a single parameter such as K, or J is insufficient to describe accurately the stress and strain fields near the crack tip. A unique crack tip field such as the K field, see the Glossary “stress intensity factor”, or the HRR field, see the Glossary “crack tip
log stress
Fitness-for-Service Fracture Assessment of Structures Containing Cracks
Triaxiallity
76
N
1
f(Lr) Lr
log strain a W
Relative crack depth a / W
Strain hardening exponent N
Ligament yielding Lr
Triaxiallity
Fy Fx
Fx
φ
Fy
Biaxial loading Fx /Fy
Tension
σ b /σ t
Bending
Crack front position φ
Figure 4.22: Parameters which affect the triaxiality of the stress state ahead of a crack.
opening displacement” and “J-integral”, exists only under idealised conditions. As a consequence, the complete characterisation of the stress-strain field by means of fracture mechanics requires a second parameter or further parameters in addition to K, or J. Note that the proposals on how to include constraint in the SINTAP/FITNET analysis set out in this section are not intended to be mandatory. Instead, they can be used in conjunction with the general approach to estimate an additional reserve factor in cases of low constraint geometries. As a rule, the application of standard fracture toughness values will lead to conservative SINTAP/FITNET results. Again, an exception is the determination of the crack size after a proof test. Caution should also be exercised in the case of semi-elliptical surface cracks, as within the framework of a leak-before-break analysis. A special approach to the constraint issue is provided for thin-walled structures in Section 6.10. 4.4.6.2. Constraint Parameters Various parameters for quantifying the local stress triaxiality as a measure of constraint have been proposed, three of which will be briefly introduced in the following section. (a) Triaxiality parameter h The local stress triaxiality can be described by h r z = h r z e r z
(4.71)
The Input Parameters
77
with h being the hydrostatic stress h =
3 1 1 · kk = · xx + yy + zz 3 k=1 3
(4.72)
and e being the equivalent stress, for example, according to von Mises 1 e = √ · xx − yy 2 + yy − zz 2 + zz − xx 2 + 6 · xy 2 + yz 2 + zx 2 2 (4.73) The corresponding coordinate system is shown in Fig. 4.23. (a) y
σyy τyx
x
τyz τxy
τzy
τxz
τzx
σxx
σzz
Crack front z
σyy
(b) y τxy τxy
Crack
σxx
σxx τxy
τxy
r σyy
θ x
Figure 4.23: Coordinate system at the crack – (a) Three-dimensional stress field at an arbitrary point in the uncracked ligament ahead of the crack; (b) Two-dimensional presentation including the coordinates and r.
78
Fitness-for-Service Fracture Assessment of Structures Containing Cracks
Although the h parameter can characterise the actual three-dimensional stress state without any limiting assumption, it is not recommended within SINTAP/FITNET because it is not a constant but is dependent on the distance, r, to the crack tip for strain hardening materials. Additionally, it requires a full three dimensional finite element calculation for its determination. (b) T-stress The non-singular constant stress, T, derived from the linear elastic asymptotic solution K ij = √ fij + T1i 1j 2r
(4.74)
(ij – Kronecker symbol; ij = 1 for i = j and ij = 0 for i = j) represents a crack-parallel stress component that contributes to the stress field but is not included in the K-factor solution. The value of T may be calculated from elastic finite element analysis. Analytical solutions for T are available in compendia, for example, [4.21, 4.22]. An example is given in Fig. 4.24. For a given geometry but with varying stress distributions across the section, T can be obtained by the weight function method. In the literature, T values are sometimes presented as normalised by the stress intensity factor and flaw size √ B = T a K (4.75)
a
θ
0.4
σb
a /c = 0.3 λ=1
2c 0.2 λ ⋅ σb
2H
T/σ
a /t: 0.4 0 0.3 – 0.2
λ ⋅ σb σ b
t 2W
0.2 0.1
– 0.4 – 0.6
0
20
40 60 θ in degree
80
Figure 4.24: Example of a T-stress solution for a plate with a surface crack subjected to combined in- and out-of-plane bending [4.21].
The Input Parameters
79
or in terms of some nominal applied stress. In the example of Fig. 4.24, T is given as normalised by with being the nominal stress in the general K-factor expression √ KI = · a · Y
(4.76)
The most important shortcoming of the T-stress concept is that its definition is restricted to elastic material behaviour, but it has been found still to describe crack tip constraint for some degree of yielding (see (c) below). (c) Q-stress The elastic-plastic Q-stress is defined by ij r = ij ref r + Q · o ij
(4.77)
with o usually taken as, for example, the yield strength of the material, and ij ref r being a reference stress field which can be defined in different ways but usually represents the HRR field, see the Glossary “crack tip opening displacement” and “J-integral”. A shortcoming of Q is that it not only depends on the location ahead of the crack tip, and r, but also takes different values for the various stress components, for an in-depth discussion see [4.23]. In order to make the concept practicable, the Q-stress is commonly referred to only one point in the ligament, at = 0 and r = 2J/o . Q values may be calculated from elastic-plastic finite element analyses. Q is a function of the geometry, the flaw size, the type of loading, the material stress-strain curve and the magnitude of the loading. A limited number of graphical and analytical solutions are available in the literature, for example, [4.24, 4.25]. An example is given in Fig. 4.25. For linear elastic conditions Lr ≤ 05 T and Q are related by (see [4.27]) Q=
T/Y 05 · T/Y
for − 05 < T/Y ≤ 0 for 0 < T/Y ≤ 05
(4.78)
4.4.6.3. The Structural Constraint Factor ß and its Effect on Fracture Toughness A special structural constraint parameter, [4.27], is used within SINTAP/FITNET that can be obtained from both the elastic T-stress and the hydrostatic Q-stress [4.28, 4.29]. As the T-stress requires only elastic calculations, it is recommended for initial evaluations. The Q-stress is expected to provide more accurate assessments, particularly when plasticity becomes widespread, and
80
Fitness-for-Service Fracture Assessment of Structures Containing Cracks
0.5 2c 0
t
a/t = 0.25 2c/a = 6
a
θ
2W/t = 4 n = 10 (n according to Ramberg-Osgood)
– 0.5 Q
45°
–1
θ = 90° 30°
– 1.5
16.6° 2.4°
–2
0
0.02
0.04 0.06 Jlocal /(a ⋅ σo)
0.08
0.1
Figure 4.25: Q-stress solution for a tension loaded plate with a semi-elliptical surface crack (data according to [4.26]).
should be used when more refined estimates of load margins are required, or as part of sensitivity studies. Based on T, the factor is defined by T =
T Lr Y
(4.79)
or based on Q, by Q = Q Lr
(4.80)
The factor is then used to determine a constraint dependent fracture toughc c ness designated as Kmat . At high values of constraint ( Lr > 0, Kmat may be simply taken as equal to Kmat as obtained from conventional deeply cracked bend specimens. For negative levels of constraint ( Lr < 0), the influence of constraint may be broadly summarised as follows c • in the lower shelf, Kmat increases as Lr becomes more negative, • in the upper shelf, there is little influence of constraint on the fracture toughness at stable crack initiation but considerable influence on ductile tearing resistance, that is, the slope of the R-curve increases as Lr becomes more negative. • the ductile-to-brittle transition region of ferritic or bainitic steels is shifted to a lower temperature as Lr becomes increasingly negative.
The Input Parameters
81
In [4.28] the authors have shown that the increase in fracture toughness in both the brittle and ductile regimes may be represented by an expression of the form for Lr > 0 Kmat c Kmat = (4.81) k Kmat 1 + · − Lr for Lr ≤ 0 with and k being material and temperature dependent constants. An application of the -concept is provided in Fig. 4.26. Example 4.8: An example of a T-stress based solution, taken from the SINTAP/ FITNET compendium [4.29], is provided in Fig. 4.27 and Tables 4.9 and 4.10. It refers to a hollow cylinder with a closed inner circumferential surface crack under tension loading. The inner radius is Ri = 100 mm, the wall thickness is t = 10 mm. The crack size varies from 1 mm to 3 mm, 5 mm and 7 mm. The factor is obtained from a a 2 a 3 a 4 a 5 + X2 + X3 + X4 + X5 = X0 + X1 t t t t t a 6 + X6 for 0 ≤ a/t ≤ 08 and H/t = 10 (4.82) t for the present geometry. The coefficients Xi are given in Table 4.9. The and k values are assumed as = 215 and k = 2 as in Fig. 4.26. Using these data, the results in Table 4.10 and Fig. 4.27 are obtained. An alternative method for the generation of and k values was recently developed in the context of the revised R6 routine and adopted by SINTAP/FITNET.
SEN(B) specimen M(T) specimen (a /W = 0.6) M(T) specimen (a /W = 0.8)
c
Kmat in MPa · m1/2
300
200
100
k=2 α = 2.15 Kmat = 77 MPa · m1/2 c Kmat = Kmat [1 + α (–β · Lr)k]
0 –1.4 –1.2
–1.0
–0.8
–0.6
–0.4
–0.2
0
0.2
0.4
T/σY = βT · Lr
Figure 4.26: Experimental determination of the parameters and k of Eqn (4.81) (lower bound) of a mild steel at −50 C (according to [4.30]).
82
Fitness-for-Service Fracture Assessment of Structures Containing Cracks
140
130 a
Ri Ro
110 t 2H
100
a = 1 mm 3 mm 5 mm 7 mm
c
Kmat in MPa · m1/2
120
90
80
Kmat (standard test) = 77 MPa · m1/2 70
0.2
0.4
0.6
0.8
1.0
Lr c Figure 4.27: Example 4.8: Constraint corrected toughness values as Kmat a function of crack depth, a, and ligament yielding, Lr .
Table 4.9: Hollow cylinder with a closed inner circumferential surface crack under tension loading : Polynomial coefficients for determining . The solution is restricted to H/t = 10 Ri /t
X0
X1
X2
X3
X4
X5
X6
2.5 5.0 10 20
−051 −051 −051 −051
−05247 −04074 −03490 −03175
52871 40608 33111 27925
−19103 −13768 −11111 −93521
36294 27014 24588 23048
−35242 −28024 −27936 −26860
13536 11330 11687 10808
Based on extensive finite element analyses [4.30] and empirical correlations of the coefficients and k with the E/Y ratio, the strain hardening exponent n, given by /0 = /0 n 0
for ≤ 0 for > 0
(4.83)
The Input Parameters
83
c Table 4.10: Example 4.8: Constraint corrected toughness values Kmat as a function of crack depth, a, and ligament yielding, Lr
Crack depth a (mm)
(based on Tstress) Eqn (4.82)
Ligament yielding factor Lr
Constraint corrected c toughness Kmat in 1/2 MPa·m
1 1 1 1 1
−05211 −05211 −05211 −05211 −05211
02 04 06 08 10
788 842 932 1058 1220
3 3 3 3 3
−04771 −04771 −04771 −04771 −04771
02 04 06 08 10
785 830 906 1011 1147
5 5 5 5 5
−03696 −03696 −03696 −03696 −03696
02 04 06 08 10
779 806 851 915 996
7 7 7 7 7
−02806 −02806 −02806 −02806 −02806
02 04 06 08 10
775 791 817 853 900
(with o being the limit of proportionality which is related to o by Young’s modulus, o = o /E) and the exponent m in the Weibull distribution of the Beremin model [4.31], given by m P = 1 − exp − w u
(4.84)
(with u and m being fit parameters of the model and w being the Weibull stress) have been generated and summarised in tables. The Weibull exponent m is obtained by matching experimental cleavage fracture toughness data to values obtained from the Beremin model. In order to obtain reliable estimates of m, experimental data that cover two significantly different constraint levels are necessary.
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Fitness-for-Service Fracture Assessment of Structures Containing Cracks
Table 4.11: Look-up table for and k defined with respect to T/Y (according to [4.30]) E/Y
n Eqn (4.83)
m Eqn (4.84) 5
10
15
20
k
k
k
k
500 600 700
5 5 5
0797 0894 0962
214 200 180
1097 1125 1156
141 141 141
0873 0870 0877
128 129 131
0786 0780 0785
130 133 136
400 500 600
10 10 10
0768 0914 1049
232 236 230
2591 2680 2781
203 203 205
3689 3847 4084
190 192 196
4147 4387 4804
176 179 186
200 300 400
20 20 20
0539 0712 0929
169 208 256
3941 3993 4042
260 260 260
8815 9353 9575
295 303 307
1163 1350 1525
260 280 303
A selection of and k values obtained by this method are provided in Table 4.11, which is part of the tables in the R6 and SINTAP/FITNET compendia. 4.4.6.4. T-Stress Corrected Toughness Values in the Context of the Master Curve Approach T-stress based corrections have been proposed for use in the Master Curve approach, as described in Section 4.4.5.2, in [4.32] and as adopted in the FITNET document. In order to avoid confusion with the temperature T, the authors use the term Tstress for designating the T-stress. For Tstress < 0 the toughness Kmat can be corrected by
√ √ −Tstress c Kmat ≈ 20 MPa m + Kmat − 20 MPa m · exp 0019 · (4.85) 10 MPa with Kmat denoting the standard, plane strain, toughness determined by deeplyc the constraint corrected toughness. The transicracked bend specimens, and Kmat tion temperature To can be corrected by
Tstress Tstress T0c ≈ T0 + or T0c ≈ T0 + (4.86) 10 MPa/K 10 MPa/o C
The Input Parameters
85
In Eqn (4.86) To is the transition temperature determined from standard bend specimens and T0c is its constraint corrected counterpart.
4.4.7. Reference Toughness Based on Charpy Data 4.4.7.1. General Aspects In practical applications there are cases in which no fracture toughness data are available, as in the assessment of a component that was constructed decades ago and for which no data can be generated retrospectively due to lack of material. For such situations SINTAP/FITNET contains a number of empirical correlations for estimating the fracture toughness from Charpy data. Note, however, that the price to be paid for this kind of analysis can be over-conservative results. Numerous correlations between Charpy and fracture toughness data, usually for a limited class of materials, are available in the scientific literature and in compendia, for a limited overview see, for example, [4.33]. Because they are purely empirical, the equations should only be applied to those materials for which they were obtained. SINTAP/FITNET contains lower-bound correlations for a wide range of steels, primarily with yield strengths between 250 and 600 MPa, for the lower and upper shelf and for the ductile transition range. These are given below for CV in Joule, B in mm, Y and E in MPa, toughness in MPa · m1/2 and temperature in C. 4.4.7.2. Lower Shelf and Lower Transition The proposed correlation for steel ! Kmat = 12 CV − 20 · 25/B025
(4.87)
is based on the Master Curve approach for a 5% percentile of fracture toughness at a Charpy energy of CV = 27 J. Its application is restricted to 3J ≤ CV < 27J. The Kmat estimate should be based on a minimum of three Charpy tests. Note that Eqn (4.87) is more conservative than most other empirical correlations for steel, which are usually mean curves, but is in line with empirical lower bound correlations [4.34]. If neither fracture toughness data nor Charpy test data are available, a conservative lower bound toughness of Kmat = 20 MPa · m1/2 can be used for ferritic steels.
86
Fitness-for-Service Fracture Assessment of Structures Containing Cracks
4.4.7.3. Upper Shelf Correlations are provided for KJ02 values corresponding to 0.2 mm stable crack extension and for R curves [4.33]. KJ02 can be estimated as a lower bound Kmat = KJ02 =
0133·C0256 1 128 Vus · E · 053 C · 02 Vus 1000 · 1 − 2
(4.88)
In Eqn (4.88), CVus designates upper-shelf Charpy energy in J. Alternatively, KJ02 can be determined by Kmat = KJ02 ≈ 119 · C0545 Vus
(4.89)
for steels. The application range of Eqns (4.88) and (4.89) is restricted to 170 MPa ≤ Y ≤ 1000 MPa and 20 J ≤ CVus ≤ 300 J and to cases for which brittle material behaviour can be excluded. Note, however, that even fully ductile Charpy specimen behaviour, that is, 100% shear fracture appearance, cannot automatically guarantee that the structure will also behave in a ductile manner at the same temperature. In particular, caution should be exercised for thick-walled components and for some low carbon and low sulphur steels. A lower bound R-curve (5% percentile) can be estimated by J = J a = 1 mm · am
(4.90)
with J a = 1 mm being the J-integral corresponding to a stable crack extension
a of 1 mm approximated by J a = 1 mm =
053 · C128 Vus · exp
T − 20 − 400 C
(4.91)
(for T in C) and the empirical exponent m being determined by m=
0133 · C0256 Vus · exp
T − 20 C Y − − + 003 2000 C 4664 MPa
(4.92)
The correlation was developed from 112 multi-specimen test data sets covering material properties 171 MPa ≤ Y ≤ 985 MPa and 20 J ≤ CVus ≤ 300 J and test temperatures −100 C ≤ T ≤ 300 C.
The Input Parameters
87
4.4.7.4. Ductile-to-Brittle Transition The estimates in the ductile-to-brittle transition range are based on empirical correlations between the transition temperature T0 and the Charpy transition temperature T27J (or T28J . Based on the general relationship T0 = T27J − 18 C standard deviation ± 15 C
(4.93)
a lower bound estimate is given by T0 = T27J − 3 C
(4.94)
The fracture toughness Kmat for an arbitrary specimen thickness B (or crack front length ) can then be determined by Kmat in MPam1/2 = 20 + 77 · exp 0019 · T − T27J − 3 C
025 025 B 1 × · ln 25 1−P
(4.95)
For cases in which no transition temperature T27J (or T28J is available, T27J can conservatively be determined by
C CV · CVus − 27J T27J = TCV − · ln (4.96) 4 A · CVus − CV with C being the slope of the CV -T transition curve in C, conservatively corrected, however, by a factor of 2, and where A = 27J. C can be estimated by C ≈ 34 C +
Y C − Vus 351 143
(4.97)
where T27J and TCV in C; Y in MPa; CV and CVus in J. TCV is the temperature at which Charpy data are available and CVus is the upper-shelf Charpy energy as above. If the upper-shelf energy is unknown, it can be assumed to be twice the highest measured impact energy value. If only the upper-shelf energy is known, Eqn (4.96) is replaced by
C 19 · CVus − 27J (4.98) T27J = Tus − · ln 4 27J with Tus being the lowest test temperature.
88
Fitness-for-Service Fracture Assessment of Structures Containing Cracks
4.4.7.5. Sub-Sized Charpy Specimens When the component thickness is less than 10 mm, sub-sized Charpy specimens may be used. Note that the use of smaller specimens yields a shift Tss in the transition temperature that can be estimated using
Tss = −514 · ln 2B/10025 − 1
(4.99)
with B being the thickness (<10 mm) of the sub-sized specimens. An upper-shelf estimate can be conservatively taken as CVus Charpy specimen ≥ CVus B ·
10mm B
(4.100)
with CVus (B) being the upper-shelf energy for the specimen of thickness B.
Chapter 5
The Model Parameters
5.1. The Stress Intensity Factor (K-Factor) 5.1.1. Sources for Analytical K-Factor Solutions The primary source for analytical stress intensity factor solutions recommended in this book is the FITNET compendium [5.1] which lists a large number of solutions for flat plates, round bars, spheres, pipes and cylinders, tubular joints, nozzles and weldments. Numerous K-factor solutions can also be found in a number of further compendia – some of which will briefly be introduced below – and in open literature. Comprehensive collections of solutions are available in (a) The Murakami handbook [5.2]. With about 4 500 pages this compendium probably contains the world’s largest collection of K-factor solutions. (b) The Tada–Paris–Irwin Handbook [5.3]. The third edition of this handbook was published in 2000. (c) The Rooke–Cartwright Handbook [5.4]. This is the oldest of the collections listed, dating back to 1974. (d) The BS 7910 Compendium [5.5]. The collection in the British Standard follows a K-factor format based on stress components, such as membrane and bending stresses, from which total K factors can be determined by superposition. (e) The API 579 Compendium of the American Petroleum Institute [5.6] gives a collection of K solutions for crack configurations that are likely to occur in pressurised components. (f) The French CEA-A16 Guide Compendium [5.7] contains solutions for plates, pipes, pipe elbows, piping tees and nozzles for a very wide range of dimensions. (g) The NASGRO Compendium [5.8] is part of the NASGRO computer software and contains K solutions of components and details typical of aerospace applications. (h) The R6 Compendium of British Energy Generation [5.9] (see also [5.10]) lists solutions for a number of standard geometries such as plates and cylinders.
90
Fitness-for-Service Fracture Assessment of Structures Containing Cracks
(i) The DNV (Det Norske Veritas) Compendium [5.11], an update of the Swedish SA/FoU Report 91/01 [5.12], also contains K solutions for standard geometries. (j) The ETM Compendium [5.13] of the German GKSS Research Centre contains K solutions for standard geometries, mostly for individual load cases such as tensile forces, bending moments, internal pressure or combinations of these. (k) The IIW Compendium [5.14] of the International Institute of Welding contains K-factor solutions for typical weldment geometries. (l) The EPRI Compendium [5.15] of the American Electric Power Research Institute contains K solutions for standard geometries. (m) The IWM-VERB Compendium [5.16] of IWM Freiburg in Germany is part of the VERB 7 computer software and contains K solutions mainly for standard geometries. (n) The German FKM Compendium [5.17] contains K solutions for standard geometries. (o) Further compendia of weight function solutions [5.18, 5.19].
5.1.2. Types of Analytical K-Factor Solutions The types of solutions discussed below are arranged relative to their application, not by the methods by which they have been obtained. It is possible to distinguish between • solutions for specific geometries and with specific loading, • solutions for specific geometries but with arbitrary loading, and • solutions based on linearized stress profiles. 5.1.2.1. K-factor Solutions for Specific Geometries and Specific Loading Specific geometries are flat and curved plates, pipes, pressure vessels, nozzles, etc. Specific loading may be tensile forces, bending moments, internal pressure and others. An example of a K solution for both a specific geometry and a specific loading is provided in Eqn (5.1). It refers to a hollow cylinder with an axial semi-elliptical crack at the inner side such as shown in Fig. 5.1. A stress intensity factor solution for internal pressure is provided in [5.20]. For the deepest point of the crack, K is obtained as KAI
2 t Ro + Ri2 a = 097 a · Y · · 2 + 1 − 05 2 Ri Ro − Ri t √
(5.1)
The Model Parameters
91
x 2c a Ro Ri t
Figure 5.1: Geometry used in Examples 5.1, 5.2 and 5.5: Hollow cylinder with an axial semi-elliptical crack at the inner surface.
with the hoop stress being = p · Ri t The geometry function Y is a 2 a 4 1 + M3 Y= √ M1 + M2 t t Q
(5.2)
(5.3)
with M1 = 113 − 009 a/ c M2 = −054 +
089 02 + a/ c
(5.4) (5.5)
and M3 = 05 −
1 + 14 1 − a/ c 24 065 + a/ c
(5.6)
The elliptical crack shape factor is given by Q = 1 + 1464
a 165 c
(5.7)
92
Fitness-for-Service Fracture Assessment of Structures Containing Cracks
For the surface points of the crack, K is
KIB = KIA · 11 + 035 a/ t2 · a/ c
(5.8)
Example 5.1: A tube with dimensions Ri = 100 mm, R0 = 110 mm and t = R0 – Ri = 10 mm contains an internal crack with a = 1 mm and 2c = 2 mm. If the applied load is an internal pressure of 30 MPa the K-factor at the deepest point of the crack is determined as 200 MPa · m1/2 . 5.1.2.2. K-Factor Solutions for Specific Geometries but with Arbitrary Loading in the Thickness Direction Frequently, the loading is provided as a three-dimensional stress field derived, for example, by finite element calculations in the un-cracked section. However, the K solutions introduced in this section are based on variable stress profiles in the thickness direction only, that is, the stress profile is assumed to be constant parallel to the surface. This assumption is reasonable in many cases, such as for pressurised tubes. An example of the latter is the hollow cylinder of Fig. 5.1. In this case the stress distribution in the axial direction can be assumed to be uniform. For a stress profile in the thickness direction given by a polynomial = x/ a =
3
j ·x/ aj = 0 +1 x/ a+2 x/ a2 +3 x/ a3
for 0 ≤ x ≤ a
j=0
(5.9) the K-factor can be determined by: KI =
√
a ·
3
j · fj a/t a/c Ri /t = 0 · f0 + 1 · f1 + 2 · f2 + 3 · f3 (5.10)
j=0
[5.10, 5.11]. The solution is a shortened version of solutions in [5.21] and [5.22]. Its geometry function fj is given in Tables 5.1 and 5.2. Equation (5.10) is obtained by the weight function method. Note that the stresses acting on the un-cracked section at the position of the imaginary crack are of particular importance for the analysis. Thus, in order to improve the quality of the polynomial approximation, it can be of advantage to restrict the included stresses to a region close to, but larger than, the crack area. Example 5.2: From a finite element analysis of a stiffened tubular structure, a stress profile across the wall without a crack is given in Table 5.3.
The Model Parameters
93
Table 5.1: Geometry functions of Eqn (5.10) for the deepest point of the crack. Component geometry according to Fig. 5.1 a/c = 1; Ri /t = 4
a/t
00 02 05 08
f0A
f1A
f2A
f3A
f0A
f1A
f2A
f3A
0659 0643 0663 0704
0471 0454 0463 0489
0387 0375 0378 0397
0337 0326 0328 0342
0659 0647 0669 0694
0471 0456 0464 0484
0387 0375 0380 0394
0337 0326 0328 0339
a/c = 04; Ri /t = 4
a/t
00 02 05 08
a/c = 04; Ri /t = 10
f0A
f1A
f2A
f3A
f0A
f1A
f2A
f3A
0939 0919 1037 1255
0580 0579 0622 0720
0434 0452 0474 0534
0535 0382 0395 0443
0939 0932 1058 1211
0580 0584 0629 0701
0434 0455 0477 0523
0353 0383 0397 0429
a/c = 02; Ri /t = 4
a/t
00 02 05 08
a/c = 1; Ri /t = 10
a/c = 02; Ri /t = 10
f0A
f1A
f2A
f3A
f0A
f1A
f2A
f3A
1053 1045 1338 1865
0606 0634 0739 0948
0443 0487 0540 0659
0357 0406 0438 0516
1053 1062 1359 1783
0606 0641 0746 0914
0443 0489 0544 0639
0357 0417 0440 0504
This stress profile is approximated by the polynomial of Eqn (5.9), which produces the coefficients 0 = 3158 MPa, 1 = −3554 MPa, 2 = 01049 MPa and 3 = −5838 · 10−3 MPa. x/ a = 3158 − 3554 x/ a + 01049 x/ a2 − 5828 · 10−3 x/ a3 MPa For a crack depth-to-wall thickness ratio of a/t = 02, a crack shape a/c = 1 and a tube radius to thickness ratio Ri /t = 10 (Fig. 5.1), the geometry functions are taken from Tables 5.1 and 5.2 as (a) Deepest point of the crack f0A = 0647 f1A = 0456 f2A = 0375 f3A = 0326 (b) Surface points of the crack f0B = 0726 f1B = 0126 f2B = 0047 f3B = 0024
94
Fitness-for-Service Fracture Assessment of Structures Containing Cracks
Table 5.2: Geometry functions of Eqn (5.10) for the surface points of the crack. Component geometry according to Fig. 5.1 a/c = 1; Ri /t = 4
a/t
00 02 05 08
f0B
f1B
f2B
f3B
f0B
f1B
f2B
f3B
0716 0719 0759 0867
0118 0124 0136 0158
0041 0046 0052 0062
0022 0024 0027 0032
0716 0726 0777 0859
0118 0126 0141 0163
0041 0047 0054 0063
0022 0024 0028 0033
a/c = 04; Ri /t = 4
a/t
00 02 05 08
a/c = 04; Ri /t = 10
f0B
f1B
f2B
f3B
f0B
f1B
f2B
f3B
0673 0670 0803 1060
0104 0107 0151 0229
0032 0037 0059 0095
0016 0018 0031 0051
0673 0676 0814 1060
0104 0109 0153 0225
0032 0037 0060 0092
0015 0018 0031 0049
a/c = 02; Ri /t = 4
a/t
00 02 05 08
a/c = 1; Ri /t = 10
a/c = 02; Ri /t = 10
f0B
f1B
f2B
f3B
f0B
f1B
f2B
f3B
0516 0577 0759 1144
0069 0075 0134 0250
0017 0022 0051 0103
0009 0010 0027 0056
0516 0578 0753 1123
0069 0075 0131 0241
0017 0022 0050 0099
0009 0010 0026 0053
Table 5.3: Stress profile across the un-cracked wall of a stiffened tubular structure. The coordinate x is oriented in the crack depth direction and measured from the free surface The component dimensions (without the stiffener) refer to Example 5.1 (Ri = 100 mm, R0 = 110 mm, t = 10 mm). The crack dimensions are a = 2 mm and 2c = 4 mm x in mm
x/a
in MPa
x in mm
x/a
in MPa
0 1 2 3 4 5
0 05 10 15 20 25
316 312 309 306 303 300
6 7 8 9 10
30 35 40 45 50
297 394 291 288 289
The Model Parameters
95
Extrapolation of the tabulated values is performed as described in Section 5.1.4. The resulting stress intensity factors are determined as (a) Deepest point of the crack KIA = 1607 MPa · m1/2 (b) Surface points of the crack KIB = 1814 MPa · m1/2 Weight function solutions such as Eqn (5.10) can be given for various types of stress distributions across the wall such as uniform, square root, linear, power law, quadratic, cubic or even polynomial. In some cases, solutions for specific load cases are combined with solutions for polynomial stress distributions. The K solution of the following example is provided in [5.7], see also [5.23]. Example 5.3: This example refers to a thick-walled hollow cylinder (Fig. 5.2) subjected to internal pressure, which results in a constant axial stress p , and a thermal stress gradient t (x/t). Since the cylinder is supported on two bearings its dead weight results in an additional global bending stress gb . For an outer radius R0 = 200 mm, an inner radius Ri = 100 mm, crack dimensions a = 2 mm and 2c = 4 mm and an internal pressure pi = 60 MPa, the axial stress due to the internal pressure is determined as p =
Ri2 · p = 20 MPa Ro2 − Ri2 i
(5.11)
2c
a x Ri
t Ro
Figure 5.2: Example 5.3: Geometry of the hollow cylinder.
96
Fitness-for-Service Fracture Assessment of Structures Containing Cracks
and, with an assumed global bending moment at the position of the crack of Mgb = 295 kNm, the global bending stress is gb =
Mgb 4Mgb Ro
= 50 MPa = 4 Wb Ro − Ri4
(5.12)
The axial stress arising from the thermal gradient is assumed to be given by a finite element analysis as summarised in Table 5.4. Its profile across the wall is approximated by a polynomial x/ t =
3
i x/ ti = 0 + 1 x/ t + 2 x/ t2 + 3 x/ t3
(5.13)
i=0
with coefficients i of 0 = 30 MPa, 1 = −9333 MPa, 2 = −375 MPa and 3 = 2083 MPa. Figure 5.3 gives an overview of all the stress components to which the hollow cylinder is subjected. Table 5.4: Assumed axial stresses due to the thermal gradient across the wall of the hollow cylinder x in mm
x/t
ax (thermal Gradient) in MPa
0 20 40 60 80 100
0 02 04 06 08 10
30 10 −12 −35 −58 −80
50 MPa
σgb
20 MPa
σax
–80 MPa
30 MPa
σlb
–80 MPa
–50 MPa
20 MPa
30 MPa
Figure 5.3: Example 5.3: Loading of the thick walled hollow cylinder.
The Model Parameters
The stress intensity factor is determined by an expression a a 2 a 3 √ √ KI = 0 f0 + 1 f1 + 2 f2 + 3 f3 a + Fb gb a t t t
97
(5.14)
Equation (5.14) includes terms for the thermal stress gradient t and the global bending stress gb , but not for the constant axial stress p resulting from the internal pressure. This can, however, be taken into account by adding the constant axial stress component 0 from the thermal gradient with p which also refers to a constant axial stress. Equation (5.14) can then be re-written as a a 2 a 3 √
√ + 3 f3 a +Fb gb a (5.15) + 2 f2 KI = 0 + p f0 + 1 f1 t t t with (0 + p = 30 + 20 MPa = 50 MPa. The fi and Fb functions of Eqn (5.15) are given in Table 5.5 [5.23]. For the given crack dimensions, ratios of a/c = 1 and a/t = 002 are obtained which refer to geometry functions of (a) Deepest point of the crack f0 = 06562 f1 = 04640 f2 = 03840 f3 = 03368 and Fb = 06514 (b) Surface points of the crack f0 = 07452 f1 = 01284 f2 = 00476 f3 = 00246 and Fb = 07436 The final result for the stress intensity factor is determined as (a) Deepest point of the crack KIA = 511 MPa · m1/2 (b) Surface points of the crack KIB = 588 MPa · m1/2 5.1.2.3. K-Factor Solutions for Specific Geometries but with Arbitrary Loading in the Thickness and Length Directions The K solutions introduced in this section are based on a variable stress field in both the thickness and length directions of the un-cracked section. An example is provided by a solution in [5.24]. This is derived for a semi-elliptical surface crack in a semi-infinite body such as shown in Fig. 5.4.
98
Table 5.5: Geometry functions fi and Fb functions of Eqn (5.14) for Ro /Ri = 2
1
1/2
1/4
a/t
Deepest point of the crack
Surface points of the crack
f0
f1
f2
f3
Fb
f0
00 01 02 04 06 08 00 01 02 04 06 08
0657 0653 0651 0653 0656 0655 0883 0893 0914 0993 1120 1309
0465 0460 0457 0452 0445 0443 0569 0570 0576 0599 0673 0704
0385 0380 0377 0371 0364 0362 0451 0449 0452 0462 0479 0514
0338 0332 0330 0324 0318 0315 0386 0383 0384 0389 0397 0420
0657 0629 0602 0562 0521 0480 0883 0864 0855 0871 0912 0987
0744 0750 0755 0784 0832 0898 0704 0717 0732 0797 0901 1043
00 01 02 04 06 08
1022 1048 1101 1308 1674 2358
0632 0639 0657 0727 0850 1093
0491 0492 0502 0536 0598 0726
0415 0414 0419 0439 0476 0557
1022 1015 1031 1137 1332 1676
0568 0588 0608 0667 0717 0732
f1
f2
f3
Fb
0129 0126 0127 0134 0146 0162 0119 0114 0119 0138 0169 0210
00478 00467 00471 00501 00551 00612 00406 00409 00430 00525 00669 00857
00247 00242 00243 00259 00285 00314 00204 00207 00220 00275 00357 00463
0744 0742 0736 0748 0769 0803 0704 0708 0710 0734 0762 0797
00737 00737 00798 00979 0111 0112
00215 00221 00251 00338 00396 00391
000956 00101 00118 00169 00200 00195
0568 0575 0563 0499 0360 0141
(Continued)
Fitness-for-Service Fracture Assessment of Structures Containing Cracks
a/c
Table 5.5: (Continued)
1/8
1/16
1088 1123 1210 1514 2007 – 1105 1154 1256 – – –
0664 0674 0705 0815 1002 – 0673 0689 0729 – – –
0512 0515 0532 0590 0691 – 0518 0525 0547 – – –
0431 0431 0441 0476 0541 – 0435 0438 0452 – – –
1088 1085 1128 1287 1595 – 1105 1115 1169 – – –
0432 0447 0458 0454 0378 – 0282 0334 0333 – – –
00393 00370 00401 00360 000205 – 00202 00166 00150 – – –
00 01 02 04 06 08
1122 1163 1230 1427 1778 2591
0683 0696 0723 0798 0933 1258
0526 0531 0546 0589 0664 0851
0441 0443 0454 0482 0532 0657
1122 1138 1210 1472 1994 3230
– – – – – –
– – – – – –
000808 000841 000986 000641 −00150 – 000277 000259 000153 – – – – – – – – –
000291 0432 000329 0413 000412 0330 000160 00115 −00128 −0428 – – 0000713 0282 0000862 0236 0000114 000213 – – – – – – – – – – – –
– – – – – –
The Model Parameters
0
00 01 02 04 06 08 00 01 02 04 06 08
99
100
Fitness-for-Service Fracture Assessment of Structures Containing Cracks
σyy (x, z)
θ x
y a
x
z
2c
z
z = c sin (θ) x = a cos (θ)
σyy (x, z)
Figure 5.4: Semi-infinite body containing a semi-elliptical surface crack subjected to an arbitrary stress field across the section.
The K solution which is valid for = 03 and 05 ≤ a/c ≤ 2 is given as a set of analytical equations for a two-dimensional stress distribution yy x z = 0nm z/ cn x/ am
0 ≤ n + m < 4
(5.16)
which is characterised by ten 0nm coefficients according to the ten possible combinations for n and m (Table 5.6) z 0 x 0 z 0 x 1 z 0 x 2 · + 001 · + 002 · c a c a c a z 0 x 3 z 1 x 0 z 1 x 1 + 003 · + 010 · + 011 · c a c a c a z 1 x 2 z 2 x 0 z 2 x 1 + 012 · + 020 · + 021 · c a c a c a z 3 x 0 + 030 · (5.17) c a
yy x z = 000
Table 5.6: Combination of n and m for which geometry functions FI according to Eqn (5.18) exist n
m
n
m
n
m
0 0 0 0
0 1 2 3
1 1 1
0 1 2
2 2 3
0 1 0
The Model Parameters
101
Note that the square brackets [ ] in Eqns (5.16) to (5.35) stand for “function of ” both in this and in the following equations. The KI solution as a function of the angle defining the point at the crack front, such as illustrated in Fig. 5.4, is given as a superposition of 10 K components √ KI = 0nm · FI n m a/ c · a
(5.18)
or KI =
√
a · 000 FI00 + 001 FI01 + 002 FI02 + 003 FI03
+ 010 FI10 + 011 FI11 + 012 FI12 + 020 FI20 + 021 FI21 + 030 FI30
(5.19)
The geometry function FI [m, n, , a / c ] is obtained from FI m n a/c = F0 1 + −1 + H · T m + n
(5.20)
H m n a/ c = Gm Tm + Gn Tn + Gnm − Gn − Gm · Tm×n
(5.21)
with
and Ti =
2 arctan 1000 · i
(5.22)
Note, that i in Eqn (5.21) represents n, m, (n · m) or (n + m). Ti is acting as a step function such that Ti = 0 for i = 0 and Ti = 1 for i = 0. The quantity F0 in Eqn (5.20) is determined by F0 a/ c = C0 + C2 2 + C4 4
(5.23)
C0 a/ c = 1225 − 08512 a/ c + 03414 a/ c2 − 00561 a/ c3
(5.24)
with
C2 a/ c = −054781 + 097969 a/ c − 052601 a/ c2 + 010557 a/ c3 (5.25)
102
Fitness-for-Service Fracture Assessment of Structures Containing Cracks
and C4 a/ c = 011569 − 018205 a/ c + 009851 a/ c2 − 002134 a/ c3 (5.26) The function Gm is obtained from Gm m a/c = Nm + Pm cosm
(5.27)
Nm m = 03 − 0165 · m + 0025 · m2
(5.28)
Pm m a/ c = 0493 − 0096 · m + 0009 · m2
+ a/ c · 0105 + 00645 · m − 00105 · m2
− a/ c2 · 00204 + 00039 · m − 000069 · m2
(5.29)
with
and
The further functions are Gn n a/ c = Nn + Pn · sinn
Pn n a/ c = 12728 − 0439 · n + 009 · n2
− a/ c · 05643 − 024675 · n + 00693 · n2
+ a/ c2 · 01478 − 007781 · n + 002489 · n2
(5.30)
(5.31)
Nn n = 00375 · 1 + −1n
(5.32)
Gnm n m a/ c = Nnm + Pnm sinn · cosm
(5.33)
Pnm nma/ c = 0575−00313·m −n−011863·m −n2 +a/ c· −00866+006706·m −n+005393·m −n2 +a/ c2 · 0021−00157·m −n−001193·m −n2 (5.34) and Nmn n = 0015 · 1 + −1n
(5.35)
103
The Model Parameters
Table 5.7: Example 5.4: Stresses normal to the crack plane according to Fig. 5.4, for example, arising from a thermal gradient across the wall x in mm
z in mm
in MPa
x in mm
z in mm
in MPa
x in mm
0 1 2 5 10
−20 −20 −20 −20 −20
820 775 730 630 500
0 1 2 5 10
0 0 0 0 0
800 750 710 605 480
0 1 2 5 10
z in mm
in MPa
20 20 20 20 20
755 705 655 545 435
Example 5.4: A stress field in the x and z directions according to Fig. 5.4 is provided in Table 5.7. The crack dimensions are assumed as a = 5 mm and 2c = 14 mm. With the coefficients in Table 5.8a (columns 1 and 4–14) the stress coefficients of Eqn (5.17) are obtained by multi-dimensional quasi-linear regression [5.25]. The application of the least squares method yields a set of linear equations, the solutions of which are as follows 000 = 79776 MPa
001 = −22730 MPa
002 = 2602 MPa
003 = 431 MPa
010 = −1260 MPa
011 = −6880 MPa
012 = 3375 MPa
020 = −1660 MPa
021 = −0074 MPa
and 030 = 0177 MPa Based on these, the FI functions given in Table 5.8b and the stress intensity factors are obtained for the deepest point of the crack ( = 0) and for both surface points ( = +90 and −90 ) by Eqn (5.19) as K = 0 = 6392 MPa · m1/ 2 K = +90 = 7244 MPa · m1/ 2 and K = −90 = 7081 MPa · m1/ 2
Table 5.8a: Example 5.4: Matrix which forms the set of linear equations the solutions of which are the onm -coefficients of Eqn (5.17) (“→” designates the mathematical limit) 1
2
i
x in mm
z in mm
3
4
1
0
−20
820
→1
0
0
0
2
1
−20
775
1
02
004
0008
−2857
3
2
−20
730
1
04
016
0064
4
5
−20
630
1
10
100
1000
5
10
−20
500
1
20
400
8000
6
0
0
800
→1
7
2
0
750
8
1
0
710
in MPa
5
6
7
8
z 0 x 0 z 0 x 1 z 0 x 2 z 0 x 3 c a c a c a c a
→0
→0
→0
1
→ 02
→ 004
1
→ 04
→ 016
9
10
11
12
13
14
z 1 x 0 z 1 x 1 z 1 x 2 z 2 x 0 z 2 x 1 c a c a c a c a c a
z 3 x 0
→ −2857
→ −23323
0
0
→ 8163
0
c
a
−0571
−0114
8163
1632
−23323
−2857
−1142
−0457
8163
3265
−23323
−2857
−2857
−2857
8163
8163
−23323
−2857
−5714
−11438
8163
16326
−23323
→0
0
→0
0
0
→0
→ 0008
0
0
0
0
0
0
→ 0064
0
0
0
0
0
0
9
5
0
605
1
→ 10
→ 100
→ 1000
0
0
0
0
0
0
10
10
0
480
1
→ 20
→ 400
→ 8000
0
0
0
0
0
0
11
0
20
755
→1
0
0
0
→ 2857
0
0
→ 8163
0
12
1
20
705
1
02
004
0008
2857
0571
0114
8163
1632
23323
13
2
20
655
1
04
016
0064
2857
1142
0457
8163
3265
23323
14
5
20
545
1
10
100
1000
2857
2857
2857
8163
8163
23323
15
10
20
435
1
20
400
8000
2857
5714
11428
8163
16326
23323
→ −23323
The Model Parameters
105
Table 5.8b: Example 5.4: FI -functions according to Eqn (5.19)
in
0 +90 −90
in 0 +90 −90
FI00
FI01
FI02
FI03
FI10
0.7707 0.7497 0.7498
0.5143 0.1203 0.1203
0.4093 0.0527 0.0527
0.3457 0.0226 0.0226
0.0005 0.5216 −0.5206
FI11
FI12
FI20
FI21
FI30
2.45E-4 2.39E-4 2.39E-4
1.64 E-4 6.92 E-6 3.25 E-4
0.0580 0.4708 0.4708
0.0231 0.0226 0.0226
1.6 E-4 0.3873 −0.3870
Stress in circumferential direction in MPa
u in mm
300
200
–100 100
0 100
Crack
300 200
400 0 0
100
200
300
v in mm
Figure 5.5: Example 5.5: Assumed stress distribution across the nozzle section from finite element analysis.
Example 5.5: This example refers to a nozzle loaded by a thermal gradient. The assumed two-dimensional stress field is shown in Fig. 5.5 and summarised in Table 5.10. The figure represents a section of the nozzle geometry which is shown as an idealisation in Fig. 5.6. The component dimensions are: Radius R = 2000 mm, radius r = 250 mm, tmin = 350 mm, two cracks are assumed to be quarter-circular with a = 50 mm and a = 100 mm.
106
Fitness-for-Service Fracture Assessment of Structures Containing Cracks
r u′
tm
in
v A B C u
a
R
Figure 5.6: Example 5.5: Quarter-circular corner crack in the nozzle of a cylindrical vessel.
Table 5.9: Example 5.5: Assumed stress field across the nozzle section u in mm
v in mm
in MPa
u in mm
v in mm
in MPa
u in mm
v in mm
in MPa
0 0 0 0 0 0
0 25 50 75 100 125
500 425 410 400 350 310
50 50 50 50 50 50
0 25 50 75 100 125
415 315 255 225 200 175
100 100 100 100 100
0 25 50 75 100
400 250 190 150 100
25 25 25 25 25 25
0 25 50 75 100 125
425 360 305 270 250 225
75 75 75 75 75
0 25 50 75 100
408 285 225 185 150
125 125 125
0 25 50
360 210 155
With the data of Table 5.9, the circumferential stress field is approximated using = A00 + A10 u + A20 u2 + A30 u3 + A01 v + A02 v2 + A03 v3
(5.36)
with A-coefficients of A00 = 549744 MPa A10 = −4282 MPa/mm A20 = 0038 MPa/mm2 A30 = −146 E-4MPa/mm3 A01 = −5153 MPa/mm A02 = 0047 MPa/mm2 and A03 = 171 E-4 MPa/mm3 .
The Model Parameters
107
The stress intensity factor is determined from √ KI = a · A00 + A10 + A01 · a + A20 + A02 · a2 + A30 + A03 · a3 · f00 + −A10 · a − 2A20 · a2 − 3A30 · a3 · f10 + −A01 · a − 2A02 · a2 − 3A03 · a3 · f01 + A20 · a2 + 3A30 · a3 · f20 + A02 · a2 + 3A03 · a3 · f02 −A30 · a3 · f30 − A03 · a3 · f03 · fc (5.37) [5.12] using the f-coefficients given in Table 5.10. The resulting K factors are Point A) KIA a = 50 mm = 15293 MPa · m1/2 KIA a = 100 mm = 17595 MPa · m1/2 Point B) KIB a = 50 mm = 11936 MPa · m1/2 KIB a = 100 mm = 12869 MPa · m1/2 Point C) KIC a = 50 mm = 13186 MPa · m1/2 , KIC a = 100 mm = 14377 MPa · m1/2 . 5.1.2.4. Solutions Based on Linearized Stress Profiles These type of solutions are sometimes designated as code solutions since they are relevant for the design and assessment codes such as BS 7910 [5.5] or the ASME (American Society of Mechanical Engineers) Boiler and Pressure Vessel Code [5.26]. They are usually based on bending and membrane stress components determined as described in Section 4.1.3. Table 5.10: Example 5.5: Coefficients of Eqn (5.37) for the positions A, B and C at the crack front (Fig. 5.6) a/r
fcA
fcB
fcC
ij
fijA
fijB
fijC
00 01 02 03 04 05 10 15 20 25 30
064 064 064 064 064 064 064 064 064 064 064
064 060 057 055 054 053 051 049 047 046 046
064 059 055 054 052 052 049 047 046 045 043
00 10 20 30 01 02 03
140 098 080 058 098 065 055
108 100 077 050 100 077 050
140 098 065 055 098 080 058
Fitness-for-Service Fracture Assessment of Structures Containing Cracks
σ stress σ
250
a = 3 mm
108
200 150
σ = 140 MPa
100 x a
linearized stress profile
50 t 0
2c
0
0.25
0.5 x/t
1
0.75
x t
θ
A a 2c
σ
B
2W
Figure 5.7: Example 5.6: Semi-elliptical surface crack in a plate subjected to a non-linear stress profile. Table 5.11: Example 5.6: Assumed stress profile across the wall of the plate of Fig. 5.7 x in mm 0.0 0.5 1.0 2.0
in MPa
x in mm
in MPa
x in mm
in MPa
250 200 180 150
3.5 5.0 7.0 9.0
140 140 130 110
11.0 13.0 15.0
100 90 80
Example 5.6: The geometry and stress profile is illustrated in Fig. 5.7, the latter summarised in Table 5.11. The thickness of the plate t is 15 mm, the crack dimensions are a = 3 mm and 2c = 6 mm. A K solution based on a linearized stress profile is provided in [5.27, 5.28]. The stress intensity factor is given by √ a KI = · F · m + Hb (5.38) E k for 0 ≤ a/c ≤ 1; 0 ≤ a/t < 1; 0 ≤ ≤ and a plate width larger than twice the crack length 2c.
The Model Parameters
The geometry function F is given by F = M1 + M2 a/ t2 + M3 a/ t4 · g · f · fW
109
(5.39)
with M1 , M2 and M3 from Eqns (5.4), (5.5) and (5.6). The further parameters are H = H1 + H2 − H1 · sinp
(5.40)
p = 02 + a/ c + 06 a/ t
(5.41)
with
H1 = 1 − 034 a/ t − 011 a/ c · a/ t
(5.42)
H2 = 1 + G1 a/ t + G2 a/ t2
(5.43)
G1 = −122 − 012 a/ c
(5.44)
G2 = 055 − 105 a/ c075 + 047 a/ c15 g = 1 + 01 + 035 a/ t2 · 1 − sin 2
(5.45) (5.46)
1 4 f = a/ c2 cos2 + sin2 /
(5.47)
a
1/2 a/ t fw = sec 2W
(5.48)
and E k =
Q Q according to Eqn 5.7
(5.49)
The given stress profile is linearized as m = −25 MPa and b = 275 MPa. The resulting K-factor is (a) Deepest point of the crack ( = 90 ) KIA = 1140 MPa · m1/2
(b) Surface points of the crack ( = 0 ) KIB = 1626 MPa · m1/2 For comparison, using the weight function solution provided in [5.29] which is based on a polynomial fit of the stress profile as described in 4.1.3, values of the stress intensity factor KIA = 1005 MPa · m1/2 and KIB = 1537 MPa · m1/2 were
110
Fitness-for-Service Fracture Assessment of Structures Containing Cracks
obtained. Thus, the stress linearization yielded slightly conservative results, that is, it overestimated the applied K-factor.
5.1.3. Superposition of K Factors The superposition principle allows the determination of K factors for complex configurations by algebraic summation of stress intensity factors for basic loading cases as long as the crack opening modes are identical. An example is that, for mode I loading, an overall KI can be obtained by KI = KI load case 1 + KI load case 2 +
(5.50)
The basic load cases can be tension, bending, local bending at notches, etc., or they can be loading by internal pressure or by dead weight. Note that simple superposition is not possible for K factors of different crack opening modes (mixed mode) for which an equivalent K has to be determined as described in Section 6.8. The K-factor solutions based on linear, quadratic, polynomial, etc. descriptions of the stress profile as described above were derived by application of the superposition principle. Equation (5.14), for example, can be re-written as KI = KI 0 +KI 1 +KI 2 +KI 3 +KI gb a √ a 2 √ a 3 √ √ √ = 0 f0 a +1 f1 a +2 f2 a +3 f3 a +Fb gb a t t t (5.51) with KI 0 , KI 1 , KI 2 , KI 3 and KI gb being the partial K-factor solutions.
5.1.4. Treatment of Geometry Factor Solutions Available in Table Format If the geometry function is given in table format, as in most of the examples above, the correct value has to be found by interpolation – the user should decide how to do this. One possible method [5.7] is explained as follows. The geometry function has been designated as V in general terms. It can depend on one, two or three parameters, x, y and z, which can be the crack shape a/c of a surface crack, the crack depth-to-wall thickness ratio a/t, the ratio of wall thickness to the inner radius t/Ri of a hollow cylinder etc.
The Model Parameters
111
(a) If V depends only on one parameter, x V x =
1 xi+1 − x · Vi + x − xi · Vi+1 xi+1 − xi
(5.52)
(b) If V depends on two parameters, x and y V x y =
1
xi+1 − xi · yj+1 − yj
xi+1 − x · yj+1 − y · Vij + xi+1 − x · y − yj Vij+1 × + x − xi · yj+1 − y · Vi+1j + x − xi · y − yj Vi+1j+1 (5.53)
(c) If V depends on three parameters, x, y and z Vx y z 1
xi+1 − xi · yj+1 − yj · zk+1 − zk ⎧ ⎫
xi+1 − x · yj+1 − y · zk+1 − z · Vijk + xi+1 − x · y − yj · zk+1 − z · Vij+1k ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ + x − xi · y j+1 − y · zk+1 − z · Vi+1jk + x − xi · y − yj · zk+1 − z · Vi+1j+1k × ijk+1 ij+1k+1 + xi+1 − x · yj+1 − y · z − zk · V + xi+1 − x · y − yj · z − zk · V ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ + x − xi · yj+1 − y · z − zk · Vi+1jk+1 + x − xi · y − yj · z − zk · Vi+1j+1k+1 =
(5.54)
5.1.5. Individual Determination of K Factors by Finite Element or Comparable Methods In cases where no analytical stress intensity factor solution is available, its individual determination by numerical approaches such as the finite element or the boundary element method is an alternative, even if it is then used within the analytical SINTAP/FITNET procedure. Various techniques are available for deriving K from the elastic stress–strain field ranging from the rather simple displacement correlation methods via energy difference to those that correlate K with the numerically determined contour J-integral. No detailed description of these methods will be given here, for more information see, for example, [5.30, 5.31].
112
Fitness-for-Service Fracture Assessment of Structures Containing Cracks
5.2. Net Section Yield Load FY , Reference Stress ref and Ligament Yielding Lr 5.2.1. Methods for the Generation of Yield Load Solutions The net section yield load FY (frequently called the limit load), or its associated parameters reference stress ref and Lr ratio Lr = F/FY = ref /Y
(5.55)
represent a key parameter for the accuracy of any SINTAP/FITNET analysis. The result, whether a critical load or a critical crack size, is significantly dependent on the conservatism or non-conservatism of the yield load. Unfortunately, the available yield load solutions vary greatly in quality since they have been obtained over decades by different methods. For some geometries, various papers and documents provide half a dozen or more yield load solutions which all produce different results. In the past, limit loads were generally determined as maximum loads that a given structure made of perfectly plastic material could sustain. When such a limit load is reached the deformations become unbounded. However, real materials usually show elastic–plastic and not perfectly plastic deformation behaviour. As a consequence a distinction must be made between two limiting conditions (a) The load at which the whole ligament becomes plastic. This is roughly correlated with the load above which the load-deformation characteristic of the component section becomes non-linear. This book designates this type of limit load as the yield load. (b) The load at which the component fails due to plastic collapse. This limit load is higher than the yield load and will be designated here as the plastic collapse load. Many yield load solutions are determined as limit loads based on a perfectly plastic material with a limiting stress equal to the yield strength of the material. In contrast, the collapse load is determined as the limit load for a perfectly plastic material with a limiting stress equal to what shall be designated a flow stress, f , which is usually defined as the average of the yield and tensile strengths f = 05 · Y + Rm
(5.56)
In this way, an approximation of the effect of work hardening on ligament deformation is taken into account.
The Model Parameters
113
Traditionally, solutions are obtained by two bounding theorems – a lower bound, characterised by a statically admissible stress field that satisfies equilibrium and yield, and an upper bound, based on a kinematically admissible strain-rate field satisfying compatibility and the flow rule [5.32]. The yield load solutions available in the compendia are commonly generated for plane strain or plane stress considerations and Tresca or von Mises yield criteria. As a rule, the lowest values are obtained for the lower-bound theorem, plane stress and Tresca’s criterion, whereas the highest values refer to the upper-bound theorem, plane strain conditions and von Mises criterion. Recent yield load solutions are often based on finite element analyses [5.33–5.40]. This is advantageous with respect to the possibility of threedimensional analyses, the treatment of complex structures and a yield load definition closer to SINTAP/FITNET. The last point needs some discussion. In the context of a SINTAP/FITNET analysis, FY , with respect to its original definition, is a reference load such as Po in the EPRI approach (see Section 2.5). It should be defined such that the crack driving force in terms of the J-integral or CTOD is predicted as closely as possible by the SINTAP/FITNET equations. Although the yield loads as defined above can be used as (mostly conservative) estimates of this reference load, the two quantities are not identical. Therefore, some authors have determined reference load solutions from finite element J solutions in recent papers, such as [5.35, 5.41]. In an investigation on a thin-walled curved and stiffened plate, two of the present authors used the Option 1 SINTAP/FITNET f(Lr ) function for defining a reference load at F/FY = Lr = 1 [5.42]. According to Eqn (6.25) this refers to a ratio Jep /Je = 22118 or ep /e = 22118 with “ep” standing for elastic-plastic and “e” for elastic. Both the elastic and the elastic-plastic crack driving force were obtained by finite element analyses. The coexistence of a number of different limit load solutions for identical geometries in the compendia and open literature is a source of uncertainty to the user and additional systematic investigations would be desirable. In general, the user is referred to the original sources to establish the suitability of the yield-load solutions that are to be applied.
5.2.2. Global vs. Local Yield Load For a cracked structure, the terms “global” and “local” yield loads must be defined. The term “global” refers to the entire structure including the cross section containing the crack, whereas “local” refers to a defined local region in the ligament ahead of the crack. In the case of through cracks a global yield load is usually applied in a SINTAP/FITNET analysis. Care is, however, required for
114
Fitness-for-Service Fracture Assessment of Structures Containing Cracks
very large ligament lengths, where even through cracks may have to be analysed based on local yield loads. In the case of partial penetration flaws, the SINTAP/FITNET procedure uses local yield loads as input parameters. Since ligament yielding usually will occur prior to global yielding the local yield load is less than, or at the maximum equal to, the global yield load for a given structure. However, when such an assessment leads to unacceptable results due to the over-conservatism of the local yield load solution, it is possible to re-characterise the crack as fully penetrating (see Section 4.2.5). Alternatively, a more exact local yield load solution can be used. It should be noted that in some recent studies the use of the global yield load has been justified for specific cases [5.53, 5.54].
5.2.3. Conservatism in Yield Load Determination The conservatism or non-conservatism of a SINTAP/FITNET analysis depends strongly on the conservatism of the yield load solution applied. In most applications conservative means that the yield load solution is under-estimated. There exist, however, cases where conservatism is reversed (a) If SINTAP/FITNET is applied in the context of a proof test philosophy, information is needed on the maximum crack size that would have survived the overload. This implies that an upper-bound yield load has to be used instead of a lower-bound value, which means that a global instead of a local yield load has to be adopted for surface flaws. (b) A complex situation arises in leak-before-break analyses of pressurised components. The aim of such an analysis is to establish whether the time span between the detection of leakage and the final failure of the component is large enough for intervention to occur. If the crack length at break-through is overestimated due to an underestimated yield load the residual lifetime up to the overall failure of the structure will also be underestimated which, in terms of the final result is conservative. On the other hand a larger crack size allows a larger leak rate. If the detection of the leak depends on the leak rate, early detection is predicted which does not occur in reality. This would be non-conservative. Generally, therefore, a leak-before-break analysis should be based on best-estimate yield loads and material properties, with sensitivity studies to determine if a prediction of detectable leakage prior to a guillotine break is strongly dependent on any of the inputs to the SINTAP/FITNET assessment. This approach is adopted by R6. Lower-bound values are based on the lower-bound theorem, plane stress conditions and the Tresca yield criterion and, in the case of embedded or surface
The Model Parameters
115
Out of plane membrane forces
Shear forces
Figure 5.8: Loading conditions under which even plane stress yield load solutions can be non-conservative.
cracks, on the local solution, whereas upper-bound values refer to the upperbound theorem, plane strain conditions, the von Mises yield criterion and the global solution respectively. Note, however, that even plane stress solutions may be non-conservative in the presence of high shear forces or high out-of-plane membrane forces (Fig. 5.8).
5.2.4. Sources for Analytical Yield Load Solutions The primary source for analytical yield load solutions recommended here is the FITNET compendium [5.43] which lists a large number of solutions for plates, bars, spheres, pipes and cylinders, tubular joints, nozzles and strength mismatch components. In addition to the FITNET compendium, numerous yieldload solutions can be found in a number of other compendia – some of which will briefly be introduced below – and in the open literature. (a) Probably the most comprehensive source of yield load solutions is the compendium provided by Miller [5.32] which formed the basis for early versions of the R6 procedure. As the compendium dates back to 1988, it is mostly based on analytical work.
116
Fitness-for-Service Fracture Assessment of Structures Containing Cracks
(b) R6 Revision 4 [5.10, 5.44, 5.45, 5.52] contains a compendium of yield load solutions that may be regarded as an updated version of that developed within SINTAP and of the Miller compendium. A number of finite element based solutions are introduced for standard geometries, particularly with partialpenetrating defects, that are less conservative than the older approaches. (c) The BS 7910 Compendium [5.46] gives all solutions in terms of the net section stress ref . (d) Similarly, the API 579 Compendium of the American Petroleum Institute [5.47] provides the user with net section stresses ref . (e) The DNV Compendium [5.48], and its older version SA/FoU 91/01 [5.49], provide the yield load in terms of the Lr ratio. (f) The ETM Compendia [5.13] and [5.50] of the GKSS Research Centre contain yield load solutions for homogeneous and strength mismatch components. The solutions for the homogeneous components are mainly based on specific loads such as tension forces, bending moments or internal pressure. (g) The EPRI Compendium [5.15] of the American Electric Power Research Institute contains reference load solutions Po for standard geometries that were originally defined on the basis of the EPRI approach, but nevertheless are close to the yield solutions needed in SINTAP/FITNET. (h) The IWM-VERB Compendium [5.16] of IWM Freiburg in Germany is part of the VERB 7 computer software and contains yield load solutions mainly for standard geometries. (i) Similarly, the German FKM Compendium [5.17] contains yield load solutions for standard geometries.
5.2.5. Types of Analytical Yield Load Solutions for Homogenous Components 5.2.5.1. Yield Load Solutions for Specific Geometries and Specific Loading As in the case of the K-factor solutions, specific geometries are flat and curved plates, pipes, pressure vessels, nozzles etc. Specific loading may be tensile forces, bending moments, internal pressure or others. Example 5.7: A local and a global yield solution for a tension loaded flat plate with a semi-elliptical surface crack, such as shown in Fig. 5.9, are provided by Eqns (5.57 – 5.60) and (5.62 – 5.63). A finite element based local yield load solution of this configuration, which has been adopted by many flaw assessment procedures [5.43, 5.45, 5.46, 5.48],
The Model Parameters
117
2c a
2W
t
Figure 5.9: Example 5.7: Geometry of the tension loaded plate with semi-elliptical surface crack.
is provided in [5.33]. Unfortunately, each of the referenced documents uses its own nomenclature. In terms of the yield load FY , it can be written as FY = 2W · t · Y · 1 − with =
⎧ ac ⎪ ⎨ t c + t
for W ≥ t + c
⎪ ⎩a · c t W
for W ≤ t + c
(5.57)
(5.58)
Replacing the load F by the membrane stress m = F/2 · W · t, Eqn (5.57) can be rewritten in terms of the reference stress ref ref =
1 1 − m
(5.59)
or in terms of the Lr ratio Lr =
1 m 1 − Y
(5.60)
The solution is applicable up to an a/t ratio of 0.8. The plate should be sufficiently large, compared to the crack length 2c, to exclude edge effects. Note that Eqn (5.60) was replaced by Lr =
1 − 157 m 1 − 2 Y
(5.61)
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Fitness-for-Service Fracture Assessment of Structures Containing Cracks
in [5.48]. A global yield load solution is provided in [5.34] FY = 2W · t · Y ·
d
1 + 2 + d1
for a/t ≤ 1
(5.62)
with = ·
= a/ t
= c/W
and d1 = 1 − 2 + 2 −
(5.63)
Again, equations for ref and Lr can be obtained by simple transformation according to Eqn (5.55). The yield loads of a plate of width 2W = 200 mm and thickness t = 20 mm for different crack depths, a, and crack geometries, a/c, are shown in Table 5.12 and Fig. 5.10 for Y = 300 MPa. Example 5.8: A thick-walled hollow cylinder with an outer radius R0 = 110 mm, an inner radius Ri = 55 mm and a wall thickness t = 55 mm, and with internal semi-elliptical surface cracks (crack geometry: a/c = 02 and 1.0, crack depth: a = 0 to 55 mm) is subjected to internal pressure (Fig. 5.11). The yield strength is assumed as Y = 500 MPa. Local yield solutions for this geometry are provided by Eqns (5.64 – 5.76). A local yield solution has been provided in [5.44] and also in [5.10, 5.43, 5.45]. Without considering crack face pressure, the limit pressure is given by 1 + 05 t R 1 s 1 − a/ t · ln pY = c + s1 1 − a/ t 1 1 − 05 t R 1 + 05 t R · Y + c · ln (5.64) 1 − 05 t R + a R with R being the mean radius R = 05 R0 + Ri and s1 being a R a a·c − 1 − 05 t R · Mg · ln 1 + s1 = 1 − 05 t R R R
(5.65)
The Model Parameters
119
Table 5.12: Example 5.7: Local and global yield loads of a tension loaded plate with a semi-elliptical surface crack (Fig. 5.9; 2W = 200 mm; t = 20 mm) as a function of crack depth, a, and crack geometry, a/c a in mm
a/c
0 1 2 3 4 5 6 7 8 9 10 11 12
3/4 3/4 3/4 3/4 3/4 3/4 3/4 3/4 3/4 3/4 3/4 3/4 3/4
Local yield Global yield load (Eqn 5.57) load (Eqn 5.62) in kN in kN → 12000 11984 11939 11867 11769 11649 11509 11350 11177 10990 10793 10588 10378
12000 11962 11859 11700 11495 11250 10971 10664 10330 9975 9600 9208 8800
a/c
Local yield Global yield load (Eqn 5.57) load (Eqn 5.62) in kN in kN
1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2
12000 11977 11909 11800 11653 11473 11262 11024 10762 10481 10184 9876 9560
12000 11945 11800 11585 11314 11000 10650 10271 9867 9442 9000 8543 8073
1200 Crack geometry: a/c = 0.75
Yield load FY in kN
1100
Crack geometry: a/c = 0.5 Plate geometry: 2W = 200 mm t = 20 mm Yield strength: σY = 300 MPa
1000
2c a
900
2W 800
0
2
Global yield load Local yield load
t 4
6
8
Crack depth a in mm
10
12 0
2
4
6
8
10
12
Crack depth a in mm
Figure 5.10: Example 5.7: Local and global yield loads of a tension loaded plate with semi-elliptical surface crack (Fig. 5.7; 2W = 200 mm; t = 20 mm) as a function of crack depth, a, and crack geometry, a/c.
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Fitness-for-Service Fracture Assessment of Structures Containing Cracks
a
2c
t Ri
Ro
Figure 5.11: Example 5.8: Geometry of the hollow cylinder with an internal axial surface defect subjected to internal pressure.
Taking into account crack face pressure, the limit pressure is given by 1 + 05 t R 1 s 1 − a/ t · ln pY = c + s2 1 − a/ t 2 1 − 05 t R 1 + 05 t R 1 − 05 t R · ln · Y + c· 1 − 05 t R + a R 1 − 05 t R + a R (5.66) with
! 1 + 05 t R 1 − 05 t R · Mg · ln s2 = R 1 − 05 t R ! 1 + 05 t R 1 − 05 t R a · ln − (5.67) − 1 − 05 t R + a R 1 − 05 t R + a R R a·c
and
" # # Mg = $1 +
a R
a/ c2 · 1 − 05 t R
with = 105
(5.68)
In a slightly modified R6 nomenclature, Eqns (5.64) to (5.68) are written as D R R Ro · Y (5.69) pY = s1 · ln o + c ∗i · ln Ri R1 Ri + a s1 + c
The Model Parameters
121
with s1 =
Mg · Ri ·
a · c · 1 − a/ t ∗ ln Ro Ri − Ri R1 · ln Ro Ri + a − a
(5.70)
The parameter D is defined by the yield condition 1√ for Tresca D= 2/ 3 for von Mises
(5.71)
(the compendia solutions of SINTAP/FITNET and R6 are given for the yield criterion of Tresca) and
R1∗
=
Ri Ri + a
excluding crack face pressure including crack face pressure
(5.72)
Note, that the SINTAP/FITNET solutions (Eqns 5.64 and 5.66) are based on the revised Folias (bulging) factor with = 105 whereas R6 [5.45] uses a value of = 161. % c2 Mg = 1 + 161 (5.73) Ri a Recently, based on finite element results, an improved version of Eqn (5.70) was proposed in [5.51] (see also [5.38]). & ' ⎧ R Ro ⎪ D · ln o · Y if pI ≥ D · ln · Y ⎨ Ri Ri pY = (5.74) & ' & ' & ' & ' ⎪ Ro ⎩ D · Y s1 · ln Ro + c Ri + a · ln Ro · Y if pI < D · ln s1 + c
with &
Ri pI = D · R1 replacing s1 by s1
Ri
'
R1
Ri + a
Ri
& ' & ' & ' 1 Ri + a Ri + a Ro ln + ln · Y Mg Ri Ri Ri + a
'& ' & ' Ri Ri + a Ro p − D · Y · ln a I R1 Ri Ri + a & ' s1 = c 1 − Ro t − pI D · Y · ln Ri
(5.75)
&
(5.76)
122
and
Fitness-for-Service Fracture Assessment of Structures Containing Cracks
R1∗
by R1 ,
R1 =
⎧ ⎨ Ri
excluding crack face pressure
⎩ R + a including crack face pressure i 2 In addition, Eqn (5.73) is changed to % c2 Mg = 1 + 125 a · Ri + a
.
(5.77)
The finite element analyses used for deriving Eqn (5.74) cover Ro /Ri ratios up to 2. The resulting local yield loads according to Eqn (5.74) are reproduced as a function of crack depth in Table 5.13 and Fig. 5.12. Example 5.9: Global yield loads have to be determined for the thick-walled hollow cylinder of Example 5.8. Solutions are provided by Eqns (5.78–5.81). The global yield solution used in SINTAP/FITNET has been provided in [5.44]. Without considering crack-face pressure the limit pressure is given by a R 1 + 05 t R · Y + ln (5.78) pY = 1 − 05 t R · Mg 1 − 05 t R + a R with Mg according to Eqn (5.68). Taking into account crack-face pressure, it is determined by a R 1 − 05 t R + pY = 1 − 05 t R · Mg 1 − 05 t R + a R 1 + 05 t R · Y × ln (5.79) 1 − 05 t R + a R In a slightly modified R6 nomenclature, the above equations are written as a Ri Ro · Y + ∗ · ln (5.80) pY = D · Ri · Mg R1 Ri + a with D being given by Eqn (5.71), Mg by Eqn (5.73) and R1∗ by Eqn (5.72). Again, the R6 solution is based on a Folias factor with = 161, Eqn (5.73), whereas FITNET/SINTAP uses the revised value of = 105.
The Model Parameters
123
Table 5.13: Examples 5.8 and 5.9: Local and global yield loads of the thick-walled hollow cylinder of Fig. 5.11. The results were obtained by Eqns (5.74) and (5.81) with and without considering crack face pressure Crack depth a in mm
Crack geometry a/c
Local yield pressure in MPa
Global yield pressure in MPa
Excl. crack face pressure
Incl. crack face pressure
Excl. crack face pressure
Incl. crack face pressure
5 10 15 20 25 30 35 40 45 50 55
02 02 02 02 02 02 02 02 02 02 02
3466 3453 3272 3024 2718 2364 1966 1527 1051 542 0
3376 3137 2844 2522 2180 1826 1465 1100 733 366 0
3466 3466 3352 3188 2991 2764 2510 2231 1928 1602 1256
3386 3183 2946 2690 2424 2152 1877 1600 1323 1047 772
5 10 15 20 25 30 35 40 45 50 55
10 10 10 10 10 10 10 10 10 10 10
3466 3466 3466 3466 3466 3466 3466 3466 2625 1457 0
3466 3466 3466 3287 3013 2661 2239 1755 1216 629 0
3466 3466 3466 3466 3466 3466 3466 3466 3466 3351 3124
3466 3466 3466 3397 3254 3084 2894 2689 2473 2249 2018
An improved version of Eqn (5.79) has recently been proposed in [5.51] (see also [5.38])
& ' & ' & ' Ro Ri 1 Ri +a Ri +a Ro pY = min D·ln
D· ·ln + ·ln Ri R1 Mg Ri Ri Ri +a
1 · −2R1 +a+ 4R1 ·R1 +a+2a2 ·Y (5.81) + 2R1
124
Fitness-for-Service Fracture Assessment of Structures Containing Cracks
400
Yield pressure pY in MPa
(a) Global yield load
(b) Local yield load excluding crack face pressure
300
with crack face pressure
200
with crack face pressure
a 2c 100
t Ri Ro
400
Crack geometry: a/c = 0.2
Crack geometry: a/c = 0.2
0
Yield pressure pY in MPa
excluding crack face pressure
(c) Global yield load
300
(d) Local yield load
with crack face pressure
200
with crack face pressure
excluding crack face pressure
excluding crack face pressure 100
Crack geometry: a/c = 1.0
Crack geometry: a/c = 1.0 0
0
10
20
30
40
50
0
10
Crack depth in mm 0
0.2
0.4
0.6
a/t
20
30
40
50
Crack depth in mm 0.8
1 0
0.2
0.4
0.6
0.8
1
a/t
Figure 5.12: Examples 5.8 and 5.9: Local and global yield loads of the thick-walled hollow cylinder of Fig. 5.11. The results were obtained by Eqs. (5.74) and (5.81) with and without considering crack face pressure.
In Eqn (5.80), the coefficient R1∗ of Eqn (5.79) is replaced by R1 according to Eqn (5.76), and Mg is replaced by Mg according to Eqn (5.77). The finite element analyses used for deriving Eqn (5.81) are valid for Ro /Ri ratios up to 2. The resulting global yield loads according to (Eqn 5.80) are reproduced as a function of crack depth in Table 5.13 and Fig. 5.12.
The Model Parameters
125
5.2.5.2. Solutions Based on Linearized Elastic Stress Profiles In cases where no specific yield load solutions are available, the so-called “general plate model” can be used to obtain estimates based on the elastically calculated membrane and bending stresses of the defect-free cross sections such as described in Section 4.1.4. From an application point of view, this method is comparable to the K-factor determination by weight function solutions. The membrane and bending stresses are determined for the components to be assessed, whereas the yield load is obtained for a substitute geometry; in the present case, the plate. Plate solutions are generally presented for either plane stress or plane strain conditions. Section 5.2.3 covers the problem of conservatism for this condition. Special caution is advisable in using plate solutions based on restrained bending – in such cases it should be proved that the moments can be redistributed away from the section containing the crack without causing an even more onerous condition in other parts of the structure. Example 5.10: The flat plate with a semi-elliptical surface crack of Fig. 5.7 is subjected to a combined membrane and bending load. The local and global yield loads are provided by Eqns (5.82 – 5.88) and (5.89 – 5.91). A local yield load solution is provided in [5.33] FY = 2W · t · Y ·
1 − 2 ( 2 · g + 2 · g 2 + 1 − 2
for a/ t ≤ 06 (5.82)
with defined as in Eqn (5.58). The factor is the bending to membrane load ratio = Mb /F · t =
1 / 6 b m
(5.83)
and g = 1 − 20 · 3 · a/2c075
(5.84)
Equation (5.82) can be re-written in terms of ref : ref
( 2 2 2 = g · b + g · b + 9 · 1 − m 3 · 1 − 2 1
(5.85)
126
Fitness-for-Service Fracture Assessment of Structures Containing Cracks
In [5.47], Eqn (5.85) is used for loading cases with bending restraint, whereas a modified version ( 1 2 2 2 ref = b + 3 · m · + b + 3 · m · + 9 · 1 − m (5.86) 3 · 1 − 2 is applied to cases with negligible bending restraint (e.g., pin-jointed). Eqn (5.82) can also be re-written in terms of Lr 1 b b 2 2 2 Lr = g · + g · + 1 − m 3 3 1 − 2 Y
(5.87)
The solution is applicable up to an a/t ratio of 0.6. The plate should be large enough compared to the crack length 2c to exclude edge effects. Note that Eqn (5.87) is replaced by 2 1 1 − 158 · b + 1 − 316 · b + 1 − 314 m2 (5.88) Lr = 3 3 1 − 2 Y in [5.48]. An extension of the global yield load of Eqn (5.62) to combined loading is provided in [5.34] and modified in [5.35] FY = 2W ·t ·Y
×
⎧ d ⎪ ⎪ ( 1 ⎪ ⎪ ⎨ 2+ + 2+2 +d
for ≤ 0 1
⎪ ⎪ ⎪ ⎪ ⎩
2+ ·1−/−+
(
d2
2 2+ ·1− − +d2 ·/−
for ≥ 0
(5.89) with = ·
= a/t
= c/W
d1 as in Eqn (5.63), as in Eqn (5.83),
and
d2 = 1 − · 2 − · 1 − / − + 2 1 −
(5.90)
( 0 = − − 05 + − 052 + 1 − 05
(5.91)
The plate geometry of Example 5.10 is identical to that of Example 5.7 (2W = 200 mm and t = 20 mm). The results of the analyses are summarized in Table 5.14 and Fig. 5.13.
The Model Parameters
127
Table 5.14: Example 5.10: Yield tensile force versus yield bending moment results for the plate of Fig. 5.7, but subjected to combined membrane and bending loads a, mm a/c 7
7
3/4
1/2
Global solution (Eqn 5.89) Local solution (Eqn 5.82) (Eqn 5.83) / (tens) / (bend) / (tens) / (bend) mY mY bY bY mY mY bY bY 00010 00020 00040 00080 00160
09980 09960 09920 09841 09684
00040 00080 00158 00314 00619
09980 09959 09919 09838 09678
00041 00081 00162 00322 00633
00320 00640 01280 02560 05120
09379 08799 07758 06109 04070
01199 02249 03966 06246 08321
09367 08777 07720 06054 04010
01225 02296 04040 06335 08392
10240 20480 40960 81920 163840 00010 00020 00040 00080 00160 00320 00640 01280 02560 05120 10240 20480 40960 81920 163840
02311 01204 00609 00305 00153 09980 09960 09920 09841 09684 09378 08797 07755 06107 04068 02311 01204 00609 00306 00153
09449 09845 09958 09988 09996 00040 00079 00158 00314 00618 01197 02247 03961 06238 08311 09441 09840 09955 09987 09996
02266 01178 00595 00298 00149 09979 09959 09918 09836 09675 09361 08766 07701 06027 03982 02247 01167 00589 00295 00148
09486 09861 09965 09991 09998 00041 00082 00164 00325 00639 01236 02316 04069 06368 08415 09495 09864 09965 09991 09998
128
Fitness-for-Service Fracture Assessment of Structures Containing Cracks
1.0
Crack geometry: a/c = 0.5 and a/c = 0.75
σmY/smY (tension only)
0.8
Crack depth: a = 7 mm Local and global limits
0.6
0.4
2c a 0.2
2W
t
Plate geometry: 2W = 200 mm t = 20 mm Yield strength: σY = 300 MPa
0 0
0.2
0.4
0.6
0.8
1.0
σbY/sbY (bending only)
Figure 5.13: Example 5.10: Yield tensile force versus yield bending moment curve for the plate of Fig. 5.7 but subjected to combined membrane and bending loading.
5.2.5.3. Individual Determination of Yield Loads by Finite Element Analysis or Model Tests In cases where no analytical yield load solution is available, an individual determination using numerical approaches such as the finite element method is an alternative. No comment will be given here about general aspects of the finite element method as such. However, an important question, which has briefly to be addressed, is how to define the limit load. The cracked structure may be analysed using a small-displacement finite element model assuming an elastic–perfectly plastic material. If the analysis is performed for a stepwise load increase, the maximum load attained defines the yield load/yield strength ratio FY /Y . Another possibility is the application of a non-linear analysis to a very low work hardening material law. In that case the yield load refers to that load above which the resulting load-deformation characteristics shows a significantly reduced slope. Note, however, that while these approaches are applicable to global analyses of through cracks, they do not allow a distinction between local and global yield loads in the case of partially penetrating cracks. Unfortunately, no unique philosophy exists on how to define local yield loads in a finite element analysis.
The Model Parameters
129
w
w
a 2c
Figure 5.14: Piecewise partition in sections of different continuous stress fields at plastic limit [5.51].
A frequently used definition is that a certain region at the crack plane needs to have become plastic. This is illustrated in Fig. 5.14 as a hatched area, the size of which will vary from section to section. The quantity s1 in Eqns (5.74) and (5.75) is a local yield load correction factor related to w by s1 = w · 1 − a/ t
(5.92)
[5.51]. Another philosophy is followed in [5.36] in which the authors define the local yield load of a surface crack “as the load needed to cause local crackligament yielding somewhere along the crack front”. It was already mentioned in Section 5.2.1 that two of the present authors defined the yield load of a curved and stiffened thin-walled plate with a through crack by the Jep /Je or ep /e ratio at Lr = 1, according to Eqn (6.25), this being 2.2118. Both the elastic-plastic J or CTOD (Jep , e ) and its elastic counterpart (Je , e ) were obtained using finite elements. This approach does not follow either a local or a global philosophy which is of advantage if the ligament length ahead of a through crack is so large that the definition of a global yield load becomes meaningless. On the other hand it is probably not suited to the determination of yield loads for partial penetrating cracks. A final potential method of yield load determination is the utilization of tests on scaled models of the component under consideration. It is advantageous to use materials that show almost elastic-perfectly plastic deformation behaviour.
5.2.6. Equivalent Yield Load Solutions for Strength Mismatch Components Components that consist of material sections of different strength (e.g., bi-materials or weldments) are designated as strength mismatch components.
130
Fitness-for-Service Fracture Assessment of Structures Containing Cracks
Within SINTAP/FITNET, strength mismatch has to be taken into account when the yield strengths of the materials involved differ by more than 10%. No detailed discussion on the mismatch effect will be performed here, this will be provided in Section 6.11. Basically it is dependent on parameters such as the yield strength mismatch factor M M = YW /YB
(5.93)
with YW being the yield strength of the weld metal and YB that of the base material, the strain hardening coefficient of the two materials, the geometry and dimensions of the component and of the weldment, the crack dimensions, geometry and location with respect to the weld. All of these factors affect the patterns of deformation that develop under applied loads, as illustrated in Fig. 5.15 [5.50]. The mismatch deformation patterns affect the overall deformation behaviour of the component which is different to that of the individual parts, the base material and the weld metal. Within SINTAP/FITNET this effect is taken into account by replacing the base material yield load, FYB , or the all weld metal yield load, FYW , by an equivalent mismatch yield load in the following, designated as FYM . A compendium of FYM solutions was first provided in [5.50] and then used in other documents such as [5.10, 5.52] and the FITNET compendium [5.43].
(a) Base plate
Undermatching (b) Base plate
Weld Weld Crack
(c)
Crack Overmatching (d)
Base plate
Base plate Weld
Weld
Crack
Crack
Figure 5.15: Classification of plasticity deformation patterns for strength mismatched plates [5.50]; (a) Undermatching, deformation confined to weld metal; (b) Undermatching, deformation penetrating to the base plate; (c) Overmatching, deformation penetrating to the base plate; (d) Overmatching, base plate deformation.
The Model Parameters
131
Example 5.11: Four welded plates of the dimensions 2W = 200 mm, B = 10 mm and 2H = 10 mm are subjected to tension loading as illustrated in Fig. 5.16. The yield strengths of the different materials are: material A Y = 150 MPa, material B Y = 300 MPa, material C Y = 450 MPa. The yield strength mismatch ratios are – (a) and (c) M = 05, (b) and (d) M = 15. The equivalent yield load FYM for plane stress conditions originally provided in [5.50] in SINTAP/FITNET nomenclature is given by Case (a) Crack in the centre line of the weld, undermatching (M < 1) FYM = YB ⎧ ⎪ ⎧ ⎪ √ M √ √ ⎫ ⎨ ⎨ M · 2/ 3 − 2 − 3 / 3 · 143/ ⎬ × min ⎪ ⎪ ⎩ ⎭ ⎩ 1 − 1 − M · 143/
for 0 ≤ ≤ 143 for > 143 (5.94)
with = W − a/H
(5.95)
FYB = 2 · B · W − a · YB
(5.96)
and
Case (b) Crack in the centre line of the weld, overmatching (M > 1) ⎧ W/W − a ⎪ ⎪ ⎨⎧ ⎨M FYM = FYB · min 24 · M − 1 1 M + 24 ⎪ ⎪ · · ⎩⎩ 25 25
for all for ≤ 1
(5.97)
for > 1
with 1 = 1 + 043 · exp −5 · M − 1 · exp − M − 1/5 and and FYB according to Eqns (5.95) and (5.96).
(5.98)
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Fitness-for-Service Fracture Assessment of Structures Containing Cracks
(a)
(b)
Material B Weld
Material B Weld
Material A 2a
Material C 2a
2H
2H
Material B
Material B
2W
2W B
B (c)
(d)
Material B Weld
Material B Weld
Material A 2a
Material C 2a
2H
2H
Material B 2W
Material B 2W
B
B
Figure 5.16: Example 5.11: Geometry of the welded M(T) plates Cases (a) and (c) are undermatched whereas cases (b) and (d) are overmatched structures.
The Model Parameters
133
Plane Strain
2a Overmatching
2.0
FYM /FYB
2W
1.5
2H
M = 1.0; 1.2; 1.4; 1.6; 1.8; 2.0
1.0
Undermatching 0.5
M = 1.0; 0.8; 0.6; 0.4; 0.2 0
0.8 1
2
4
6
8 10
20 24
(W-a)/H
Figure 5.17: Example 5.11: Strength mismatch correction factor for a tension loaded M(T) plate with a crack in the centre line of the weld (a/W = 05). Table 5.15: Example 5.11: Equivalent strength mismatch yield loads, FYM , for mismatch factors of M = 05 and M = 15 and cracks in the centre line of the crack and at the fusion line. In addition, conservative lower bound values are provided based on the lower of the yield strengths of the base material and weld metal a, mm
0 10 20 30 40 50 60 70 80 90
a w
0 01 02 03 04 05 06 07 08 09
Crack in the centre line of the weld
Crack at the fusion line
Conservative lower bound
M = 05
M = 15
M = 05
M = 15
Y (material A)
Y (material B)
34309 30845 27381 23917 20453 16989 13525 10060 6596 3132
60000 56429 50309 44189 38069 31949 25829 19709 13589 7469
32850 29565 26280 22995 19710 16425 13140 9855 6570 3285
60000 59080 52516 45951 39387 32822 26258 19693 13129 6564
30000 27000 24000 21000 18000 15000 12000 9000 6000 3000
60000 54000 48000 42000 36000 30000 24000 18000 12000 6000
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Fitness-for-Service Fracture Assessment of Structures Containing Cracks
700
FYM, crack location: fusion line FYM, crack location: weld centre Lower bound value: FY*
Equivalent yield load FYM in kN
600
Over matching (M = 1.5)
500
FY* = FY based on σYB or σYW (whichever is lower)
400
Under matching (M = 0.5)
300
200
100
0 0
20
40
60
80
100
0.8
1
Crack length a in mm 0
0.2
0.4
0.6
a/t
Figure 5.18: Example 5.11: Equivalent strength mismatch yield loads, FYM , for mismatch factors of M = 05 and M = 15 and cracks in the centre line of the weld and at the fusion line. In addition, conservative lower bound values are provided based on the lower of the yield strengths of the base material and weld metal.
Case (c) Crack at the interface between base material and weld metal, undermatching (M < 1) FYM = FYB ·M ·1095 − 0095 · exp − 1 − M/0108 M
for all (5.99)
with and FYB according to Eqns (5.95) and (5.96). Case (d) Crack at the interface between base material and weld metal, overmatching (M > 1) W/W − a for all FYM = FYB · min 1095 − 0095 · exp − M − 1/0108 for all (5.100) with and FYB according to Eqns (5.95) and (5.96).
The Model Parameters
135
An example for the correction function FYM /FYB = f[a/W, M, (W−a)/a] is graphically illustrated in Fig. 5.17. The results of the case studies (a) to (d) are summarised in Table 5.15 and shown in Fig. 5.18. In addition to the equivalent mismatch yield loads FYM , homogeneous yield loads based on the lower of the yield strengths of the base material and weld metal are introduced, these could be used as conservative lower-bound solutions if no mismatch corrected solutions are available.
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Chapter 6
Structural Assessment 6.1. Acceptable or Critical Conditions of a Component The failure of a component may be due to different causes such as fracture, corrosion damage, buckling, erosion, cavitation, etc. As already mentioned in Section 3.1, of this range only fracture is considered by the SINTAP/FITNET method. The fracture mechanisms covered are cleavage and micro-ductile, the failure modes include stable and unstable fracture as well as plastic collapse. A component can be assessed for the following three cases (see Section 3.2): (1) Assessment of a postulated or detected crack for a given load with respect to the criticality of the component (2) Determination of the critical or acceptable crack size for a given load and a given material (3) Specification of the minimum toughness of the material to ensure that the component is not at risk for a given loading and crack size The steps of SINTAP/FITNET analysis for these cases are illustrated in Fig. 6.1 and will be explained in more detail within Section 6.5. The examples of this section will, however, be restricted to primary loading and homogeneous components. Analysis including secondary loading such as residual stresses will be introduced in Section 6.6, constraint issues in Section 6.7, mixed mode loading in Sections 6.8 and 6.10, thin wall structures in Section 6.10, strength mismatch in Section 6.11, misalignment in Section 6.12 and reliability issues in Section 6.13. The critical or end-of-life state of a component can either refer to • • • •
stable crack initiation or unstable crack growth subsequent to very limited stable crack extension, or unstable crack growth subsequent to substantial stable crack extension, or plastic collapse.
Whilst the assessment for stable crack initiation, unstable crack growth after small stable crack extension and plastic collapse are performed within one failure
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Fitness-for-Service Fracture Assessment of Structures Containing Cracks
Component geometry
Loading
Single loads: tension, bending, internal pressure etc.
– plates – cylinders – weldments – tubular joints – substitute geometries
Primary loading
– etc.
Secondary loading
Stress profiles across the uncracked sections
E.g. residual stresses: Measurements, compendium, conservative assumptions
Crack type: Through crack, surface crack, corner crack, embedded crack
Crack dimensions
Crack location and orientation (with respect to the principal stress planes)
Variation of crack size
Variation of primary loading
Determination of yield load FY (or σref)
Determination of ligament yielding f(Lr)
For secondary stresses: Determination of V
Determination of the K-factor
Variation of fracture toughness
(a)
FAD analysis
CDF analysis
(b) (c)
no
yes Critical state ?
(a) Critical crack size (b) Critical load (c) Required toughness
Figure 6.1: Flow chart of SINTAP/FITNET analysis. (a) Determination of critical crack size; (b) Determination of critical load; (c) Determination of minimum required fracture toughness.
Structural Assessment
139
assessment diagram (FAD) or crack driving force (CDF) analysis route, the assessment for unstable crack growth subsequent to substantial stable crack extension requires a different kind of analysis.
6.2. Assessment Based on the FAD Philosophy 6.2.1. The FAD The FAD is given by a function Kr = fLr
(6.1)
with Kr being the crack driving force in terms of the stress intensity factor K normalised by the fracture toughness of the material Kmat Kr = K Kmat (6.2) and f(Lr ) is a function of the ligament yielding parameter Lr Lr = F FY = ref Y
(6.3)
The latter terms have been explained in more detail in Section 5.2. What is important at this stage is that the f(Lr ) function is used independently of the individual component geometry and loading. It depends, however, on the deformation behaviour of the material and is different for the various assessment options of SINTAP/FITNET that will be introduced in Section 6.4.
6.2.2. The Assessment Point (or Path) The assessment point (Kr , Lr ) is different to the FAD line in that it is a function of the component geometry, the applied loads, crack size and shape. Whether a component is safe or potentially unsafe depends on the relative location of the assessment point with respect to the FAD. The information necessary for determining the assessment point comprises (1) The load either in terms of forces, moments, pressure etc. or stress distributions. This issue has been discussed in Section 4.1. (2) The stress intensity factor K for the given crack size. Its determination has been explained in Section 5.1. (3) The net section yield load FY or its equivalents in terms of the net section stress ref or the ligament yielding parameter Lr for the given crack size as outlined in Section 5.2.
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Fitness-for-Service Fracture Assessment of Structures Containing Cracks
(4) The toughness of the material in terms of Kmat , a general term that can have different meanings (see Section 4.4): (a) It can be identical to the plane strain fracture toughness KIc . (b) It can be identical to the resistance against stable ductile crack initiation, Ji , i , J02/BL , 02/BL etc. but expressed in terms of the K-factor. In cases where the toughness shows considerable scatter either a minimum (or maximum, depending on the application) toughness should be used as a “design value” or a complete statistical analysis should be performed for which guidance is given. If the toughness is originally available in terms of the J-integral or the crack tip opening displacement , KJ or K values can be formally determined by KJ = J · E
1 − 2 1/ 2
(6.4)
or K = m · · Y · E
1 − 2 1/ 2
(6.5)
respectively. In Eqn (6.5), m is a constraint factor which, for structural steels, is conservatively (25% percentile) estimated as m = 15 within SINTAP/FITNET. Note that this value is only appropriate for highly constrained standard specimens, that is, deeply notched bend specimens with almost square ligaments. It should not be applied for low strain hardening materials (strain hardening exponent N < 005) and for materials other than steel. If individual values for m are established experimentally, these should be used instead of 1.5. If the fracture resistance is given in terms of J-R or -R curves, the formal conversion is performed as KJ a = J a · E
1 − 2 1/ 2
(6.6)
or K a = m · a · Y · E respectively.
1 − 2 1/ 2
(6.7)
Structural Assessment
141
6.2.3. Types of FAD Analysis Three types of FAD analysis are illustrated in Figs 6.2 and 6.3. In the first example (Fig. 6.2a) the crack size and material toughness Kmat are assumed to be constant values whereby Kmat can be specified as a KIc value or the resistance against initiation of stable ductile crack extension in terms of K (Eqns 6.4 or 6.5). As the load increases the assessment point moves towards the FAD line. The failure condition is then given by the intercept point between the FAD curve and the path of the assessment point. For loads above the intercept, the component is designated as “potentially unsafe”, taking into account that the analysis gives a conservative estimate rather than exact information. (b)
(a) Potentially unsafe
Potentially unsafe 1 Fracture
Increasing load
Kr = K/Kmat
Kr = K/Kmat
1
Fracture
Increasing crack size
Safe
Safe
Lr = F/FY = σ ref/σ Y
1 Lrmax
Lr = F/FY = σ ref /σ Y
1 L max r
Figure 6.2: Types of FAD analysis – (a) Crack size and toughness are constant values, but the applied load is increasing; (b) Applied load and toughness are constant values, but the crack size is increasing. The Lr max limit assigns the plastic collapse limit.
Unstable crack initiation Load 3 Load 2 Load 1
Kr = K/Kmat
1
Load 3 > Load 2 > Load 1
Lr = F/FY = σ ref/σ Y
1 L max r
Figure 6.3: FAD analysis for determining the instability load of a component. For three constant applied loads the stable crack extension is simulated according to the R-curve of the material. The Lr max limit assigns the plastic collapse limit.
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Fitness-for-Service Fracture Assessment of Structures Containing Cracks
In the second example (Fig. 6.2b), both toughness and applied load are given as constant values. The crack size is increased in a stepwise manner. Again the assessment point moves towards the FAD line and intersects it at “failure”. In the third example (Fig. 6.3) various different constant applied loads are considered. For each of them an assessment path is determined taking into account stable ductile crack extension. Because the crack is growing according to the R-curve [KJ = f a], both Kr and Lr co-ordinates of the assessment point will be affected by the increasing crack size due to stable crack extension. Kr will be affected by means of an increasing K value for the increasing crack size and an increasing Kmat value according to the R-curve, and Lr by means of the decreasing yield load FY with increasing crack size. The load whose assessment point path is tangent to the FAD line refers to the instability load of the component. The point where the curve touches the FAD defines the associated stable crack extension.
6.2.4. Non-unique Solutions So-called non-unique solutions are obtained when an assessment path crosses the FAD line more than once (Fig. 6.4). This is possible for component geometries and applied load situations for which the stress intensity factor does not monotonically increase with increasing crack size, for mechanical properties varying through the section and for residual stress relaxation at larger ligament yielding [6.1]. Care has to be exercised for (1) In-thickness variations in stresses, for example, near to stress concentrations (2) In-thickness variations in material properties such as toughness, for example, in welded structures (3) Interaction effects between primary and secondary stresses, for example, in weldments (variation of the V factor with Lr , see Section 6.6) (4) Crack re-characterisation (Section 4.2.5) when both the original and the re-characterised crack geometry are plotted in one assessment path (5) Growth of elliptical or semi-elliptical cracks during which the more critical condition moves along the crack front, for example the highest stress might initially occur at the surface points but this could later change to the deepest point, or vice versa It is possible to miss the most critical solution to a defect assessment for one load and one crack size if multiple solutions exist. This can, however, simply be avoided if the complete assessment path is determined, starting from an assumed or measured defect size, if appropriate. In this case the first solution is the right one.
Structural Assessment
Solution 2
Potentially unsafe
Solution 3
Kr = K/Kmat
1
Solution 1
143
Increasing crack size
Safe
Lr = F/FY = σ ref /σ Y
1 Lr max
Figure 6.4: FAD approach: Non-unique critical crack size conditions.
6.3. Assessment Based on the CDF Philosophy 6.3.1. The CDF Functions The CDF functions are given by J = Je · f Lr −2
(6.8)
Je = K 2 E
(6.9)
with
in terms of the J-integral and by = e · f Lr −2
(6.10)
e = K 2 E · Y
(6.11)
with
in terms of the crack tip opening displacement, . Note that, for the FAD and CDF approaches, the f(Lr ) functions of Eqns (6.1), (6.8) and (6.10) are identical. However, in contrast to the FAD line, which is fully defined by f(Lr ), the CDF functions are defined for specific component and loading geometries. The information necessary for determining the CDF functions are (1) The load, as discussed in Section 4.1. (2) The stress intensity factor K as a function of crack size, as discussed in Section 5.1.
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Fitness-for-Service Fracture Assessment of Structures Containing Cracks
(3) The yield load FY or its equivalents in terms of net section stress ref or the ligament yielding parameter Lr similarly as a function of crack size as outlined in Section 5.2. The toughness of the material is not included in the CDF functions but is compared with the CDF in a separate step.
6.3.2. The Determination of the Critical Condition
J or δ
The fracture toughness is given as Jmat or mat which is a general term that can be specified as the resistance against stable ductile crack initiation, Ji , i , J02/BL , 02/BL , etc., or as R curves J- a or - a. If the toughness of the material is given as a single value the critical condition is reached when the crack driving force is identical to or exceeds Jmat or mat as illustrated in Fig. 6.5. No conversion of the toughness values to Kmat is necessary because the crack driving force is expressed in terms of J or . When the toughness is provided as an R-curve, the critical condition for unstable crack growth which terminates previous stable crack extension can be determined as illustrated in Fig. 6.6. In Fig. 6.6a, functions of J or versus crack depth are plotted for different constant applied loads. For each of these, J or is determined by Eqn (6.8) or (6.10) for stepwise increased crack depths. The diagram is then completed by the J- a or - a curve with the origin at a given original crack depth in the component. The load whose CDF curve is tangent to the R-curve is the instability load of the structure. Note that the crack driving force J or at instability is always geometry dependent and does not refer to J or at the maximum load in a test specimen.
Fracture
Jmat or δ mat Increasing load or crack size
Lr = F/FY = σref/σ Y
Lrmax
Figure 6.5: Determination of the critical condition of a component following the CDF philosophy when the toughness is given by a single value such as Ji , i , J02/BL , 02/BL etc. The Lr max limit assigns the plastic collapse limit.
Structural Assessment
145
(b)
(a) Load 3
J or δ
Maximum load
Load
Load 2 (Maximum load) Load 1 R-curve (material)
CDF curves
Start of unstable crack growth (instability point) Load 3 > Load 2 > Load 1 ao Δa
Stable crack extension Δa
Crack depth a
c
Δa
Figure 6.6: Types of CDF analysis for determining the instability load of a component – (a) CDF curves are determined for constant applied loads and compared with the R-curve of the material (ao marks the original crack size in the component); (b) The load versus stable crack extension curve is determined following the route in Fig. 6.8.
This type of diagram can be used in a simple way for assessing the effects of various parameters on the instability load as demonstrated in Fig. 6.7 for an improved R-curve behaviour in terms of a steeper slope (Fig. 6.7a) and for a smaller original crack depth in the component (Fig. 6.7b). The latter could, for example, be the benefit of an improved non-destructive inspection (NDI) strategy. In addition to the critical load, the crack size at instability and the (b)
(a)
Load 3
Load 3 Load 2
Load 2
Load 1
J or δ
J or δ
Load 1
Improved NDI (smaller initial crack size)
Improved R-curve behaviour
Load 3 > Load 2 > Load 1
Load 3 > Load 2 > Load 1 ao
Δa
Crack depth a
ao,1 ao,2
Crack depth a
Figure 6.7: The effect of various parameters on the instability load of the component– (a) Toughness of the material in terms of steepness of R-curve; (b) Original crack size ao in the component.
146
Fitness-for-Service Fracture Assessment of Structures Containing Cracks
corresponding amount of stable crack extension can be determined by this type of diagram. A second way to determine the maximum load that a component can sustain, that is, its instability load, is provided in Fig. 6.6b with the analysis steps illustrated in Fig. 6.8. In this case the complete load versus stable crack extension characteristics of the component are predicted.
Input data J or δ
2ao
R-curve Δa Stepwise increase of crack size a by discrete increments Δai
F=0 a = a + Δai
Stepwise increase of applied load F by discrete increments ΔFi
F = F + ΔFi
R-curve J or δ (ao + Δa) Increasing F
J or δ
ao
ao + Δa
Determination of J (a,F) or δ (a,F) by SINTAP/FITNET equations
? J(a,F) ≥ J(ao + Δa) R-curve ? δ(a,F) ≥ δ(ao + Δa)
no
R-curve
yes Load F
ao
Δa
Figure 6.8: Flow chart for determining the applied load versus stable crack extension characteristics in Fig. 6.6 (b).
Structural Assessment
147
6.4. The f(Lr ) Function According to the Different Analysis Levels 6.4.1. General Remarks The characteristic feature of the SINTAP/FITNET procedure of following a hierarchical structure of various assessment options with increased complexity and decreased conservatism has already been briefly addressed in Section 3.3. The options differ with respect to their f(Lr ) functions in Eqns (6.1), (6.8) and (6.10). These will be provided for Options 0, 1 and 3 in this section. The information for Option 2 will be given in Section 6.11 where the strength mismatch issue will be discussed in detail.
6.4.2. Option 0 (“Basic Option”) Since the only strength parameter which has to be available at Option 0 is the yield strength, Option 0 is defined only up to the yield load FY or, accordingly, up to Lr = 1. However, for materials showing continuous yielding behaviour, with no yield plateau, the tensile strength Rm has empirically been correlated to the proof strength Y = Rp02 (Fig. 6.9) by an upper bound correlation for the ratio Rp02 /Rm , leading to the lower bound estimate 25 Rm = Rp02 · 1 + 2 · 150 Rp02 Rp02 in MPa
(6.12)
1.0
Rp0.2 /Rm
0.9 0.8 0.7
Upper bound correlation
0.6 0.5 200
400
600
800
1000
Rp0.2 in MPa
Figure 6.9: Relationship between yield strength and yield to tensile strength ratio of materials showing continuous yielding (no yield plateau).
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Fitness-for-Service Fracture Assessment of Structures Containing Cracks
which allows the Lr range for this class of materials to be extended beyond Lr = 1. Taking into account the additional information, the f(Lr ) function of Option 0 is provided as (1) Option 0A: Materials expected to display a yield plateau (Y = ReL ): −1 2 f Lr = 1 + 05 · Lr2 /
for 0 ≤ Lr ≤ Lr max
(6.13)
with Lr max = 1
(6.14)
(2) Option 0B: Materials not expected to display a yield plateau (Y = Rp02 ): −1 2 fLr = 1 + 05 · Lr2 / · 03 + 07 · exp −06 · Lr6 for 0 ≤ Lr ≤ Lr max (6.15) with 25 Lr max = 1 + 150 Rp02 Rp02 in MPa
(6.16)
Eqn (6.16) is obtained by inserting Eqn (6.12) into the general expression for the plastic collapse limit Lr max Lr max = 05 · Y + Rm Y
(6.17)
in this special case.
6.4.3. Option 1 (“Standard Option”) As with Option 0, a distinction has to be made between materials with and without yield plateau. However, the reasons are different. If one assumes, as in the ETM method (Section 2.8), that the crack driving force versus Lr curve roughly follows the stress–strain curve of the material, it can be shown that the crack driving force is underestimated when the yield plateau is not taken into account (Fig. 6.10). In order to avoid this mistake a step function for Lr is introduced at Lr = 1 for materials displaying a yield plateau. If it is uncertain whether a material or a particular material condition shows discontinuous yielding or not (see also the guidance in Section 4.3.7), treating it as discontinuous provides conservative results.
Structural Assessment
149
Strain/yield strain
Stress-strain curve No yield plateau
Yield plateau
Crack driving force
Stress/σ Y FAD or CDF curve
Stress/σ Y
Yield plateau taken into account Yield plateau not taken into account
Lr = F/FY
Lr = F/FY
Figure 6.10: The effect of the step function (Eqn 6.19) on the crack driving force of materials with a yield plateau.
The f(Lr ) function of Option 1 is provided as (1) Option 1A: Materials expected to display a yield plateau (Y = ReL ): −1 2 for 0 ≤ Lr ≤ 1 fLr = 1 + 05 · Lr2 /
−1/ 2 1 for Lr = 1 fLr = + 2
(6.18) (6.19)
with
= 1 + E · ReL
(6.20)
and fLr = f Lr = 1 · LrN−1/ 2N
for 1 ≤ Lr ≤ Lr max
The Lüders’ strain is conservatively estimated by R
= 00375 1 − eL ReL in MPa 1000
(6.21)
(6.22)
150
Fitness-for-Service Fracture Assessment of Structures Containing Cracks
and the strain hardening coefficient N by N = 03 · 1 − ReL Rm
(6.23)
The plastic collapse limit Lr max is given as Lr max = 05 · ReL + Rm ReL
(6.24)
(2) Option 1B: Materials not expected to display a yield plateau (Y = Rp02 : −1 2 f Lr = 1 + 05 · Lr2 / · 03 + 07 · exp − · Lr6
for 0 ≤ Lr ≤ 1 (6.25)
with
= min
0001 E Rp02 06
f Lr = f Lr = 1 · LrN−1/ 2N
(6.26) for 1 ≤ Lr ≤ Lr max
(6.27)
No step function is introduced at Lr = 1 as in Option 1A. The strain hardening coefficient N is determined by N = 03 · 1 − Rp02 Rm
(6.28)
and the plastic collapse limit Lr max by Lr max = 05 · Rp02 + Rm ReL
(6.29)
The term in Eqn (6.25) is introduced in order to provide a conservative estimate of f(Lr ) in the strain range between (Rp02 /E) and (Rp02 /E + 002. This becomes necessary since the proof strength Rp02 refers to a significantly larger strain than ReL . It belongs to the lower tail of the strain hardening branch of the stress–strain curve rather than assigning a transition between elastic and plastic deformation. The task of is to avoid any effect of strain hardening at and below Lr = 1.
Structural Assessment
151
6.4.4. Option 3 (“Stress–strain Defined Option”) The Option 3, f(Lr ) function is based on the true stress–strain curve of the material as input information. It is given by
E · ref 1 Lr2 f Lr = + ref 2 E · ref ref
−1/ 2 for 0 ≤ Lr ≤ Lr max
(6.30)
with Lr max = 05 · Y + Rm Y
(6.31)
No distinction is made between materials with and without a yield plateau. The basic principle of the analysis is illustrated in Fig. 6.11. For a given applied load F the Lr ratio in the component is determined by a SINTAP/FITNET analysis. It is then used as the input parameter for determining a reference stress ref by ref = Lr · Y
(6.32)
Alternatively, ref can be available as the input for the given component geometry and loading (see Section 5.2).
Lr Component information
σref = Lr . σY True stress strain curve
εref
Figure 6.11: SINTAP/FITNET Option 3: Determination of (ref ,ref points on the true stress-strain curve dependent on the ligament yielding parameter Lr of the component.
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Fitness-for-Service Fracture Assessment of Structures Containing Cracks
(a) Ferritic steel 1.2
max
Lr
Option 0A
Option 3
0.8
1000
Stress in MPa
Function f(Lr)
(Option 0A)
Option 1A
1.0
0.6
0.4
0.2
max
true
Lr
800
(Options 1A & 3)
600
engineering
400
Option 3
200 0
0
4
12
8
16
Option 1A
Strain in % 0 0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
Ligament yielding Lr = F/FY = σref /σY (b) Austenitic steel 1.2
Option 3 Option 1B Option 0B
0.8
1000
Stress in MPa
Function f(Lr)
1.0
0.6
0.4
0.2
max
Lr
(Option 0B)
true
800
max
Lr
600 400
(Options 1B & 3)
engineering
200 0
0
15
30
45
Option 3
60
Option 1B
Strain in % 0 0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
Ligament yielding Lr = F/FY = σref /σY
Figure 6.12: f(Lr ) function for the various analysis options. (a) ferritic steel; (b) austenitic steel. The f(Lr ) function is used as an universal function independent of the specific component geometry and loading. It depends, however, on material. An exception is option 0 which provides a geometry independent lower bound function.
Structural Assessment
153
The reference stress then defines a corresponding reference strain ref on the true stress–strain curve. In principle, for a given applied load, only one pair (ref ref has to be determined to define the critical condition. It is, however, frequently advantageous to determine the CDF as a function of the applied load. In this case f(Lr ) will be determined point-wise following the true stress–strain curve. The application of Option 3 requires stress–strain curves of relatively high quality. In particular, the yield strength region should be adequately represented including data points at least at Lr = 07, 0.9, 0.98, 1.00, 1.02, 1.10 and 1.20. Note that Option 3 provides an additional module for describing strength mismatched components. This will be introduced in Section 6.11.4. A comparison between fLr functions of the various analysis options for a ferritic and an austenitic steel is shown in Fig. 6.12.
6.5. Examples for SINTAP/FITNET Analysis 6.5.1. Determination of the Critical Load Example 6.1: The tension loaded plate with a surface crack of Example 5.7 (Fig. 5.9) is assumed to be made of an aluminium alloy with a resistance against stable crack extension of Ji = 10 N/mm which might have been established fractographically. The plate dimensions are 2W = 200 mm and t = 10 mm; the cross section containing a semi-elliptical crack of depth a = 3 mm and geometry a/c = 1 is shown in Fig. 6.13. The engineering and true stress–strain curves are provided in Table 6.1 and Fig. 6.14; the tensile test data are: yield strength Y = 242 MPa, tensile strength Rm = 350 MPa, Young’s modulus E = 703 GPa
θ
a
t
2c 2W
Figure 6.13: Examples 6.1 to 6.4: The cross section of the tensile loaded plate containing a surface crack.
154
Fitness-for-Service Fracture Assessment of Structures Containing Cracks
Table 6.1: Example 6.1: Engineering and true stress-strain curves of the material. The data refer to an aluminium alloy. The true stress and strain are obtained by t = 1 + and t = ln1 + Engineering curve
True curve
Engineering curve
True curve
in MPa
in %
t in MPa
t in %
in MPa
in %
t in MPa
t in %
200 213 223 232 241 254 267 278 288 296
0.284 0.323 0.367 0.430 0.543 0.861 1.380 1.895 2.410 2.921
200.6 213.7 223.8 233.0 242.3 256.2 270.7 283.3 294.9 304.6
0.284 0.322 0.367 0.429 0.541 0.858 1.370 1.878 2.381 2.879
310 318 325 336 343 348 350
4.441 5.452 6.462 8.478 10.488 11.995 12.998
323.8 335.3 346.0 364.5 379.0 389.7 395.5
4.345 5.309 6.262 8.138 9.974 11.328 12.22
500
true curve
Stress in MPa
400
300
engineering curve
200
100
0
0
2
4
6
8
10
12
14
Strain in %
Figure 6.14: Examples 6.1–6.4: Engineering and true stress-strain curves of the material.
and Poisson’s ratio = 033. The material does not display a yield plateau. The information required is the critical load that the plate can sustain. For the given yield and ultimate tensile strength, the plastic collapse limit Lr max is determined by Eqns (6.29) and (6.31) as Lr max = 122 at Options 1B and 3. The Option 1B strain hardening exponent is N = 0.09. The corresponding f(Lr ) functions are determined by Eqns (6.25), (6.27) and (6.30) and summarised in Table 6.2.
Structural Assessment
155
Table 6.2: Example 6.1: fLr functions according to Options 1B and 3 ( = 029) Lr 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70
Option 1 fLr
Option 3 fLr
1000 0999 0998 0994 0990 0985 0978 0970 0961 0951 0940 0927 0912 0895 0875
1000 0999 0998 0994 0990 0985 0978 0971 0962 0953 0943 0932 0921 0909 0896
Lr 075 080 085 090 095 100 105 110 115 120 Lrmax : 1.22
Option 1 fLr
Option 3 fLr
0852 0826 0795 0759 0718 0672 0525 0414 0330 0266 0252
0883 0870 0851 0824 0787 0726 0631 0546 0478 0426 0415
Applying these f(Lr ) functions, both an FAD and a CDF analysis are carried out. The K-factor was determined by the following equations [6.2]: √ 1 m · a · Fs E k Fs = M1 + M2 · a/ t2 + M3 · a/ t4 · g · f · fW
113 − 009 · a/ c for a/ c ≤ 1 M1 = √ c/ a · 1 + 004 · c/ a for a/ c > 1 −054 + 089 02 + a/ c for a/ c ≤ 1 M2 = for a/ c > 1 02 · c/ a4 for 05 − 1 065 + a/ c + 14 · 1 − a/ c24 M3 = 4 −011 · c/ a for for 1 + 01 + 035 · a/ t2 · 1 − sin 2 g= 2 2 for 1 + 01 + 035 · c/ a · a/ t · 1 − sin KI =
(6.33) (6.34) (6.35)
(6.36) a/ c ≤ 1 a/ c > 1 a/ c ≤ 1 a/ c > 1
(6.37)
(6.38)
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Fitness-for-Service Fracture Assessment of Structures Containing Cracks
1 4 a/ c2 · cos2 + sin2 / f = 1 4 c/ a2 · sin2 + cos2 /
for a/ c ≤ 1
(6.39)
for a/ c > 1
and 1/ 2 ·c · a/ t fW = sec 2W
(6.40)
The elliptical integral is approached by ⎧ 1 2 ⎪ ⎨ 1 + 1464 · a/ c165 / for a/ c ≤ 1 E k = 1/ 2 ⎪ ⎩ 1 + 1464 · c/ a165 for a/ c > 1
(6.41)
In the present case the highest K factors were reached at the surface points of the crack, that is at = 0. A local yield load was determined by the approach introduced in Example 5.7 (Eqn 5.57) or, alternatively, by Eqns (5.59) and (5.60) for ref and Lr . Table 6.3 gives the ligament yielding parameter Lr , the corresponding Kr (FAD approach) and the Option 1B and 3 applied J values (CDF approach) as a function of a stepwise increased applied load F. Table 6.3: Example 6.1: Ligament yielding Lr , Kr and Option 1B and 3 applied J values as a function of a stepwise increased applied load F for a crack depth a = 3 mm and a crack geometry a/c = 1. Kr was based on the K values at the surface points of the crack ( = 0) F in kN 0 50 100 150 200 250 300 350 400 450 500 550
Lr
J in N/mm (Option 1B)
J in N/mm (Option 3)
0 0111 0222 0333 0444 0555 0666 0777 0888 0999 1110 1221
0 0044 0178 0412 0765 1267 1977 3024 4704 7748 27693 88289
0 0044 0178 0412 0763 1252 1908 2767 4016 6642 15285 31334
Kr 0 0066 0132 0198 0263 0329 0395 0461 0527 0593 0659 0725
Structural Assessment
(a)
157
(b) 100 550 1.0
Option 3
80
F in kN
Option 1B
550
Option 1B
J-integral in N/mm
0.8
Kr
450 0.6
500 350 400
250
0.4
300
150 0.2
50
2c
60
a
2W
40
t
F in kN
400 200 300 350 450 250 150
20
200
500
550 Option 3
500 Jmat = 10 N/mm
100 0
0 0
0.4
0.8
Lr
1.2
1.6
0
0
0.4 0 50 100
0.8
1.2
Lr
Figure 6.15: Example 6.1: Determination of the critical load for a crack depth a = 3 mm and a crack geometry a/c = 1. (a) FAD approach; (b) CDF approach.
The results are illustrated graphically in Fig. 6.15 for the FAD and CFD approach. The critical load is Fc = 4602 kN for Option 1B and Fc = 4745 kN for Option 3. Note that the a/c ratio was assumed to be constant in this example. In a real application, such a simplification would usually be inadmissible since the crack when extending, for example, by fatigue, would change its geometry.
6.5.2. Determination of the Critical Crack Size Example 6.2: The tension loaded plate with a surface crack of Example 6.1 is subjected to a membrane stress of 150 MPa (applied load F = 300 kN). The desired information is the critical crack depth a when the crack geometry is assumed to be a/c = 02. The K-factor and yield load solutions are identical to those applied in Example 6.1 above. Table 6.4 gives the ligament yielding value of Lr , the corresponding Kr (FAD approach) and the Option 1 and 3 applied J values (CDF approach) as a function of crack depth a. The results are illustrated graphically in Fig. 6.16 for the FAD and CFD approach. The critical crack depth is ac = 405 mm for Option 1B and ac = 430 mm for Option 3.
158
Fitness-for-Service Fracture Assessment of Structures Containing Cracks
Table 6.4: Example 6.2: Ligament yielding Lr , Kr and Option 1 and 3 applied J values as a function of crack depth a for membrane stress of 150 MPa (applied load F = 350 kN) and a crack geometry a/c = 02 a in mm 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0
Lr
J in N/mm (Option 1B)
J in N/mm (Option 3)
0641 0662 0689 0720 0756 0797 0845 0900 0964 1039 1127
1284 2034 2922 4012 5398 7219 9713 13320 18951 37604 104567
1248 1966 2798 3793 5006 6510 8491 11303 15736 27434 53776
(a)
Kr 0322 0402 0476 0549 0624 0703 0787 0876 0972 1074 1184
(b) 100 5.5
1.0 3.5
5.0 Crack depth a in mm
4.5 4.0
J-integral in N/mm
0.8
Option 3 Option 1B
3.5
Kr
0.6 3.0 2.5 2.0
0.4
1.5
3.0
60
Option 3 2.5 5.5
40 2.0
1.0
4.5
1.5
2c
0.2
Option 1B
80
5.0
20
a
1.0 Jmat = 10 N/mm
2W
0
0
4.0
t
0.4
0.8
Lr
1.2
1.6
0
0
0.4
0.8
1.2
Lr
Figure 6.16: Example 6.2: Determination of the critical crack depth for a applied load F = 150 kN and a crack geometry a/c = 02. (a) FAD approach; (b) CDF approach.
6.5.3. Determination of the Required Minimum Toughness Example 6.3: The tension loaded plate of Example 6.1 contains a surface crack of depth a = 3 mm and geometry a/c = 1. The question is to determine the minimum required toughness of the material such that the component is able to sustain a membrane stress of 250 MPa (applied load F = 500 kN).
Structural Assessment
159
Table 6.5: Example 6.3: Minimum required toughness for the tension loaded plate of Example 6.1 for a crack depth a = 3 mm, a geometry a/c = 1 and a membrane stress of 250 MPa (applied load F = 500 kN) Assessment Option
1B 3
Minimum required toughness Kmat (required) in MPa · m1/2
Jmat (required) in N/mm
45.5 34.5
26.2 15.1
mat (required) in mm 0.07 0.04
With K-factor and yield load solutions identical to those applied in Example 6.1 this is determined as summarised in Table 6.5. For the conversion between K, J and , Eqns (6.4) and (6.5) are used.
6.5.4. Determination of the Instability Load (R-Curve Analysis) Example 6.4: The tension loaded plate of Example 6.1 contains an original surface crack of depth a = 3 mm and geometry a/c = 1. The question is to determine the instability load when the R-curve of Table 6.6 and Fig. 6.17 is taken as fracture toughness characteristics. Only Option 1B will be applied. Again, the K-factor and yield load solutions of Example 6.1 are used. Table 6.7 summarises the assessment paths (Kr Lr ) for three different applied loads, Table 6.6: Example 6.4 (FAD approach): Crack resistance (R) curve of the material. For its use in the FAD approach the J- a curve is formally transformed to a KJ - a curve by 1/2 KJ = J · E/1 − 2
a in mm 0 005 012 03 05 07 10 15 20 25
J in N/mm 0 20 40 58 68 75 85 97 108 120
KJ in MPa · m1/2
a in mm
J in N/mm
KJ in MPa · m1/2
0 3972 5618 6764 7324 7692 8189 8748 9231 9730
30 35 40 50 60 70 80
130 142 152 172 192 210 230
10127 10584 10951 11649 12307 12871 13470
160
Fitness-for-Service Fracture Assessment of Structures Containing Cracks
J-integral in N/mm
250 200 150 100 50 Ji (fractography) ≈ 10 N/mm
0
0 1 2 3 4 5 6 7 8 Stable crack extension Δa in mm
Figure 6.17: Example 6.4: J-R-curve of the material. Table 6.7: Example 6.4: Assessment paths points (Lr ,Kr for three applied loads. The original crack depth was ao = 3 mm, the crack geometry was simplified taken as a/c = 1 Load = 470 kN
a in mm 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00
Load = 520 kN
Load = 570 kN
Lr
Kr
a in mm
Lr
Kr
a in mm
Lr
Kr
1.046 1.048 1.050 1.053 1.055 1.058 1.060 1.063 1.065 1.068 1.071 1.073 1.076 1.079 1.082 1.085 1.087 1.090 1.093 1.096
0.442 0.341 0.307 0.294 0.282 0.272 0.269 0.266 0.263 0.260 0.259 0.258 0.258 0.257 0.256 0.256 0.255 0.254 0.254 0.253
0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00
1.157 1.159 1.162 1.165 1.167 1.170 1.173 1.176 1.179 1.182 1.185 1.188 1.191 1.194 1.197 1.200 1.203 1.206 1.210 1.213
0.489 0.377 0.340 0.325 0.312 0.301 0.297 0.294 0.291 0.288 0.287 0.286 0.285 0.284 0.283 0.283 0.282 0.281 0.281 0.280
0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00
1.268 1.271 1.274 1.277 1.280 1.283 1.286 1.289 1.292 1.295 1.298 1.302 1.305 1.308 1.312 1.315 1.319 1.322 1.326 1.330
0.536 0.413 0.373 0.356 0.342 0.33 0.326 0.322 0.319 0.316 0.315 0.313 0.312 0.311 0.311 0.310 0.309 0.309 0.308 0.307
Structural Assessment
(a)
161
(b) 300 F = 570 kN
1.0
F = 520 kN
Kr
F = 470 kN
0.6
F = 520 kN F = 570 kN
0.4
J-integral in N/mm
Option 1 FAD
0.8
200
F = 470 kN
R-curve 100
2c a
0.2 ao = 3 mm 2W
0
0
a /c = 1.0
t
0.4
0.8
1.2
1.6
0
0
2
ao
4
6
8
10
12
Crack depth a
Lr
Figure 6.18: Example 6.4: Determination of the instability load of the component. (a) FAD assessment; (b) CDF assessment according to Fig. 6.6a.
Fig. 6.18a shows these in conjunction with the Option 1B FAD curve. The resulting instability load is approximately Fmax = 520 kN. The same result is obtained by the CDF analysis (Table 6.8; Figs 6.18b and 6.19). Note that, as in the cases above, the assumption of a constant a/c ratio during crack extension is often inadmissible in a real application.
Table 6.8: Example 6.4 (CDF approach): Crack Driving Force curves for three applied loads. The original crack depth was a0 = 3 mm, the crack geometry was simplified taken as a/c = 1 Load = 470 kN a in mm
J in N/mm
1 2 3 4 5 6
247 564 1305 3078 7753 21340
Load = 520 kN
Load = 570 kN
a in mm
J in N/mm
a in mm
1 2 3 4 5 6
727 1928 4463 10529 26519 72998
1 2 3 4 5 6
J in N/mm 2221 5889 13634 32167 81022 223023
162
Fitness-for-Service Fracture Assessment of Structures Containing Cracks
600
Maximum load 500
Load in kN
400
300
2c a 200
2W
100
t
Option 1B analysis 0
0
0.5
1.0
1.5
2.0
2.5
3.0
Stable crack extension Δa in mm
Figure 6.19: Example 6.4: Determination of the instability load of the component. CDF assessment according to Figs 6.6 (b) and 6.8.
6.6. Combined Primary and Secondary Stresses 6.6.1. General Remarks The terms primary and secondary stresses have already been introduced in Section 4.1.2. In SINTAP/FITNET, the specification differs from their general use insofar as primary stresses are defined as those stresses that contribute to plastic collapse and secondary stresses are defined as those stresses that do not. The latter can, however, contribute to fracture. Typical secondary stresses are residual stresses and thermal stresses although thermal loading can also produce primary stresses. It was also mentioned in Section 4.1.2 that secondary stresses that are self-equilibrating over the entire component may sometimes contribute to plastic collapse in the net section containing the crack, when, for example, there is significant elastic follow-up from the surrounding structure due to global boundary restraint effects in complex multi-component structures. If it is not clear which kind of classification is appropriate, the treatment as primary stress is, at least, conservative. In a SINTAP/FITNET analysis, secondary stresses are considered in the determination of the K-factor whereas the yield load, or its associated parameters
Structural Assessment
163
ref and Lr , are not affected by them. For small-scale yielding conditions, the mode I K-factor can simply be determined by KI = KIp + KIs
(6.42)
with the indices “p” and “s” indicating “primary” and “secondary”. However, for contained and net-section yielding, the situation becomes more complex. Because of plasticity and relaxation effects the resulting K-factor no longer equals the sum of KIp and KIs . As a consequence, when a component is loaded by a combination of primary and secondary stresses an additional interaction or correction term is applied in SINTAP/FITNET following a concept that was originally introduced in [6.3]. Most commonly this term is designated . SINTAP/FITNET has introduced an alternative correction term, V, [6.50] based on the same concept as that yields completely identical results but is slightly more transparent from a didactical point of view. Therefore the following introduction makes use of the V term only.
6.6.2. The Correction Term V in the FAD and CDF Approaches In the FAD approach, Eqn (6.2), is replaced by Kr =
KIp + V · KIs Kmat
(6.43)
In the CDF approach, Eqns (6.8), and (6.10), are substituted by
2 1 KIp + V · KIs J= · E f Lr
(6.44)
p
2 KI + V · KIs 1 · = Y · E f Lr
(6.45)
and
where E = E in plane stress and E = E/1 − 2 in plane strain or axi-symmetry. The V term can be interpreted as a correction term for K compared with simple superposition according to Eqn (6.42). Its magnitude depends mainly on the primary and secondary stresses, on the crack size and on Lr . The last effect
164
Fitness-for-Service Fracture Assessment of Structures Containing Cracks
Correction factor V
Stress relaxation effects
1
Plasticity effects due to residual stresses
Lr
1
Figure 6.20: The V correction term as a function of the ligament yielding parameter Lr (schematic).
is schematically illustrated in Fig. 6.20 in which the magnitude of V is the result of two competing effects. In the contained yielding range approximately up to Lr = 1, the plasticity corrected K is larger than KI = KIp + KIs because of ligament yielding effects and therefore V > 1. Beyond this range the plasticity corrected K becomes smaller than KIp + KIs due to relaxation effects. Note that there is a wide range where both effects are superimposed.
6.6.3. The Determination of V The determination of V is based on theoretical analysis [6.3] backed up by the results of finite element analysis results on a range of geometries. The values of V have been made available in table format in [6.4] such that V is obtained from V=
Kps KIs
·
(6.46)
with KIs being the mode I stress intensity factor due the secondary stresses, Kps a plasticity corrected value of KIs and an auxiliary function defined in the tables. KIs is determined as described in Section 5.1.2.2, usually based on a residual stress distribution in polynomial format or as described in Section 5.1.2.3 for secondary stress distributions varying in two directions. For the determination of
Structural Assessment
165
Kps , various approaches exist of which only the most simple and least expensive will be explained here. It is based on a plastic zone size correction Kps a =
a 1/ 2 eff · KIs a a
(6.47)
with the plastic zone corrected crack depth aeff being obtained by aeff
s
2 1 KI a = a+ · 2 Y
(6.48)
where = 1 in plane stress; = 3 in plane strain. Note that Eqn (6.48) may yield overconservative results for elastic stresses not much greater than yield. In some cases aeff will formally exceed the section thickness. The simplified procedure of Eqn (6.47) leads always to values of Kps greater than KIs and hence should not be used in cases where it is known that Kps < KIs , where significant plastic relaxation of stress occurs. In that case it is conservative to set Kps equal to KIs , and preferably to follow more complex routes for V determination. Once Kps has been established, the function can be determined from Table 6.9 for the primary loading K value, KIp , and the ligament yielding parameter Lr . A simplified route for determining V is given in R6 for Kps /KIp /Lr ≤ 4 [6.4]: ⎧ p ⎨ 1 + 02 · Lr + 002 · KIs KI Lr · 1 + 2 · Lr V = 31 − 2 · Lr ⎩ 1
for Lr < Lr∗ for L∗r < Lr < 105 for Lr > 105 (6.49)
Lr∗ < 1 is a function of Kps /KIp /Lr determined by the intersection of the first two lines in Eqn (6.49).
6.6.4. Welding Residual Stress Profiles 6.6.4.1. Introduction Whilst the procedure described in this section is generally applicable to all kinds of secondary stress fields, SINTAP/FITNET provides special information on as-welded residual stress distributions. In an appendix of FITNET [6.5] conservative residual stress profiles are provided for a range of components such as plate butt and plate seam welds, plate T-butt welds, pipe butt welds, set-in nozzles
166
Table 6.9: Tabulation of as a function of Lr and Kps /KIp /Lr (according to [6.4]) KIp /Lr
0
0.02
0.04
0.06
0.08
0.1
0.2
0.3
0.4
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1.000 1.000 1.001 1.001 1.002 1.002 1.004 1.005 1.007 1.008 1.010 1.012 1.027 1.049 1.082 1.126 1.176 1.215 1.215 1.133
1 1.000 1.000 1.001 1.001 1.002 1.003 1.004 1.006 1.008 1.010 1.012 1.014 1.029 1.052 1.085 1.128 1.175 1.214 1.212 1.130
1 1.000 1.001 1.001 1.001 1.003 1.004 1.005 1.007 1.009 1.011 1.013 1.015 1.031 1.054 1.087 1.129 1.175 1.212 1.210 1.128
1 1.000 1.001 1.001 1.002 1.003 1.004 1.006 1.008 1.010 1.012 1.014 1.017 1.033 1.057 1.089 1.130 1.175 1.211 1.208 1.125
1 1.000 1.001 1.002 1.002 1.004 1.005 1.007 1.009 1.011 1.013 1.015 1.018 1.035 1.059 1.091 1.131 1.174 1.210 1.206 1.123
1 1.001 1.002 1.003 1.004 1.006 1.008 1.011 1.013 1.016 1.019 1.022 1.026 1.045 1.068 1.096 1.131 1.171 1.204 1.198 1.112
1 1.001 1.003 1.004 1.006 1.009 1.012 1.015 1.018 1.022 1.025 1.029 1.033 1.051 1.071 1.096 1.129 1.169 1.200 1.191 1.102
1 1.002 1.004 1.006 1.008 1.012 1.016 1.020 1.023 1.027 1.031 1.034 1.038 1.054 1.071 1.095 1.128 1.168 1.198 1.185 1.094
1 1.003 1.006 1.008 1.011 1.016 1.020 1.024 1.028 1.032 1.035 1.038 1.041 1.055 1.071 1.095 1.129 1.169 1.196 1.180 1.087
1 1.001 1.019 1.026 1.031 1.039 1.045 1.050 1.054 1.057 1.060 1.063 1.066 1.080 1.099 1.126 1.161 1.195 1.210 1.178 1.070
1 1.019 1.031 1.040 1.047 1.058 1.066 1.072 1.077 1.082 1.086 1.090 1.094 1.113 1.135 1.164 1.196 1.224 1.228 1.184 1.067
1 1.023 1.038 1.048 1.056 1.068 1.077 1.084 1.090 1.096 1.101 1.106 1.110 1.133 1.157 1.187 1.218 1.242 1.241 1.190 1.069
1 1.026 1.042 1.053 1.061 1.074 1.084 1.092 1.099 1.105 1.111 1.116 1.121 1.146 1.173 1.203 1.234 1.256 1.252 1.199 1.080
1 1.028 1.044 1.056 1.065 1.078 1.088 1.097 1.104 1.111 1.117 1.124 1.128 1.155 1.184 1.215 1.248 1.269 1.267 1.218 1.098
1 1.029 1.046 1.059 1.068 1.083 1.093 1.102 1.110 1.117 1.123 1.129 1.136 1.165 1.196 1.229 1.262 1.288 1.291 1.240 1.105
1 1.030 1.048 1.060 1.071 1.087 1.098 1.108 1.116 1.123 1.130 1.137 1.144 1.175 1.209 1.246 1.284 1.314 1.316 1.259 1.104
1 1.031 1.051 1.061 1.076 1.092 1.103 1.114 1.122 1.131 1.138 1.145 1.153 1.187 1.225 1.266 1.311 1.343 1.341 1.271 1.094
1 1.032 1.047 1.062 1.081 1.099 1.112 1.112 1.132 1.142 1.149 1.158 1.166 1.205 1.248 1.292 1.337 1.365 1.355 1.272 1.073
Lr 0 0.01 0.02 0.03 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
(Continued)
Fitness-for-Service Fracture Assessment of Structures Containing Cracks
Kps
Table 6.9: (Continued) 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0
1 1 1 1 1 1 1 1 1 1
0.951 0.710 0.498 0.376 0.334 0.320 0.308 0.296 0.283 0.268
0.948 0.708 0.497 0.375 0.334 0.349 0.308 0.296 0.283 0.268
0.946 0.707 0.496 0.375 0.333 0.349 0.308 0.296 0.283 0.268
0.943 0.705 0.495 0.375 0.333 0.319 0.308 0.296 0.283 0.268
0.941 0.703 0.494 0.374 0.333 0.319 0.308 0.296 0.282 0.268
0.930 0.695 0.490 0.373 0.332 0.318 0.307 0.295 0.282 0.267
0.921 0.688 0.486 0.371 0.331 0.318 0.307 0.295 0.282 0.267
0.912 0.682 0.483 0.370 0.331 0.317 0.306 0.294 0.281 0.267
0.905 0.677 0.480 0.368 0.330 0.317 0.306 0.294 0.281 0.266
0.884 0.661 0.471 0.363 0.330 0.317 0.306 0.294 0.281 0.266
0.877 0.658 0.461 0.361 0.331 0.319 0.307 0.295 0.281 0.266
0.882 0.649 0.449 0.359 0.332 0.320 0.308 0.296 0.282 0.266
0.887 0.633 0.439 0.358 0.333 0.321 0.309 0.296 0.282 0.266
0.879 0.613 0.426 0.357 0.332 0.320 0.309 0.295 0.284 0.262
0.861 0.597 0.427 0.355 0.333 0.320 0.308 0.297 0.283 0.261
0.842 0.583 0.420 0.354 0.331 0.332 0.309 0.295 0.281 0.256
0.820 0.571 0.415 0.351 0.333 0.319 0.308 0.297 0.280 0.263
0.801 0.561 0.413 0.351 0.331 0.319 0.308 0.292 0.280 0.266
Structural Assessment 167
168
Fitness-for-Service Fracture Assessment of Structures Containing Cracks
(b)
(a) W1
W1 Transverse direction
z
z t
W2
t
W2
Longitudinal direction
Figure 6.21: Plate butt or pipe seam weld – (a) single weld; (b) double sided weld.
and set-on nozzles, repair welds, transition welds and weld T-intersections. The profiles are given for both along the surface and in the thickness direction for transverse (normal to the weld run) and longitudinal (parallel to the weld run) directions. In the following an example will be given for plate butt and pipe seam welds (Fig. 6.21). 6.6.4.2. Surface Residual Stress Profiles The surface profiles which can be applied to ferritic and austenitic steels and aluminium alloys are shown in Fig. 6.22. At the weld position the residual stresses are constant and of yield strength magnitude with the longitudinal stresses limited by the weld metal yield strength YW and the transverse stresses limited by the lower of the weld metal and base plate yield strength YB . Adjacent to the weld the stresses decline linearly and become zero at distances ro , yo , t and 2t depending on the orientation (longitudinal or transverse) and the plate thickness. Qualitatively “thick plates” means that the weld bead dimensions are small compared to the plate thickness; the quantitative criterion is ro ≤ t. Otherwise the configuration is treated as “thin plate”. The parameters ro and yo are given by ro =
1 2 k ·q / · YB v
(6.50)
yo =
1033 · k · q · YB vt
(6.51)
and
Structural Assessment
169
(a) L
σR
Thick plate (ro ≤ t)
σYW
Thin plate (ro > t)
W1
U ro
W
ro
yo
yo
Side 1: W = W1 Side 2: W = W2 (b) T
σR σY*
Unrestrained plates and pipe axial seam welds
t
W
t
Restrained plates
2t
W1
2t
Side 1: W = W1 Side 2: W = W2
Figure 6.22: Surface residual stresses – (a) longitudinal; (b) transverse. For asymmetric welds side 1 is the side with the widest weld face. YB is the yield strength ReH or Rp02 of the base metal (for austenitic steels the 1% proof strength), Y ∗ refers to the lower of weld metal and base plate yield strengths.
with YB being the yield strength ReH or Rp02 of the base metal or, in the case of austenitic steels, a 1% proof strength chosen in order to account for the high work hardening at the start of plastic deformation, which results in a large variability of the material properties. The parameter k (in Nmm/J) is a material constant related to the coefficient of thermal expansion , the Young’s modulus E, the density and the specific heat c: k=
2··E 2718 · · · c
(6.52)
170
Fitness-for-Service Fracture Assessment of Structures Containing Cracks
Table 6.10: Typical material properties used in Eqns (6.50) – (6.52) Material property
Material Ferritic steels
Austenitic steels
Aluminium alloys
Coefficient of thermal expansion in C−1
12×10−6
16×10−6
24×10−6
Young’s modulus E in MPa Volumetric specific heat · c in J · mm3 / C 2· · E k= 2718 · · · c in Nmm/J
207 000
193 000
70 000
0.0038
0.0036
0.0024
153
201
164
The other parameters are the process efficiency, that is, the fraction of arc power entering the plate as heat , the arc power q (in J/sec), the weld travel speed v (in mm/sec) and the wall thickness t. Typical values for , E, · c and k are provided in Table 6.10. 6.6.4.3. Through-Thickness Residual Stress Profiles The through-thickness stress profiles are given by Longitudinal direction: ⎧ ⎪ ⎪1 z z 2 ⎪ ⎨ 095 + 1505 · − 8287 · L R YW z/ t = ⎪ t z 3 tz 4 ⎪ ⎪ ⎩ +10571 · − 408 · t t
ferritic steels (6.53) austenitic steels
Transverse direction: z z 2 z 3 − 14533 · + 83115 · RT Y∗ z/ t = 1 − 0917 · t t t z 4 z 5 z 6 − 21545 · + 24416 · − 9336 · t t t
(6.54)
Structural Assessment
(a)
(b) L
T
σ R /σYW –0.5
0
0.5
σ R /σY* 1
0
0.5
1.5
–0.5
0
0.5
1
1.5
Ferritic steels (Eqn 6.53)
Austenitic steels (Eqn 6.53)
–0.5
171
z Ferritic and austenitic steels (Eqn 6.54)
1
1.5
–0.5
0
0.5
1
t
1.5
Figure 6.23: Through-thickness residual stresses – (a) longitudinal; (b) transverse.
for ferritic and austenitic steels (Fig. 6.23). In Eqns (6.53) and (6.54) YB characterises the yield strength ReH or Rp02 of the base metal (for austenitic steels, the 1% proof strength) and Y ∗ is the lower of weld metal and base plate yield strengths. All residual stress profiles represent upper bounds to individual measured or simulated profiles. Note that these upper bound stress profiles, other than the underlying original data, are not self-equilibrating across the section. An impression of the conservatism of such reference profiles compared to experimental data is provided in Fig. 6.24 for a T-node weld [6.6]. Note that the longitudinal residual stresses usually refer to the weld metal yield strength, whereas the transverse residual stresses are normalised by the lower of
1.5 SINTAP/FITNET upper bound curve
1.0
L
0.5
z
t
T
σ R/σ Y*
T
0 –0.5
Experimental scatter band –1.0
0
0.2
0.4
0.6
0.8
1
z/t
Figure 6.24: Comparison of SINTAP/FITNET transverse residual stress profile with measured residual stress data for a T-node weld (after [6.6]).
172
Fitness-for-Service Fracture Assessment of Structures Containing Cracks
the weld metal and base plate yield strengths. Exceptions are cracks in repair welds, cracks at weld intersections and shallow cracks. For those configurations the higher of the two yield strengths has to be chosen. The yield strength at the temperature of the assessment should be inserted. If the temperature varies, the value at the lowest temperature is relevant. Note that, for conservatism reasons, the yield strength should be over- rather than under-estimated. That means, for example, that the upper yield strength ReH should be applied for materials displaying a yield plateau and not the lower yield strength ReL . Care has to be exercised when only specified minimum values of the yield strength are available.
6.6.5. Examples of Analysis for Combined Primary and Secondary Stresses Example 6.5: The tension loaded plate with a surface crack shown in Examples 5.7 and 6.1 is assumed to be made of an austenitic steel with a resistance against stable crack extension of Ji = 100 N/mm. The plate dimensions are 2W = 200 mm and t = 10 mm; the cross section containing a semi-elliptical crack of a depth a = 3 mm and geometry a/c = 1 is shown in Fig. 6.25. The engineering and true stress–strain curves are provided in Table 6.11 and Fig. 6.26; the tensile test data are: yield strength Y = Rp02 = 240 MPa, tensile strength Rm = 617 MPa, Young’s modulus E = 195 GPa and Poisson’s ratio = 03. The material does not display a yield plateau. In the first step the critical load will be determined for the plate without a weld. The welded plate will be treated in the second step assuming no strength σm
a
2c
σm
t
Figure 6.25: Example 6.5: Geometry of the welded plate subjected to tension.
Structural Assessment
173
Table 6.11: Example 6.5: Engineering and true stress-strain curves of the material. The true stress and strain are obtained by t = 1 + and t = ln1 + Engineering curve in MPa 150 171 186 200 209 216 235 247 260 273
True curve
Engineering curve
True curve
in %
t in MPa
t in %
in MPa
in %
t in MPa
t in %
008 010 012 015 018 020 030 041 059 079
1505 1716 1864 1998 2096 2168 2359 2482 2616 2747
008 010 012 015 018 020 030 041 059 079
281 307 337 362 452 529 571 605 617
100 193 334 476 1175 2179 3195 4596 5997
2839 3127 3487 3791 5049 6439 7536 8832 9870
100 191 329 465 1111 1971 2773 3782 4698
1000
Stress in MPa
True curve 800 600
Engineering curve
400 200 0
0
10
20
30
40
50
60
Strain in %
Figure 6.26: Example 6.5: Engineering and true stress-strain curves of the material.
mismatch, that is, the stress–strain curves of the weld metal and base plate are identical. What is taken into account is the transverse residual stress distribution across the wall thickness provided by Eqn (6.54). (1) Determination of the critical load without taking into account welding residual stresses: For the given yield and ultimate tensile strengths the plastic collapse limit Lr max is determined by Eqns (6.29) and (6.31) as Lr max = 179 for Options 1B and 3. The Option 1B strain hardening exponent is N = 018. The f(Lr ) functions are determined by Eqns (6.25), (6.27) and (6.30) and are summarised in Table 6.12.
174
Fitness-for-Service Fracture Assessment of Structures Containing Cracks
Table 6.12: Example 6.5: fLr functions according to Options 1B and 3 ( = 06) Lr
Option 1 fLr
Option 3 fLr
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1000 0998 0990 0978 0961 0937 0903 0853 0782 0682
1000 0998 0990 0978 0962 0942 0921 0865 0795 0705
Option 1 fLr
Lr 10 11 12 13 14 15 16 17 Lrmax : 1.785
0559 0452 0372 0311 0264 0226 0196 0171 0154
Option 3 fLr 0586 0459 0360 0290 0248 0220 0200 0185 0175
Applying these f(Lr ) functions, both an FAD and a CDF analysis were carried out. The K-factor was determined for the plate cross section of Fig. 6.27 by the following equations [6.7]: Deepest point of the crack (point A): KIA =
a 0
x · MIA x a dx
B
(6.55)
A 2c
B a
x
2W t
Figure 6.27: Example 6.5: Plate geometry.
Structural Assessment
Surface points of the crack (point B): a KIB = x · MIB x a dx
175
(6.56)
0
x 1/ 2 A x a = 1 + M1 · 1 − a 2 a − x x x 3/ 2 A A + M2 · 1 − + M3 · 1 − a a
x 1/ 2 x x 3/ 2 2 B B B B + M2 · + M3 · MI x a = √ 1 + M1 · a a a ·x 2
MIA
(6.57) (6.58)
24 M1A = √ 4Y0 − 6Y1 − 5 2Q
(6.59)
M2A = 3
(6.60)
M3A = 2 · √ · Y0 − M1A − 4 2Q
(6.61)
M1B = √ 30 · F2 − 18 · F1 − 8 4Q
(6.62)
M2B = √ 60 · F2 − 90 · F1 + 15 4Q M3B = − 1 + M1B + M2B Y0 = B0 + B1 ·
a 2 t
+ B2 ·
a 4
B0 = 110190 − 0019863 · B1 = 432489 − 149372 ·
t a
c a
(6.63) (6.64)
(6.65)
− 0043588 ·
+ 194389 ·
a 2
c a 2
− 852318 ·
(6.66) a 3
c c c a a 2 a 3 B2 = −303329 + 996083 · − 12582 · + 53462 · c c c a 2 a 2 Y1 = A0 + A1 · + A2 · t t
(6.67) (6.68) (6.69)
176
Fitness-for-Service Fracture Assessment of Structures Containing Cracks
A0 = 0456128 − 0114206 · A1 = 3022 − 108679 ·
a c
A2 = −228655 + 788771 · Q = 1 + 1464 · F1 = · F2 = ·
a c a c
a 165 c
a c
− 0046523 ·
+ 1494 · a c
a 2 c
a 2 c
− 68537 ·
− 110675 ·
a 2 c
(6.70) a 3 c
+ 516354 ·
a ≤ 1 c
for
(6.71) a 3 c
(6.72) (6.73)
(6.74)
(6.75)
= 114326 + 00175996 · = 0458320 − 0102985 · = 0976770 − 0131975 ·
a t a t a t
+ 0501001 · − 0398175 · + 0484875 ·
a 2 t a 2 t a 2 t
(6.76)
(6.77)
(6.78)
and = 0448863 − 0173295 ·
a t
− 0267775 ·
a 2 t
(6.79)
For membrane stress loading, = m , Eqns (6.55) and (6.56) can be solved as Deepest point of the crack: a x · MIA x a dx = m · K0A (6.80) KIA = 0
with K0A
=
a 0
MIA
x a dx =
2a 2 A 1 A A · 2 + M1 + M2 + M3 3 2
Surface points of the crack: a x · MIB x a dx = m · K0B KIB = 0
(6.81)
(6.82)
Structural Assessment
with K0B
=
a 0
177
MIB x a dx
a 2 B 1 B B =2 · 2 + M1 + M2 + M3 3 2
(6.83)
In the present case the highest K factors were reached at the surface points of the crack (point B). A local yield load was determined by the approach introduced in Example 5.7 (Eqn 5.57) or alternatively by Eqns (5.59) and (5.60) for ref and Lr . Table 6.13 gives the ligament yielding parameter Lr , the associated values of Kr (FAD approach) and the Option 1B and 3 applied J values (CDF approach) as a function of a stepwise increased applied load F. The results are illustrated graphically in Fig. 6.28 for the FAD and CDF approaches. The critical load is Fc = 733 kN for Option 1B and Fc = 747 kN for Option 3. Note that the a/c ratio was assumed to be constant in the example. In a real application such a simplification would usually be inadmissible since the crack when extending, for example, by fatigue, would change its shape. (2) Determination of the critical load taking into account welding residual stresses: Since secondary stresses such as welding residual stresses do not affect the yield and collapse loads, the parameters Lr and Lr max as functions of the load are identical to those calculated in (1). What is, however, different is Kr (FAD Table 6.13: Example 6.5: Ligament yielding Lr , K-factor at the surface points of the crack, Kr and Option 1B and 3 applied J values as a function of increased applied load F for a crack depth a = 3 mm and a crack geometry a/c = 1 excluding welding residual stresses. Kr is based on the K values at the surface points of the crack (point B) F in kN 0 100 200 300 400 500 600 700 800 900
Lr 0 0224 0448 0671 0895 1192 1343 1567 1791 2014
KIB in MPa · m1/2 0 369 738 1107 1476 1845 2215 2584 2953 3322
Kr
J in N/mm (Option 1B)
J in N/mm (Option 3)
0 0025 0050 0076 0101 0126 0151 0177 0202 0227
0 0065 0282 0757 2151 8411 27288 73814 174782 373850
0 0065 0280 0735 2026 8211 31698 73050 134066 208170
178
Fitness-for-Service Fracture Assessment of Structures Containing Cracks
(a)
(b) 400
F in kN: 800
1.0
Option 3
Kr 0.4
F in kN 800
0.2
100 200
300
400
2c
Option 3 2W
900 0.8
1.2
1.6
2.0
t
0
700
Jmat = 100 N/mm
200 300 400
600 700 0.4
800
a
200
100
500
0 0
J-integral in N/mm
Option 1B
0.6
0
Option 1B
300
0.8
700
600 500
0 100 0
0.4
0.8
Lr
1.2
1.6
2.0
Lr
Figure 6.28: Example 6.5: Determination of the critical load for a crack depth a = 3 mm, a crack geometry a/c = 1 and a crack resistance of Ji = 100 N/mm. (a) FAD approach; (b) CDF approach. Welding residual stresses are not considered.
approach) and J (CDF approach). In a first additional step K factors have to be provided for the secondary stress profile according to Eqn (6.54). Since this is given as a sixth order polynomial x = 0 + 1 · x + 2 · x2 + 3 · x3 + 4 · x4 + 5 · x5 + 6 · x6
(6.84)
Equations (6.55) and (6.56) will also be presented in this format. Note that the i coefficients in Eqn (6.84) are not identical to those in Eqn (6.54) but have to be transformed to the new format. In the present example this yields values 0 = 2850 MPa, 1 = −2613 MPa, 2 = −4142 MPa, 3 = 2369 MPa, 4 = −614 MPa, 5 = 0696 MPa and 6 = −00275 MPa. The K-factor is determined by: Deepest point of the crack (point A): KIA =
a 0
x · MIA x a dx = 0 · K0A + 1 · K1A + 2 · K2A
+ 3 · K3A + 4 · K4A + 5 · K5A + 6 · K6A
(6.85)
Structural Assessment
with K0A according to Eqn (6.81), K1A
K2A
K3A
K5A
=
=
x · MIA
a
x
· MIA
2
0
=
0
=
=
K4A
a
a
x
· MIA
3
0
x
· MIA
4
0
2a7 32 1 A 32 A 1 A · + M + M + M (6.88) 35 4 1 315 2 20 3
2a9 256 1 A 256 A 1 A · + M + M + M (6.89) 315 5 1 3465 2 30 3
a
x
· MIA
5
0
x a dx =
2a11 512 1 A 512 A 1 A · + M + M + M (6.90) 693 6 1 9009 2 42 3
and K6A
=
a
x
6
0
· MIA
(6.86)
2a5 16 1 A 16 A 1 A · + M + M + M (6.87) 15 3 1 105 2 12 3
x a dx =
x a dx =
2a3 4 1 A 4 A 1 A · + M + M + M 3 2 1 15 2 6 3
x a dx =
a
x a dx =
179
x a dx =
2a13 2048 1 A 2048 A 1 A · + M + M + M 3003 7 1 45045 2 56 3 (6.91)
Surface points of the crack (points B): a KIB = x · MIB x a dx = 0 · K0B + 1 · K1B + 2 · K2B + 3 · K3B 0
+ 4 · K4B + 5 · K5B + 6 · K6B
(6.91)
with K0B according to Eqn (6.83), K1B
K2B
K3B
=
=
=
a 0
x · MIB x a dx
x
2
· MIB x a dx
x
3
· MIB x a dx
0
a5 2 1 B 2 B 1 B =2 · + M + M + M 5 3 1 7 2 4 3
a 0
(6.92)
a
a3 2 1 B 2 B 1 B =2 · + M + M + M 3 2 1 5 2 3 3
=2
a7 2 1 B 2 B 1 B · + M1 + M2 + M3 7 4 9 5
(6.93)
(6.94)
180
Fitness-for-Service Fracture Assessment of Structures Containing Cracks
a9 2 1 B 2 B 1 B = x =2 · + M + M + M 9 5 1 11 2 6 3 0
a a11 2 1 B 2 B 1 B B 5 B K5 = x · MI x a dx = 2 · + M + M + M 11 6 1 13 2 7 3 0
K4B
a
4
· MIB x a dx
and K6B
=
(6.96)
a
x 0
(6.95)
6
· MIB x a dx
a13 2 1 B 2 B 1 B =2 · + M + M + M 13 7 1 15 2 8 3
(6.97)
A K solution based on a sixth order polynomial description of the stress distribution was chosen because the residual stress profile was given in that format. Note, however, that there is usually not much benefit from using such high order polynomials, and particularly if, as is often the rule, no adequate K solutions are available. An alternative approach is to transfer the residual stress profile to a polynomial of a lower order. Since the requirement is to describe the stresses along the assumed crack position as well as possible rather than along the whole cross section, a third order polynomial will be satisfactory in most cases. As an example based on the stress profile of Eqn (6.54) a table of stress values at z/t = 0, 0.1, 0.2, … , 1.0 has been generated and fitted by a third order polynomial z z 2 z 3 − 14533 · + 83115 · (6.98) RL Y∗ z/ t = 1 − 0917 · t t t The original sixth order and the re-calculated third order polynomial are compared in Fig. 6.29. Based on Eqn (6.98) an input stress profile for K determination is generated x = 0 + 1 · x + 2 · x2 + 3 · x3
(6.99)
with transformed coefficients 0 = 27987 MPa, 1 = −1619 MPa, 2 = −1938 MPa and 3 = 2135 MPa. The determination of K simplifies to a KIA = x · MIA x a dx = 0 · K0A + 1 · K1A + 2 · K2A + 3 · K3A (6.100) 0
KIB =
a 0
x · MIB x a dx = 0 · K0B + 1 · K1B + 2 · K2B + 3 · K3B
(6.101)
Structural Assessment
181
1.2
T direction
1.0 0.8
z
0.6
σTR
t 0.4
3rd order polynomial (simplified)
0.2 0
6th order polynomial (compendium)
–0.2 –0.4
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
z/t
Figure 6.29: Comparison between sixth order polynomial transverse residual stress profile (Eqn 6.54) according to the SINTAP/FITNET compendium with a re-calculated third order polynomial profile according to Eqn (6.98).
with the coefficients KiA and KiB (i = 0 to 3) given above (Eqns 6.81, 6.86–6.88 and 6.83, 6.92–6.94). The K-factors due to residual stresses KIs are: Deepest point of the crack (point A): KIs = 1143 MPa · m1/2 sixth order polynomial and KIs = 1146 MPa · m1/2 (third order polynomial); Surface points of the crack (points B): KIs = 1915 MPa · m1/2 (sixth order polynomial) and KIs = 1926 MPa · m1/2 (third order polynomial).
182
Fitness-for-Service Fracture Assessment of Structures Containing Cracks
In the present example the differences are about 0.3% and 0.6% which, in the context of the background of the rather crude determination of the upper bound reference residual stress profiles, is absolutely negligible. The further analysis will use the sixth order polynomial. However, there would only be a very small effect on the final result if the third order polynomial were taken. Since, both the primary stress K-factor, KIp , and the secondary stress K-factor, KIs , are larger at the surface points of the crack (points B) the further analysis will be exclusively based on this point. Table 6.14 gives a summary of values of Lr , Kr and the Option 1B and Option 3 applied J values as a function of increased applied load F for the present example (a = 3 mm; a/c = 1), this time taking into account the residual stresses. The results are illustrated graphically in Fig. 6.30 for the FAD and CDF approaches. The critical load is Fc = 691 kN for Option 1B and Fc = 690 kN for Option 3. These critical loads are only slightly smaller (6% and 8%) than those of the analysis without considering the residual stresses. The reason is that the failure takes place at very high ligament yielding (Lr > 14) where the residual stresses are largely released. Note that the Option 3 critical load is slightly smaller than that of Option 1B. This is because the Option 1B and 3 f(Lr ) functions are almost identical in the present case. The result seems to contradict the philosophy of lower conservatism at higher analysis options. Actually there exist cases where this is the case. Table 6.14: Example 6.5: Lr , Kr , and V functions and Option 1B and 3 applied J values as a function of increased applied load F for a crack depth a = 3 mm and a crack geometry a/c = 1 including welding residual stresses. Kr is based on the K values at the surface points of the crack (point B). The fracture resistance is Ji = 100 N/mm F in kN 100 200 300 400 500 600 700 800 900
Lr
V
0.224 0.448 0.671 0.895 1.192 1.343 1.567 1.791 2.014
1089 1149 1206 1184 0837 0420 0323 0296 0266
1260 1329 1395 1370 0968 0486 0373 0342 0377
Kr 0190 0224 0258 0280 0253 0215 0225 0246 0267
J in N/mm (Option 1B)
J in N/mm (Option 3)
3704 5573 8820 16585 33805 55042 120276 260970 518260
3704 5537 8567 15620 33003 63937 119432 200179 288580
183
Structural Assessment
(a)
(b) 400 1.0
Option 1B σm
Option 3
Kr
0.6
F in kN 0.4 100
200
300 400
900 700 800 500
0.2
J-integral in N/mm
Option 1B
200
no re
l
0
0.4
ses stres
0.8
800 Option 3
t
σm
F in kN: 700 100
3.6.8 (Fig.
a
2c
800
Jmat = 100 N/mm
500 600
)
sidua
0
900
300
0.8
100 200 300
600
no residual stresses (Fig. 3.6.8)
400 600
1.2
1.6
2.0
Lr
0 0
0.4
0.8
1.2
1.6
2.0
Lr
(c) 1.6
V
1.2 0.8 0.4 Failure
0
0
0.4
0.8
1.2
1.6
2.0
Lr
Figure 6.30: Example 6.5: Determination of the critical load for a crack depth a = 3 mm, a crack geometry a/c = 1 and a crack resistance of Ji = 100 N/mm. (a) FAD approach; (b) CDF approach. The transverse through thickness welding residual stresses (Eqn 6.54) are taken into account.
However, the effect is usually so small that it can be neglected. In principle, when an Option 3 analysis provides lower results than Option 1 analysis the latter should be trusted despite the fact that it is, against the usual rule by being more conservative. Example 6.6: The tension loaded plate of Examples 6.5 will be analysed again, but for a crack size of a = 6 mm (a/c = 1) and an assumed crack resistance of the weld of only Ji = 40 N/mm. All other data are taken from Example 6.5. The equations used for determining the K-factor and the yield load are identical to
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Fitness-for-Service Fracture Assessment of Structures Containing Cracks
Table 6.15: Example 6.6: Lr , Kr , and V functions and Option 1B and 3 applied J values as a function of applied load F for a crack depth a = 6 mm and a crack geometry a/c = 1 excluding and including welding residual stresses. Kr is based on the K values at the surface points of the crack (point B). The fracture resistance is Ji = 40 N/mm F in kN
Lr
Residual stresses excluded 100 0.269 – 200 0.538 – 300 0.806 – 400 1.075 – 500 1.344 – 600 1.613 – Residual stresses included 100 0.269 1099 200 0.538 1167 300 0.806 1221 400 1.075 0926 500 1.344 0419 600 1.613 0317
V
Kr
J in N/mm (Option 1B)
J in N/mm (Option 3)
– – – – – –
0040 0080 0120 0159 0199 0239
0165 0743 2373 11255 47526 154195
0165 0727 2297 10744 55202 146061
1260 1337 1391 1060 0480 0363
0264 0318 0369 0348 0285 0304
7232 11796 22563 53703 96979 248880
7230 11560 21837 51263 11264 23575
the previous analysis. The results are summarised in Table 6.15 and illustrated graphically in Fig. 6.31. The critical load was determined as: (1) Without taking into account welding residual stresses: Option 1: Fc = 486 kN; Option 3: Fc = 476 kN ; (2) Taking into account welding residual stresses: Option 1: Fc = 364 kN; Option 3: Fc = 374 kN. This time the critical loads were reduced due to the residual stresses by 25% at Option 1 and by 21% at Option 3. The reason for the much larger effect compared to Example 6.5 is that the failure was shifted to much smaller values of Lr (c. Lr = 10), corresponding to a much higher correction term V. Case (1) is another example where an Option 3 result is slightly more conservative than the corresponding Option 1 result. The difference is, however, of the order
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Structural Assessment
(b)
(a)
400
Kr
0.6 500
F in kN
300
100 200
0.4
with residual 600 stresses
500 es ess
600
tr
al s
0.2
u sid t re
ou
ith 100 w
J-integral in N/mm
Option 1B
200
0.4
without residual stresses
500
100
Option 1B
400
F in kN:
400
100 200 300
Jmat = 40 N/mm 400
1.2
Lr
1.6
2.0
600
Option 3
300
0.8
600
t
σm
200
0
a
2c
with
300
0.8
0
resid
σm
Option 3
1.0
ual s tress es
400
0
0
0.4
0.8
1.2
500
1.6
2.0
Lr
Figure 6.31: Example 6.6: Determination of the critical load for a crack depth a = 6 mm, a crack geometry a/c = 1 and a crack resistance of Ji = 40 N/mm. (a) FAD approach; (b) CDF approach. Calculations have been performed with and without the transverse through thickness welding residual stresses according to Eqn (6.54).
of only 2% and insignificant against the background of the usual scatter in the input data.
6.6.6. Further Remarks on Welding Residual Stress Profiles 6.6.6.1. Validity Ranges In Section 6.6.4.3 a conservative upper bound welding residual stress profile taken from the SINTAP/FITNET compendium is provided. Table 6.16 gives an overview of the validity ranges of all profiles available in the SINTAP/FITNET compilation. Note that the plate dimensions and the yield strength of Examples 6.5 and 6.6 are outside these ranges so that the results would have to be used with caution in a real application. However, they are used here to illustrate the principles of using the approaches in cases with welding residual stresses. There exist many cases where no residual stress profiles are available. An alternative, although even more conservative, approach is to assume uniformly distributed stresses, that is, membrane stress distributions equal to the yield strength. In order to reduce conservatism, individual residual stress profiles have
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Fitness-for-Service Fracture Assessment of Structures Containing Cracks
Table 6.16: Validity ranges for as-welded residual stress profiles in ferritic steels provided by the SINTAP/FITNET compendium Geometry
Plate butt welds Pipe circumferential welds Pipe seam welds T butt welds Tubular and pipe to plate welds Repair welds
Thickness in mm
Yield strength in MPa
24–300 9–84 50–85 25–100 22–50 75–152
310–740 225–780 345–780 375–420 360–490 500–590
Electric heat input per unit length in kJ/mm 1.6–4.9 0.35–1.9 Not known 1.4 0.6–2.0 1.2–1.6
to be provided by measurements or numerical simulations, or by more appropriate information from the literature. 6.6.6.2. Post Weld Mechanical Treatment The profiles provided in SINTAP/FITNET refer to the as-welded condition and will be affected by potential post-weld mechanical or heat treatment. Prior overload (or proof) tests are a typical measure that may have a beneficial effect on the welding residual stresses. These tests are performed prior to the service of the component to demonstrate its sound fabrication or to improve its structural performance. The effect of the mechanical stress relief depends, however, on whether the structure is potentially cracked prior to the treatment or not – information which frequently will not be available. The following expression is provided for the reduced residual stress magnitude: Y R = min (6.102) 14 − ref f · Y If a crack is believed to have initiated in service, after the proof test, Eqn (6.102) is replaced by Y r = min (6.103) 11 · Y − 08 · a In Eqns (6.102) and (6.103) Y and f denote the yield and flow stress under proof test conditions, for example, at the proof test temperature, and a is the
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applied stress due to the proof load. This should be taken as the lower of the membrane stress in the section or the total stress including membrane, bending and stress concentration effects. According to [6.8] the reduced residual stresses should be applied as uniform across the section, that is, as membrane stresses, in the further analysis. Note that Eqns (6.102) and (6.103) currently have limited validation and should be used with caution because they have been obtained for simplified conditions. In reality, the redistribution of residual stresses in a proof test is affected by factors such as the weld geometry, the mechanical properties of the weld metal and the base plate, the initial stress in the body and the nature of the proof test loading [6.9]. Care has also to be exercised when mechanical residual stresses are introduced by the proof test itself, for example, when an initial ovalisation of a tube is removed by the test. Another kind of post weld mechanical treatment is given by in-service fatigue loading before failure. Depending on the cyclic load magnitude, stress relief effects are also expected, the evaluation of which is, however, not an easy issue and no recommendation is given by SINTAP/FITNET.
6.6.6.3. Post Weld Heat Treatment The aim of post-weld heat treatment (PWHT) is mainly to reduce the as-welded residual stresses. Their magnitudes will depend on a number of factors such as the initial residual stress state, the weld geometry, restraint due to adjacent structures, the effect of metallurgical phase transformations during cooling, the creep behaviour of the weld metal and base plate, and the nature of the PWHT procedure (heat band size, heating and cooling rates, etc.). This complexity makes any prediction of the residual stress profile after PWHT difficult, and this is complicated further if the stress relief mechanisms cause local creep damage, prior crack tip plasticity in the case of pre-existing cracks or adverse effects on the microstructure. The basic effects of a temperature increase are a reduction in the Young’s modulus (see Section 4.3.3) and a reduction in the yield strength with the latter causing conversion of elastic into plastic strain. A simple but conservative approach which neglects creep effects is to assume a uniformly distributed (i.e., membrane) residual stress across the section, the magnitude of which is equal to the yield strength at the PWHT temperature corrected by the ratio of the Young’s modules at the assessment and room temperature. If no Young’s modulus data for different temperatures are available for a specific material the information provided in Table 4.2 can be used.
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Fitness-for-Service Fracture Assessment of Structures Containing Cracks
6.7. Constraint Effects 6.7.1. Consideration in the FAD and CDF Approaches The effect of constraint on fracture toughness has already been addressed in Section 4.4.6. In a SINTAP/FITNET analysis, a constraint corrected toughness c according to Eqn (4.81) is applied which, for its use in the CDF approach, can Kmat be transformed to Jcmat or cmat values by Eqns (6.4) and (6.5). Since the constraint decreases with increasing ligament yielding (see Section 4.4.6.1, Fig. 4.22) the constraint corrected toughness will increase with increasing Lr . The application of the constraint correction is schematically illustrated in Fig. 6.32. Whilst the CDF plot directly shows the increasing toughness (Fig. 6.32b), an additional remark is necessary with respect of the FAD approach (Fig. 6.32a). The constraint effect can be considered in the assessment point or assessment path but also by using a modified FAD line, the ordinate of which, Kr , is replaced by a constraint modified term c Kr = fLr → Kr = fLr · Kmat Kmat (6.104) As a consequence, the modified FAD can exceed values of 1 near the Kr axis as shown in the figure. The benefit of the toughness increase due to the constraint reduction is a higher critical load or crack size.
6.7.2. Example of an Assessment Including Constraint effects Example 6.7: The constraint correction will be demonstrated for a tension loaded plate with a semi-elliptical surface crack comparable to that in Example 6.1. (b)
(a) Fracture Effect of constraint reduction
1
Increasing crack size Safe
Lr = F/FY = σ ref /σY 1 L max r
J-integral of CTOD, δ
Potentially unsafe Kr
Effect of constraint reduction
Fracture
Fracture toughness (Jmat or δ mat)
Increasing crack size
Lr = F/FY = σ ref /σ Y
1 L max r
Figure 6.32: Considering constraint effects. (a) FAD approach; (b) CDF approach.
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189
The dimensions of the plate are 2W = 200 mm and t = 10 mm. However, the crack is much smaller than in Example 6.1. Its depth is a = 15 mm and its geometry a/c = 024. The material parameters are: Y = ReL = 300 MPa, Rm = 500 MPa, E = 210 GPa and = 03 typical for a mild steel. A yield plateau is expected. The toughness is assumed to be Kmat = 180 MPa · m1/2 determined from deeply cracked C(T) specimens. It is further assumed that the fracture resistance is geometry dependent. Only an Option 1A analysis will be performed in the following. The deeply cracked C(T) specimen (a/W ≥ 05) is characterised by a normalised T-stress of B = T Y · ≈ 06 (6.105) with Y being the stress intensity factor geometry function of the specimen and the applied stress. This value refers to a slightly positive T-stress. In contrast, the T-stress is negative for the component as shown in Fig. 6.33 and this means that the constraint of the component is lower than that of the specimen. For the present example (a/t = 015 and a/c = 024), a ratio T/ = −051 is obtained for the centre point of the crack for which the further analysis will be carried out.
a/t = 0.15
–0.5
a/t = 0.25 –0.6
T/σ
a/t = 0.60 a/c = 0.24 –0.7
t Φ
a c –0.8
0
20
40
60
80
Angle Φ
Figure 6.33: Example 6.7: T-stress normalised by the applied stress as a function of crack depth a/t and the position along the crack front (a/c = 024) according to [6.10].
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Fitness-for-Service Fracture Assessment of Structures Containing Cracks
c Table 6.17: Example 6.7: T-stress and constraint corrected fracture toughness Kmat c and Jmat
F in kN 0 100 200 300 400 500 600 700 800
m in MPa
T in MPa
T/Y = · Lr
0 50 100 150 200 250 300 350 400
0 −255 −510 −765 −1020 −1275 −1530 −1785 −2040
0 −0085 −017 −0255 −034 −0425 −051 −0595 −0680
c Kmat in MPa · m1/2
18000 18280 19118 20516 22474 24990 28066 31701 35895
c Jmat in N/mm
14040 14480 15839 18240 21886 27062 34133 43547 55833
With the applied stress = m = F 2W · t
(6.106)
T is determined as a function of the applied load F and normalised by the yield strength Y as summarised in Table 6.17. Since all · Lr ≤ 0 only the second part of Eqn (4.81) c Kmat = Kmat · 1 + · − · Lr k (6.107) has to be applied. The coefficients are assumed as = 215 and k = 2 as in Fig. 4.26. For the given yield and tensile strength the plastic collapse limit is Lr max = 1333, the strain hardening exponent N = 012 and the Lüders’ strain = 0026. The Option 1A f(Lr ) function is summarised in Table 6.18. The FAD assessment path Kr Lr = fm , the CDF function J = fLr and c the constraint corrected toughness Jmat = fLr ) of an Option 1A analysis are summarised in Table 6.19 and illustrated in Fig. 6.34. Note that the assessment c point Kr is defined by the toughness Kmat and not by Kmat as the constraint correction is included in the FAD curve. The critical load was determined as Fc = 6581 kN (mc = 330 MPa) without constraint correction and Fc = 7490 kN (mc = 3745 MPa) with constraint correction. This is a reduction of conservatism of almost 14% due to the consideration of the constraint effect on the fracture toughness.
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Table 6.18: Example 6.7: Constraint corrected Option 1A fLr function F in kN 0 100 200 300 400 500 600 700 800
m in MPa
Lr
Kr (no correction)
0 50 100 150 200 250 300 350 400
0 0177 0354 0531 0708 0884 1061 1238 1415
1000 0992 0970 0936 0894 0848 0183 0104 0064
c Kr = fLr · Kmat /Kmat
1000 1008 1030 1067 1117 1177 0285 0183 0127
Table 6.19: Example 6.7: FAD assessment path (Kr , Lr = f(m , CDF function J = f(Lr c and constraint corrected toughness Jmat = fLr of an Option 1A analysis F in kN 0 100 200 300 400 500 600 700 800
m in MPa
Lr
Kr assessment point)
J in N/mm (Option 1A)
c Jmat
0 50 100 150 200 250 300 350 400
0 0.177 0.354 0.531 0.708 0.884 1.061 1.238 1.415
0 0.020 0.041 0.061 0.081 0.101 0.122 0.142 0.162
0 006 024 059 115 200 6215 26199 91106
140.40 144.80 158.39 182.40 218.86 270.62 341.33 435.47 558.33
6.8. Mixed mode loading 6.8.1. General aspects In Section 4.2.6 three criteria are assumed when a mixed mode analysis has to be performed: (1) There is an angle of 20 or greater between the principal stress plane and the plane of the actual flaw. (2) The difference between the stress intensity factors of the projected crack at the two principal stress planes is small. (3) The maximum stress intensity factor and the maximum limit load belong to two different projection planes.
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Fitness-for-Service Fracture Assessment of Structures Containing Cracks
(a)
(b) 1.8
1000 700
1.6
0.8
No constraint correction
0
600
F in kN:
0.4
500 300 400 100 200
Constraint corrected toughness
800
700 800
J-integral in N/mm
c
Kr = (Kmat /Kmat) ⋅ Lr
Constraint correction
600 500
F in kN:
400
100
200
200
400
300
700
600
No constraint correction
0
0
0.4
0.8
1.2
1.6
0
0.4
Lr
0.8
1.2
1.6
Lr
Figure 6.34: Example 6.7: Determination of the critical load with and without constraint correction of fracture toughness. (a) FAD approach; (b) CDF approach.
For all other cases the crack has to be projected onto the plane of the maximum principal stress and the analysis has then to be performed as a mode I analysis as described in Section 6.2–6.4. The basic crack opening modes I, II and III are defined in Fig. 6.35. In practical applications, mixed mode loading is quite frequent, which means that a structure is simultaneously loaded by more than one mode, for example, modes I and II. Since cracks tend to extend normal to the mode I direction, mixed mode loading only occurs under special geometry and applied loading conditions, such as in
Mode I (tension)
Mode II (in-plane shear)
Mode III (out-of-plane shear)
Figure 6.35: Basic crack opening modes.
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193
fillet welded joints and out-of-plane loading, and more frequently for small than for long cracks. Note that shear and mixed mode loading tend to generate lower hydrostatic stresses but higher strains near the crack tip. This can affect both the crack driving force in the component and the fracture toughness of the material. However, in contrast to mode I loading, no universally accepted test standards exist for mixed mode loading. Furthermore, pure mode II and mode III toughness determination is faced with problems such as crack face friction or, even more fundamentally by the definition of crack extension. There is, however, some indication that mode I loading, compared to mode II, mode III or mixed mode, leads to lower bound toughness values, except for cases where crack tip plasticity is very small, for example in very high strength/low toughness steels or in common structural steels at very low temperatures. Therefore SINTAP/FITNET makes a distinction between materials displaying toughnessyield strength ratios below and above Kmat /Y = 02 m1/2 . The situation for ductile behaviour is more complex than for brittle material behaviour, as mode II initiation and crack growth values have been reported to be both higher and lower than the mode I equivalent in a number of cases [6.11]. Because the reason for this discrepancy is not currently clear, stable crack extension is excluded from SINTAP/FITNET mixed mode analysis unless it is justified by empirical evidence. Basically, the assessment approach considers the mixed mode effect by modifying the crack driving force side. This is set out explicitly below.
6.8.2. FAD Analysis The FAD lines for the different assessment options stay unchanged in accordance with Eqn (6.1) and with the f(Lr ) functions introduced in Section 6.4. The mixed mode effect is, however, considered in the assessment point, that is, in the K-factor and yield load solutions. Instead of the mode I K-factor, KI , a mixed mode equivalent K-factor, Keq , is used which is determined by 1 2 2 1 − / Keq = KI2 + KII2 + · KIII
for Kmat Y ≥ 02 m1/ 2
(6.108)
for mode I fracture toughness. Eqn (6.108) is with Kmat being the general term 1/ 2 also used for Kmat Y < 02 m when there is evidence that the critical mode II fracture toughness is larger than Kmat . Otherwise 1 2 2 1 − / Keq = K12 2 + · KIII
for Kmat Y < 02 m1/ 2
(6.109)
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Fitness-for-Service Fracture Assessment of Structures Containing Cracks
with
3/ 2 2KI + 6 KI2 + 8 · KII2 KI2 + 12 · KII2 + KI KI2 + 8 · KII2 · K12 = 8 2 · KI2 + 18 · KII2 (6.110) for KI KII ≥ 0466 and K12 = KII 07
for KI KII < 0466
(6.111)
The parameter is generally assumed as = 1 in Eqns (6.108) and (6.109) unless there is detailed evidence on the effect of Mode III loading. Similarly, a mixed mode yield load solution FY has to be used for the Lr coordinate of the assessment point. The subsequent FAD analysis is identical to the basic approach as outlined in Sections 6.2 to 6.5 or in conjunction with combined primary and secondary stresses as in Section 6.6. Although theoretical evidence points to the possibility that the approach described might be nonconservative for mode II or III loading under certain circumstances [6.11], it is generally found from experimental data to be conservative. However, since this statement is only empirically based, some caution is advisable given the present state of knowledge.
6.8.3. CDF Analysis The KI factor has to be replaced by Keq according to Eqns (6.109) to (6.111) in Eqns (6.9) and (6.11) respectively. The crack driving force in terms of J or is then compared to the mode I fracture toughness. No example of a mixed mode loaded geometry is provided here, this will be covered in Section 6.10 in conjunction with a thin wall application.
6.9. Rapid Loading and Crack Arrest 6.9.1. General Aspects The SINTAP procedure is applicable to structures subjected to quasi-static loading rates and temperatures below the creep range. Its extension in FITNET provides
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195
additional information on phenomena such as crack arrest and creep crack extension. Although no comprehensive overview on these will be provided in this book, some important aspects will be addressed. The first term that has to be defined is “quasi-static loading” as the opposite or complement to “dynamic” or “rapid loading”. Unfortunately, a classification simply based on a limit loading rate or limit strain rate is not possible, since the effect depends on many factors such as the material or the geometry of the component under consideration. Certain loading events such as the collision of cars, vehicles or ships, impacts on excavator shovels during their use, cold rolling or drop forging are dynamic. In order to decide when a loading event is quasi-static and when it is dynamic, three basic effects of impact loading have to be understood [6.12]. (1) Inertia effects These occur when the load changes abruptly or when the crack grows rapidly. A portion of the applied load is converted to kinetic energy. (2) Transient stress waves effects Stress waves are initiated by an impact. They propagate through the material and reflect off free surfaces. The effect is a time-dependent oscillation of the stresses at the crack tip and the loss of proportionality of loading, that is, the requirement that the applied load should increase proportionally to the local load at any point in the vicinity of the crack tip is violated. After some time the waves subside and a quasi-static state is reached. (3) Effects on material properties Both the deformation properties of the material, for example, its yield strength and strain hardening exponent and the fracture toughness, will be affected by a strain rate increase of several orders of magnitudes. This is because the movement of dislocations is hindered at high strain rates in a comparable manner to the effect of low temperatures in materials of body-centred cubic or hexagonal lattices. As a rule, above the ductile-to-brittle transition and as long as creep effects do not occur, yield strength and strain hardening are higher and the ultimate strain is lower for increased strain rate (Fig. 6.36). Young’s modulus is only slightly affected. The effect of higher strain rates on fracture toughness becomes manifest in a shift of the ductile-to-brittle transition to higher temperatures. At the lower shelf the increase in the loading rate causes a decrease in fracture toughness and at the upper shelf it improves the material resistance against stable crack initiation and stable crack extension (the slope of the R-curve). Note, however, that the effects on the lower shelf toughness and on the resistance against stable crack initiation are rather moderate (Fig. 6.37).
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Fitness-for-Service Fracture Assessment of Structures Containing Cracks
(a)
(b) Increase in yield strength and strain hardening
Failure stress
Tensile strength
Stress
Increase in rate Decrease in ultimate strain Moderate effect on Young’s modulus
Yield strength
Brittle
Ductile range
Strain
Ductile
Temperature
Figure 6.36: Effect of an increase in the loading rate on the stress-strain curve of materials displaying ductile to brittle transition behaviour – (a) The yield strength and the strain hardening increase, but the ultimate strain decreases for higher strain rate; (b) The effect of (a) vanishes below the ductile to brittle transition. Note that loading rate can also affect the type of yield behaviour from discontinuous to continuous.
Crack resistance
(a)
(b) Ductile-to-brittle transition: Significant shift of transition temperature
Lower shelf: No or moderate effect
Stable crack initiation: No or moderate effect
Increased section size effect
Temperature
Upper shelf
Stable crack extension: Significant increase of R-curve slope
Stable crack extension Δa
Figure 6.37: Effect of an increase in loading rate on the fracture toughness of materials displaying ductile to brittle transition behaviour – (a) The ductile to brittle transition is shifted to higher temperatures; (b) The slope of the R-curve is increased. Note that there is also an effect of section size.
6.9.2. Quasi-Static vs. Dynamic Analysis Inertia effects are negligible and the loading state of a component can be regarded as quasi-static when the rise time, tr , of the applied load is much larger than a fundamental period of the component, which refers to its period of oscillation. In SINTAP/FITNET this condition is regarded as satisfied for tr / ≥ 3. In Charpy specimens, for example, is of the order of 30 s. The determination of for real components is, however, a sophisticated task that
Structural Assessment
197
requires numerical analysis beyond the scope of SINTAP/FITNET. Factors that have to be taken into account are the component and loading geometry, the specific way the load is applied, the elastic properties of the material including its specific wave propagation velocities, etc. For shorter rise times, tr / < 3, a dynamic analysis has to be performed in order to identify the peak load which can then be used for the analysis. In addition, the strain rate dependent fracture toughness and, where required, the transition temperature should be known. It is conservative to take no benefit from the increase of the strength parameters. Dynamic fracture toughness values can be determined by a number of methods such as Hopkinson bar tests, impact response curves (using Charpy size specimens) and alternatives, which will not be discussed here (see[6.13]). A proposal ˙ = dKI dt on the transition for simulating the effect of loading rate in terms of K temperature To of the Master Curve approach ˙ To st · ln K (6.112)
To = ˙ − ln K with = 99 · exp
T0 st 190
166
109 Y + 722
(6.113)
is provided in [6.14]. In Eqns (6.112) and (6.113) To st is the reference temperature ˙ is the loading rate To (see Section 4.4.5.2) for quasi-static loading rates in C, K 1/2 expressed in terms of K-factor increase in MPa · m /s and Y is the quasi-static yield strength in MPa.
6.9.3. Crack Arrest Crack propagation is a dynamic event even for quasi-static initiation. Crack arrest can occur as a consequence of: • a reducing stress gradient (possibly arising around a stiffener for example), • a temperature gradient, • crack propagation out of an enclosed brittle zone (such as a heat affected zone of a weld), • crack propagation into a region of higher toughness (e.g., from a surface hardened layer), • a load being applied for a very short duration.
198
Fitness-for-Service Fracture Assessment of Structures Containing Cracks
Structural integrity is based on the avoidance of stable or unstable crack initiation. Crack arrest considerations can, however, be complementary or supplementary to such an assessment as a second safety barrier. A methodology outlined in R6 [6.15] which is not explained here in detail is adopted in FITNET. What is essential is that the problem is treated in a quasi-static way but modified by an enhancement factor fs to the (quasi-static) crack driving force which allows for dynamic effects at the moment of arrest. Note that this supposes a quasi-static stress field before the arrest event, which sometimes will not exist. The quasi-static crack driving force multiplied by fs is compared to the crack arrest toughness KIa , as, for example, determined according to ASTM E 1221-88 [6.16] or based on alternative crack arrest temperature concepts [6.17]. The crack is arrested when fs · KI < KIa
(6.114)
The highest dynamic stress enhancement is expected when the dynamic stress intensity factor is well below the current value of the static K at the moment of arrest. This is, for example, the case when the crack grows at high speed under increasing static K through a brittle zone. In such a case, a value fs = 15 is considered as appropriate [6.18]; in most other cases it should be conservative. The R6/FITNET method is limited to mode I crack opening and predominantly elastic conditions, Lr < 1.
6.10. Thin Wall Structures 6.10.1. General Aspects The assessment of thin wall structures has to take into account a number of special features such as • Pronounced stable crack extension prior to failure, • Constraint and other issues that make any application of standard test methods for fracture toughness impossible or overly conservative, • Buckling as a failure mechanism competitive with fracture. (1) Pronounced stable crack extension prior to failure Pronounced stable crack extension prior to failure, sometimes combined with rather low crack initiation toughness, as in the case of aluminium alloys, frequently makes it impossible to base structural integrity on the avoidance of stable crack initiation. The SINTAP/FITNET thin wall assessment is, therefore, predominantly based on R-curve analysis. The recommended crack
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tip parameter 5 is a special definition of the crack tip opening displacement which is particularly suited for describing large amounts of stable crack extension because of its connection with typical crack growth parameters such as the crack tip opening angle, CTOA, (see [6.19]). (2) Low constraint fracture toughness Thin sheet materials cannot usually be tested on the basis of the available fracture mechanics test standards as they do not meet the requirements for plane strain conditions. This is the background of present ISO and ASTM standardisation activities with the aim of providing test methods for low constraint specimens [6.20]. The methods included are mainly based on the CTOA and CTOD-5 parameters. An essential advantage of 5 over the common plastic hinge based CTOD definition (see the Glossary: “crack tip opening displacement”) is its general applicability to specimens as well as thin sheet components due to its definition at the lateral surfaces of the plates, whereas the plastic hinge based parameter is restricted to deeply notched bending geometries. The constraint issue is addressed by two requirements: (a) The specimen thickness, B, has to be chosen to be identical to that of the component or the semi-finished product in order to provide identical out-of-plane constraint and material conditions. (b) As shown by several studies [6.19, 6.21], R curves for some typical thin sheet materials such as aluminium alloys tend to be independent of the in-plane dimensions of the specimens or components if the crack length, a, and the uncracked ligament length, W-a, are greater than about four times the thickness, B: a/B
and
W − a/B ≥ 4
(6.115)
Real thin wall components will usually meet this condition, which in many cases ensures geometry independent 5 -R curves if stable crack extension does not exceed a value of (see [6.22]) 025 W-ao C(T) and SE(B) specimens (6.116)
amax = M(T) specimens W − ao − B (3) Buckling as a failure mechanism competitive to fracture. Buckling is a failure mechanism competing with ductile instability and plastic collapse, both of which are implicitly considered in SINTAP/FITNET. This should always be taken into account as a typical feature of thin wall structural behaviour.
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Fitness-for-Service Fracture Assessment of Structures Containing Cracks
6.10.2. Thin Wall Assessment Module The SINTAP/FITNET thin wall module described in this section is restricted to flat sheets without stiffeners. However, it should also be applicable to more complex geometries when the model parameters K-factor and yield load are available or are specifically determined. The procedure uses Eqns (6.10) and (6.11) in conjunction with the 5 specification of the CTOD. The parameter is uniquely defined for laboratory specimens and components by the relative displacement of two gauge points which are located 5 mm apart on a straight line going through the original crack tip (Fig. 6.38). The SINTAP/FITNET thin wall module shares the limitation of the 5 concept given by Eqn (6.116). The 5 -R-curve as the toughness input parameter has to be obtained on specimens of component thickness and crack lengths, a, and initial uncracked ligament lengths, W-a, equal to, or greater than, four times the thickness (Eqn 6.115). The f(Lr ) functions are used as given in Section 6.4 except for Option 0 which is not defined for thin wall application. Because thin sheets are manufactured by rolling, their tensile properties may tend to be anisotropic – a moderate example is provided in Fig. 6.39. The deformation pattern at the crack tip is three-dimensional and therefore not adequately
Gauge length: 5 mm
δ5 + 5 mm
Gauge points
Figure 6.38: Definition of crack tip opening displacement 5 .
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201
A1 5083 H 321
Rm(ϕ)/Rm(L)
1.0
L Rolling direction
Rm(ϕ)/Rm(L); Rp0.2(ϕ)/Rp0.2(L)
1.1
0.9
0.8
22.5°
67.5°
T
L 0.7
0
Rp0.2(ϕ)/Rp0.2(L)
45°
22.5
45
T 67.5
90
Angle to rolling direction (ϕ) in degrees
Figure 6.39: Tensile properties Rp02 and Rm of an aluminium alloy as a function of specimen orientation with respect to the rolling direction (according to [6.23]).
characterised by the tensile test results. In order to avoid non-conservatism, it is recommended that the SINTAP analysis is based on the lowest stress–strain curve which, in the example, belongs to the 45 orientation to the rolling direction. The load versus stable crack extension characteristics and the maximum sustainable load are determined as described in Sections 6.3 to 6.5 (Figs 6.3, 6.6 and 6.8). Care has to be exercised for large ligament sizes for which global limit load solutions, regardless of the fact that the cracks are through wall, are not appropriate and may yield essential overestimation of the load carrying capacity. This is an even greater problem for plane bending geometries. It was mentioned above that R curves of aluminium alloys tend to be independent of the specimen or component dimensions when Eqn (6.115) is fulfilled. Note, however, that this statement cannot be generalised for any material. Therefore, the geometry independence of the R-curve has to be checked case by case. Where no geometry independence is found, the lowest R-curve has to be used and this is usually obtained from C(T) specimens. In order to avoid extrapolation of the 5 - a curve beyond its validity limits, the stable crack extension a at the maximum load in the component should not exceed the a range which is covered by the experimental R-curve. As a rule, the width W of C(T) type test specimens should not be smaller than about 150 mm. Stable crack extension was excluded in mixed mode analysis in Section 6.8 unless it could be shown that the mode I R-curve provided a lower limit than the other opening modes. A mixed mode example is given which below has been experimentally validated. There is, however, another aspect that has to be considered for mixed mode loading cases. Mixed mode cracks that are not affected by features such as
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Fitness-for-Service Fracture Assessment of Structures Containing Cracks
weldments, notches or geometrical transitions tend to change their growth direction after a certain amount of extension. This depends on factors such as the global mixed mode ratio and the ductility of the material. According to [6.24] the effect is more distinct in brittle than in ductile materials. As a consequence, the analysis should be done for both the mixed mode crack and the Mode I crack. The lower maximum load should then be chosen as the final result.
6.10.3. Examples of Thin Wall Assessments Example 6.8: The maximum or instability load of a biaxially tensioned aluminium alloy plate (Fig. 6.40) is to be determined. The stress–strain curve of the material is provided in Example 6.1 (Table 6.1 and Fig. 6.14). Its yield strength in the L direction is Y = Rp02 = 242 MPa, its tensile strength Rm = 350 MPa, the Young’s modulus E = 703 GPa and the Poisson ratio v = 033. The material does not display a yield plateau. However, since it shows some anisotropy (Fig. 6.39) the strength values at an angle to the rolling direction of 45 have to be chosen for the analysis. These are Rp02 = 225 MPa and Rm = 329 MPa. A 5 -R-curve is
Fy
2W
Fx
Fx 2a
t
Fy
Figure 6.40: Example 6.8: Biaxial tension loaded cruciform specimen with a mode I through crack in the centre.
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203
Table 6.20: Example 6.8: 5 -R-curve obtained from C(T) specimens, the thickness of which was identical to the component thickness
a in mm 0.02 0.57 1.73 3.55 6.40 8.61 12.47 17.87 21.93 25.33
5 in mm
a in mm
5 in mm
008 025 042 061 086 106 134 172 200 220
3219 3737 4262 4864 5547 6374 6870
256 282 304 327 355 391 413
provided in Table 6.20 and Fig. 6.41. The plate dimensions are 2W = 300 mm and t = 29 mm; the cross section contains a through crack of initial length of 2c = 60 mm. The biaxial loading ratio, the ratio of the horizontal to vertical loading, is = Fx /Fy = 05.
5
CTOD-δ 5 in mm
4
3
2
1
0
0
20
40
60
80
Stable crack extension Δa in mm
Figure 6.41: Example 6.8: 5 -R-curve obtained from a C(T) specimen the thickness of which was identical to the component thickness.
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Fitness-for-Service Fracture Assessment of Structures Containing Cracks
The stress intensity factor is determined by KI =
Fy √ · a 2·B·W
(6.117)
This solution is defined for the infinite plate under tension, that is, potential effects of the geometry and dimensions of the cruciform specimen are not taken into account. Note that these are excluded or at least significantly reduced by the slits around the centre part of the specimen (6.40). The yield load is obtained by −05 Y FY = 2 · B · W · 2 + 1 − a/W−2 − 1 − a/W−1
(6.118)
[6.25] with the biaxial loading ratio being = Fx /Fy . Based on these model parameters, the SINTAP/FITNET analysis follows the philosophy described in Fig. 6.8, that is, a load versus stable crack extension curve is simulated with the 5 R-curve as the controlling input information. The results are summarised in Table 6.21 and illustrated in Fig. 6.42. Instability occurs at a load in the y direction Fmax = 1745 kN. Table 6.21: Example 6.8: Load versus stable crack extension characteristics and ligament yielding parameter Lr Load Fy in kN 18.59 141.08 154.79 162.14 166.42 169.49 171.94 173.92 174.24 174.37 174.45 174.47 174.44 174.36 174.24 174.09 173.88
a in mm
Lr
Load Fy in kN
a in mm
Lr
0001 05 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80
0104 0789 0869 0915 0943 0965 0983 0999 1005 1011 1016 1021 1025 103 1034 1038 1041
17365 17334 17298 17262 16771 16208 15566 1487 14157
85 90 95 100 150 200 250 300 350
1045 1048 1051 1054 1076 1095 1111 1125 1139
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200
Fmax = 174.5 kN 160
Load Fy in kN
Fy 120
80
Fx
Fx
40
λ = Fx/Fy = 0.5 Fy 0
0
10
20
30
40
Stable crack extension Δa in mm
Figure 6.42: Example 6.8: Load versus stable crack extension characteristics and instability load of the thin walled specimen of Fig. 6.40.
Example 6.9: The biaxially tension loaded cruciform specimen of Example 6.8 contains a 45 crack (Fig. 6.43) which requires a mixed mode analysis in order to obtain the instability load. All other input parameters are identical to those of Example 6.8. Two stress intensity factors have to be determined for mode I and II loading. The first is obtained by KI =
√ 1 + + 1 − · cos 2 a 2
(6.119)
the second by KII =
√ 1 − · sin 2 a 2
(6.120)
[6.26]. Based on KI and KII a mixed mode equivalent K-factor Keq is then determined by Eqns (6.108) to (6.111). The mixed mode yield load is obtained from 1 FY = · 2 · B · W · Y 2 + 1 2 −
(6.121)
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Fitness-for-Service Fracture Assessment of Structures Containing Cracks
Fy
2W
Fx
45°
2a
Fx
t
Fy
Figure 6.43: Example 6.9: Biaxially tension loaded cruciform specimen with a mode I/II through crack in the centre.
with =
W − a W
(6.122)
and the projected crack length a a = a · cos
(6.123)
[6.25] (Fig. 6.44). Taking into account the possibility that the crack could deviate to the mode I plane during stable extension, a mode I analysis is performed in parallel with the mode II calculations. This is done as in Example 6.8 but for the projected crack length a = 2121 mm as the initial crack size. The results of both versions are displayed in Table 6.22 and Fig. 6.45. Note that the mode I version yields a slightly lower maximum load (Fmax = 1907 kN) than the mixed mode version (Fmax = 1940 kN). Thus Fmaxc = 1907 kN is taken as the final result.
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207
Fy
λ = Fx /Fy
2a′ ϕ Fx
2a
Fx
2W
Fy
Figure 6.44: Example 6.9: Mixed mode and biaxial loading. Definition of the projected crack length a .
250
194.0 kN (mixed mode) 190.7 kN (crack path deviation to mode I) Mixed mode
Fmax = min
Load Fy in kN
200
Fy 150
Mode I
100
Fx
Fx 50
Fy 0
0
10
20
λ = Fx/Fy = 0.5 30
40
Stable crack extension Δa in mm
Figure 6.45: Example 6.9: Load versus stable crack extension characteristics and instability load of the thin walled specimen of Fig. 6.43. The mode I analysis yields a slightly lower maximum load than the mixed mode analysis.
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Fitness-for-Service Fracture Assessment of Structures Containing Cracks
Table 6.22: Example 6.9: Load versus stable crack extension characteristics and ligament yielding parameter Lr for both the mixed mode and the mode I analysis Mixed mode analysis
Mode I analysis (crack path deviation)
Load Fy in kN
a in mm
Lr
Load Fy in kN
a in mm
Lr
21.67 159.78 174.13 181.80 186.32 189.60 191.05 191.74 192.23 192.64 192.98 193.27 193.50 193.69 193.83 193.93 193.98 194.00 193.93 193.82 193.69 191.09 187.77 183.61 178.87 173.98
0001 05 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 150 200 250 300 350
0112 0828 0905 0948 0974 0994 1005 1011 1017 1022 1027 1032 1036 1040 1044 1048 1051 1054 1057 1060 1062 1081 1097 1109 1118 1127
2209 16123 17503 18224 18632 18919 18994 19034 19054 19066 19072 19072 19068 19058 19044 19026 19003 18976 18943 18905 18865 18342 17756 17091 1652 15648
0001 05 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 150 200 250 300 350
0114 0837 0912 0954 0979 0998 1006 1013 1018 1023 1028 1032 1036 104 1044 1047 1051 1054 1057 1059 1062 108 1097 111 1121 1133
6.11. Strength Mismatch 6.11.1. The Strength Mismatch Phenomenon The phenomenon of strength mismatch has already been briefly addressed in Section 5.2.6 in the context of yield load determination. Strength mismatch means that welds or bi-material joints (sometimes designated dissimilar joints) exhibit
Structural Assessment
209
substantial heterogeneity in their strength properties. A common parameter for describing this heterogeneity is the mismatch factor based on the yield strengths of the weld metal and base plate, YW and YB , of weldments (6.124) M = YW YB or on the yield strengths of the higher and lower strength materials, Y (HS) and Y (LS), in bi-material joints M > 1 (6.125) M = Y HS Y LS If the weld metal is of higher strength than the base plate (M > 1) the phenomenon is designated as overmatching (OM). If the base plate is of higher strength (M < 1) it is called undermatching (UM). Note that, whereas in common applications the mismatch factor is based exclusively on the yield strength ratio, the complete stress–strain curves (or the strain hardening exponents) of the materials involved have to be considered in fracture mechanics applications. Three different types of strength mismatch of weldments are illustrated in Fig. 6.46. Common practice is overmatching with the aim of protecting the weld metal from high deformations. In this way the risk of failure from the weld defects will be minimised. Usually, overmatching is of the order of 10–30%, however, in power beam welds, it may be as high as 300%. Similarly, undermatching applications exist in engineering practice, for example, in joints of aluminium alloys or in joints of very high strength steels. Furthermore, unintentional undermatching may occur, such as when the base plate is, in reality, of much higher strength than required by the manufacturer’s specification.
Stress
(a)
(b)
Weld metal
Weld metal
Base plate σ YW
σ YW σ YB
(c)
Base plate
σ YB
σ YW
Weld metal
Base plate
σ YB
UM Overmatching (OM) Strain
OM
Undermatching (UM) Strain
Strain
Figure 6.46: Types of strength mismatch – (a) Overmatching; (b) Undermatching; (c) Initial overmatching changes to undermatching due to different strain hardening.
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Fitness-for-Service Fracture Assessment of Structures Containing Cracks
The question of whether overmatching is really beneficial for structural integrity depends on factors such as the location of the crack with respect to the weld and the toughness of the weld and adjacent material. In general, unfavourable combinations are cracks in the weld centre line in undermatched weldments and cracks near the fusion line (e.g., in the heat affected zone) in overmatched weldments.
6.11.2. The Strength Mismatch Options There are two strength mismatch options in SINTAP/FITNET. Option 2 is a variation of Option 1 for mismatch and Option 3 contains a specific module for this purpose. In general, strength mismatch is considered in SINTAP/FITNET when the yield strength differences are more than 10% (M < 09 and M > 11). If, for example, no detailed input information is available on the strength parameters of the materials, it is conservative to base the complete analysis on the properties of the lower strength material. In the following the f(Lr ) functions of the mismatch options are presented. They are used in conjunction with the basic SINTAP/FITNET equations for the FAD (Eqn 6.1) and CDF (Eqns 6.8 and 6.10). Whilst the K-factor is unaffected by strength mismatch the effect has to be taken into account by mismatch corrected yield loads (Section 5.2.6). At Option 2 it is necessary to distinguish whether both materials (Option 2A), neither material (Option 2B), or just one material (Option 2C) exhibit a yield plateau. Note that in the following the upper yield strength ReH is applied instead of the lower yield strength ReL for determining the factor and the Lüders’ strain . The other parameters including the mismatch corrected yield loads are still based on ReL . (a) Option 2A: Both materials expected to display a yield plateau (YB = Re B YW = Re W ) −1 2 fLr = 1 + 05 · Lr2 /
−1/ 2 1 fLr = M + 2M
for 0 ≤ Lr ≤ 1
(6.126)
for Lr = 1
(6.127)
fLr = fLr = 1 · LrNM −1/ 2NM with
for 1 ≤ Lr ≤ Lr max
FYM FYB − 1 · W + M − FYM FYB · B M = M−1
(6.128)
(6.129)
Structural Assessment
B = 1 + EB B ReH B W = 1 + EW W ReH W
B = 00375 · 1 − ReH B 1000 ReH B in MPa
W = 00375 · 1 − ReH W 1000 ReH W in MPa
211
(6.130) (6.131) (6.132) (6.133)
M−1 NM = FYM FYB − 1 NW + M − FYM FYB NB
(6.134)
NB = 03 · 1 − ReL B Rm B NW = 03 · 1 − ReL W Rm W
(6.135) (6.136)
and the plastic collapse limit Lr max Lr
max
= 05 · 1 +
03 03 − NM
(6.137)
Note that Eqn (6.137) has been replaced by 05 · 1 + R R F m B eL B Lr max = YM min FYB 05 · 1 + Rm W ReL W
(6.138)
in FITNET compared to SINTAP, whereas R6-Rev.4 retained the original equation. The numerical results of both equations are almost identical. (b) Option 2B: Both materials not expected to display a yield plateau (YB = Rp02 B ; YW = Rp02 W ): −1 2 fLr = 1 + 05 · Lr2 / · 03 + 07 · exp −M · Lr6 fLr = fLr = 1 · LrNM −1/ 2NM
for 0 ≤ Lr ≤ 1 (6.139)
for 1 ≤ Lr ≤ Lr max
⎧ M−1 ⎪ ⎨ FYM FYB − 1 W + M − FYM FYB B M = min ⎪ ⎩ 06
(6.140)
(6.141)
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Fitness-for-Service Fracture Assessment of Structures Containing Cracks
B = min
0001 · R EB p02 B 06
W = min
(6.142)
0001 · R EW p02 W 06
(6.143)
The mismatch corrected strain hardening exponent, NM , and the plastic collapse limit, Lr max , are given by Eqns (6.134) to (6.136) and Eqn (6.137) or (6.138), but replacing ReL B by Rp02 B and ReL W by Rp02 W . (c) Option 2C: Base or weld material expected to display a yield plateau The Option 2C f(Lr ) curves are combinations of the Option 2A and Option 2B equations. Basically for 0 ≤ Lr ≤ 1 modified Option 1B expressions, and for Lr = 1 modified Option 1A expressions, are used. This means in detail: (c-1) Only the base plate expected to display a yield plateau (YB = Re B and YW = Rrp02 W ) −1 2 fLr = 1 + 05 · Lr2 / · 03 + 07 · exp −M · Lr6
for 0 ≤ Lr ≤ 1 (6.144)
−1/ 2 1 fLr = M + 2M
for Lr = 1
fLr = fLr = 1 · LrNM −1/ 2NM
for 1 ≤ Lr ≤ Lr max
(6.145) (6.146)
with ⎧ M−1 ⎪ ⎨ FYM FYB − 1 W M = min ⎪ ⎩ 06 W = min
0001 · R EW
p02 W
06
M − FYM FYB · B M = M−1
(6.147)
(6.148)
Structural Assessment
B = 1 + EB B ReH B
213
(6.149)
B = 00375 · 1 − ReH B 1000 ReH B in MPa
(6.150)
The mismatch corrected strain hardening exponent, NM , and the plastic collapse limit, Lr max , are given by Eqns (6.134) to (6.136) and Eqn (6.137) or (6.138), but replacing ReL W by Rp02 W . (c-2) Only the weld metal expected to display a yield plateau (YB = Rrp02 B and YW = Re W ) −1 2 fLr = 1 + 05 · Lr2 / · 03 + 07 · exp −M · Lr6
−1/ 2 1 fLr = M + 2M
for Lr = 1
fLr = fLr = 1 · Lr NM −1/ 2NM with
for 1 ≤ Lr ≤ Lr max
⎧ M−1 ⎪ ⎨ FYM FYW − 1 B M = min ⎪ ⎩ 06
for 0 ≤ Lr ≤ 1 (6.151)
(6.152)
(6.153)
(6.154)
0001 · R EB p02 B B = min 06 FYM FYB − 1 · W M = M−1
(6.155)
W = 1 + EW W ReH W
(6.156)
W = 00375 · 1 − ReH W 1000 ReH W in MPa
(6.157)
The mismatch corrected strain hardening exponent, NM , and the plastic collapse limit, Lr max , are given by Eqns (6.134) to (6.136) and Eqn (6.137) or (6.138), but replacing ReL B by Rp02 B .
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Fitness-for-Service Fracture Assessment of Structures Containing Cracks
6.11.3. Examples of Option 2 Analysis Example 6.10: A tension loaded welded plate with a through-thickness crack in the centre line of the weld, as shown in Fig. 6.47, consists of two materials, the base plate, with a yield strength YB = Rp02 B = 300 MPa and a tensile strength Rm B = 450 MPa, and the weld metal with a yield strength YW = Rp02 W = 450 MPa and a tensile strength Rm W = 550 MPa. No yield plateau is expected. The Young’s modulus and the Poisson ratio are E = 210 GPa and v = 03 for both materials. The toughness of the weld metal is assumed as Jmat = 100 N/mm, which gives Kmat = 1519 MPa · m1/2 . The plate dimensions are 2W = 200 mm, t = 10 mm and the weld height is 2H = 10 mm. The concern is the critical load for an all base material and mismatch analysis when the crack size is assumed as 2a = 30 mm. The K-factor is determined by 2 4 √ a KI = m · a · 1 − 0025 a W + 005 a W · sec · (6.158) 2 W [6.27] and the plane stress mismatch corrected yield load by pen base FYM = min FYM FYM
(6.159)
Base plate Crack
Weld metal Base plate
2H
2a
2W
t
Figure 6.47: Example 6.10: Configuration of the tension loaded plate with a through-thickness crack at the centre of the weld. The weld height is designated 2H.
Structural Assessment
215
[6.28] with the indices “pen” and “base” referring to the deformation patterns in Fig. 5.15(c) for “pen” (= penetrating to the base plate) and Fig. 5.15(d) for “base” (= base plate deformation). The solution is identical to Eqns (5.97) and (5.98) in Section 5.2.6 but this time in original ETM nomenclature. ⎧ ⎨M pen FYM = FYB · 24 M − 1 1 M + 24 ⎩ · + 25 25 = W − a H
for ≤ 1 for > 1
(6.161)
1 = 1 + 043 · exp −5 · M − 1 · exp − M − 1 5
base = 2 · W · t · YB FYM
(6.160)
for all
(6.162) (6.163)
FYB = 2 · t · W − a · YB
(6.164)
The results of the all base material analysis are summarised in Table 6.23. With N = 01 and Lr max = 1250 a critical load of Fmax = 550 kN is obtained (Fig. 6.48). Table 6.23: Example 6.10: fLr function, FAD assessment path and CDF J-integral values for the all base material analysis Load in kN 0 50 100 150 200 250 300 350 400 450 500 550 600 650 700
Lr
fLr
Kr (assessment pt.)
J in N/mm
0000 0098 0196 0294 0392 0490 0588 0686 0784 0882 0980 1078 1176 1275 1373
1000 0998 0991 0979 0962 0939 0907 0861 0795 0702 0584 0398 0269 0188 0134
0000 0036 0072 0109 0145 0181 0217 0253 0290 0326 0362 0398 0435 0471 0507
000 013 053 123 227 371 573 866 1329 2155 3841 10031 26122 63007 142371
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Fitness-for-Service Fracture Assessment of Structures Containing Cracks
(b)
(a)
400 Mismatch analysis
1.0
300
0.8 Option 1B
0.6
J-integral in N/mm
Mismatch analysis
Kr
600 500 400
0.4
F in kN 300 200
0
0
0.2
0.6
0.8
1.0
1.2
Option 1B
t
All base plate analysis
F in kN: 600
550 Jmat = 100 N/mm
All base plate analysis
0.4
600
2W
200
100
100
0.2
2H
2a
100 200
300
400 500
550 500
1.4
0
0
0.2
0.4
Lr
0.6
0.8
1.0
1.2
1.4
Lr
Figure 6.48: Example 6.10: Determination of the critical load by all base material and Option 2B mismatch analysis. (a) FAD approach; (b) CDF approach.
The Option 2B mismatch analysis yields the following results: N = 0093; Lr max = 1163 (FITNET), Lr max = 1224 (R6); critical load Fmax = 569 kN. The f(Lr ) function, the FAD assessment path and the crack driving force function are summarised in Table 6.24. Example 6.11: The welded plate of Example 6.10 is re-assessed but with the crack at the fusion line between the weld metal and base plate (Fig. 6.49). The geometry and material parameters are identical to Example 6.10. The K-factor is determined by Eqn (6.158) and the mismatch corrected yield load by Eqn (6.159), but with pen FYM = FYB · 1095 − 0095 · exp M − 1 0108
for all
(6.165)
base the parameters FYM and FYB are determined as above by Eqns (6.163) and (6.164). The Option 2B mismatch analysis yields the following results: N = 0086; Lr max = 1261 (FITNET), Lr max = 1202 (R6); critical load Fmax = 589 kN. The f(Lr ) function, the FAD assessment path and the crack driving force function are summarised in Table 6.25 and Fig. 6.50. Note that for both examples the differences between the conservative all base material analysis and the mismatch analysis were quite small.
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217
Table 6.24: Example 6.10: fLr function, FAD assessment path and CDF J-integral values for the Option 2B mismatch analysis Load in kN 0 50 100 150 200 250 300 350 400 450 500 550 600 650 700
Lr
fLr
Kr (assessment pt.)
J in N/mm
0000 0094 0187 0281 0375 0468 0562 0656 0749 0843 0937 1031 1124 1218 1312
1000 0998 0991 0981 0966 0945 0917 0878 0823 0745 0643 0486 0318 0215 0150
0000 0036 0072 0109 0145 0181 0217 0253 0290 0326 0362 0398 0435 0471 0507
000 013 053 123 225 367 561 833 1240 1913 3170 6705 18678 47931 114701
Base plate Weld metal Base plate Crack
2H
2a
2W
t
Figure 6.49: Example 6.11: Configuration of the tension loaded plate with a through-thickness crack at the fusion line between the weld metal and base plate. The weld height is designated 2H.
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Fitness-for-Service Fracture Assessment of Structures Containing Cracks
Table 6.25: Example 6.11: fLr function, FAD assessment path and CDF J-integral values for the Option 2B mismatch analysis Load in kN 0 50 100 150 200 250 300 350 400 450 500 550 600 650 700
Lr
fLr
Kr (assessment pt.)
J in N/mm
0000 0090 0179 0269 0358 0448 0538 0627 0717 0806 0896 0986 1075 1165 1255
1000 0998 0992 0982 0969 0950 0926 0892 0846 0780 0694 0587 0387 0254 0171
0000 0036 0072 0109 0145 0181 0217 0253 0290 0326 0362 0398 0435 0471 0507
000 013 053 122 224 363 551 807 1174 1743 2724 4608 12589 34426 87374
(b)
(a)
400 1.0
700 Mismatch analysis
300
0.8
2H
J-integral in N/mm
Option 1B
700
0.6
Kr
600 500
0.4
Mismatch analysis
400 600
F in kN 300 200
Option 1B
2W
200
t
All base plate analysis
F in kN: 600 Jmat = 100 N/mm
100
0.2 100
600
2a
All base plate analysis
100
500 200 300 400
550
550 500
0 0
0.2
0.4
0.6
0.8
Lr
1.0
1.2
1.4
0
0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Lr
Figure 6.50: Example 6.11: Determination of the critical load by all base material and Option 2B mismatch analysis. (a) FAD approach; (b) CDF approach.
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219
6.11.4. The Option 3 Mismatch Module The mismatch module of Option 3 uses the general f(Lr ) functions in a modified form f Lr =
E · ref M 1 Lr2 · ref M + Lr · Y M 2 E · ref M
−1/ 2
for 0 ≤ Lr ≤ Lr max
(6.166)
and Lr max = f M Y M
(6.167)
True stress
(see Section 6.4.4), in conjunction with a mismatch corrected true stress–strain curve. In Eqns (6.166) and (6.167) ref M and ref M label the reference stress and strain points on this curve, Y M designates the mismatch corrected yield strength and f M a mismatch corrected flow stress (average of yield and tensile strength). The mismatch corrected stress–strain curve is obtained from the tensile data of base plate and weld metal point-by-point described by (ref B ; ref B ) and
Weld metal Mismatch equivalent material
σ YW σ Y,M σ YB
Mismatch factor M(εp)
Base plate
M(εp) = σ ref,W(εp)/σ ref,B(εp) 0.2% Plastic strain εp
Strain
Figure 6.51: Assessment option 3, Strength mismatch module: Construction of the mismatch-corrected true stress-strain curve and the mismatch factor as a function of plastic strain.
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Fitness-for-Service Fracture Assessment of Structures Containing Cracks
(ref W ; ref W ) values (see Fig. 6.51). ref M p =
1 Mp − 1
·
FYM FYB − 1 · ref W p
+ Mp − FYM FYB · ref B p
(6.168)
Note that, in contrast to its use in Option 2, the mismatch ratio M in Eqn (6.168) is defined not only at the yield strength (here generally described as the stress at 0.2% plastic strain) but as a function of the plastic strain p of the plastic branch of the stress–strain curve: Mp = ref W p ref B p
(6.189)
Since FYM /FYB is defined for Mp it also depends on the plastic strain. The mismatch corrected yield strength is obtained as Y M =
FYM · FYB YB
(6.190)
and the mismatch corrected flow stress as f M =
FYM p · ref B p FYB
(6.191)
with p being the lower of the plastic strains at the flow stresses of base plate and weld metal.
6.11.5. Further Aspects of Strength Mismatch 6.11.5.1. Mismatch and Weld Geometry The mismatch corrected yield load solutions available in SINTAP/FITNET and other compendia refer to idealised prismatic weld geometries. However, in reality, welds have more complicated shapes such as V, X, Y etc. This poses the problem of how to define the weld strip dimension 2H in such cases. No general solution is available yet, but a promising proposal [6.29] defines an equivalent H, Heq , on the basis of the shortest distance between the crack tip and the fusion line along the slip lines emanating from the crack tip (Fig. 6.52). Note, however, that systematic investigations on this topic are still needed. In case of doubt, Heq should be chosen such that the resulting yield load is conservative.
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Heq
221
Heq
45°
45°
Figure 6.52: Proposed definition of an equivalent weld strip width Heq based on the shortest distance between the crack tip and the fusion line along the slip lines emanating from the crack tip.
6.11.5.2. Mismatch and Scatter in Deformation Properties
Strength
The mismatch assessment is implicitly based on the assumption of a constant mismatch factor across and along the weld. Reality is, however, often more complicated. For example, instead of clearly defined base plate and weld metal yield strengths, a mismatch pattern, such as schematically illustrated in Fig. 6.53, can occur which also shows variable strength properties in the heat affected zone or even in sub-areas of it. An impression of the mismatch pattern can be obtained by micro hardness measurements. Care has to be exercised when there is an
Base plate
Weld
Strength
HAZ
Figure 6.53: Possible variation of the mismatch factor across and along a weld (schematic).
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Fitness-for-Service Fracture Assessment of Structures Containing Cracks
indication of wide scatter of the mismatch factor. In such cases analyses based on the tensile properties of the material displaying the lowest strength allow conservative estimates. 6.11.5.3. Mismatch and Constraint The effect of constraint on toughness has already been discussed in Section 4.4.6 where it was also mentioned that strength mismatch is a factor that additionally affects the triaxiality of the stress state as a measure of constraint. An example for this is shown in Fig. 6.54 [6.30]. With increasing crack depth a/W the stress triaxiality increases until it reaches a plateau value. The geometry effect is, however, magnified by a mismatch factor M > 1 in the lower strength material of the bi-material joint. It was theoretically shown in [6.30] that this magnification can be up to 38% for high mismatch factors. No specific rule is provided in SINTAP/FITNET on how to deal with mismatch induced constraint effects. Its potential existence should, however, be kept in mind, in particular in toughness testing, which should be performed on specimens that have similar mismatch patterns to the component.
3.0
Maximum mismatch effect
2.0
Geometry effect
1.5
W Metal B
M=1
2.5
a
1.0 Metal A
Stress triaxiality in the lower strength material
3.5
0.5
0
0
0.2
0.4
0.6
0.8
1
a/W
Figure 6.54: Stress triaxiality (parameter h = m /Y LS=0 in the lower strength material (LS) for a highly mismatched bimaterial joint for a single edge cracked plate in pure bending compared with that for a homogeneous specimen (according to [6.30]).
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223
6.11.5.4. Mismatch and Fracture Toughness Mismatch effects have not only to be considered in the determination of the crack driving force in components but also in fracture toughness testing. Although this aspect is not covered by SINTAP/FITNET, some remarks are appropriate. The consideration of strength mismatch in fracture toughness testing is not currently very common. It is included in the GKSS test procedure EFAM GTP 02 [6.31] that forms the basis for the forthcoming European procedure ESIS P3 [6.32]. An example is provided by the following equations for the Jo integral in SE(B) (bend) specimens. The index “o” indicates in ESIS nomenclature that a correction for the stable crack extension during the test has still to be carried out. Jo is determined by Jo =
K2 A + CMOD · pl E B · W − a
(6.192)
with A being the area under the force-crack mouth opening (CMOD) record and pl a calibration function. Eqn (6.192) is applied for all H/(W-a) ratios if 09 ≤ M ≤ 125 for weld metal cracks and 09 ≥ M for heat affected zone (HAZ) cracks. Within these conditions the error in J is less than 10%. However, beyond these limits a mismatch corrected pl factor has to be applied, based on the information in Fig. 6.55. Similar solutions exist for M(T) (tension) specimens.
Weld metal crack
4.0
Heat affected zone crack
M = 0.9; 0.8; 0.65; 0.5 decreasing M
3.5
M = 0.9; 0.8; 0.65; 0.5 decreasing M
ηpl
UM
3.0
M=1
2.5
for all M > 1
increasing M M = 1.1; 1.25; 1.5; 2.0 2.0
0
0.1
0.2
0.3
H/(W-a)
0.4
0.5
0.6 0
0.1
0.2
0.3
OM 0.4
0.5
0.6
H/(W-a)
Figure 6.55: pl factors for cracks in bend specimens for weld and heat affected zone (HAZ) cracks as a function of the mismatch factor M, the height of the weld metal strip, H, and the ligament size, W-a. The solutions refer to plane strain conditions and a crack depth a/W = 05 (according to [6.31, 6.32]).
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Fitness-for-Service Fracture Assessment of Structures Containing Cracks
6.12. Weld Shape Imperfections 6.12.1. Weldment Specific K Solutions Weldments usually have shape imperfections, in particular when they are manufactured manually. The dimensions of the weld vary within relatively high tolerances, which have to be covered on the safe side in fracture mechanics analysis. A collection of K solutions for typical weldments is provided in [6.33]. An example of a typical weldment is given in Fig. 6.56. By systematic investigations on the K-factor for this configuration it was found [6.34] that the effect of the wall thickness ratio t/T was quite small (less than 5% for t/T < 2) and the same was true with respect to the weld thread, A. In contrast, the effect of the transition angle , modelled by the weld leg dimensions H and W was found to be considerable. The K-factor solution proposed is √ K = m · a · Y · Mk
(6.193)
with Y being the geometry function for the single edge notched plate and Mk being a magnification factor due to the weld. The latter is given by Mk = C ·
a k
(6.194)
t
with
2 H W H C = 08068 − 01554 · + 00794 · + 00429 · T T T
t
θ
H W
a A
T
Figure 6.56: Transverse, non-load carrying attachment.
(6.195)
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225
and
2
H H W k = −01993 − 01839 · + 00495 · + 00815 · T T T
(6.196)
for 02 ≤ H/T ≤ 1, 02 ≤ W/T ≤ 1, 15 ≤ ≤ 60 . 0175 ≤ A/T ≤ 072 and 0125 ≤ t/T ≤ 24. Note that the solution refers to a long surface crack and can be unduly conservative for other crack configurations, for example, semi-elliptical surface cracks.
6.12.2. Misalignment Another effect that has to be taken into account in a SINTAP/FITNET analysis is axial (centricity) or angular misalignment caused either by detail design or by poor fabrication or welding distortion. Some typical examples are shown in Fig. 6.57. The presence of misalignment increases (or decreases) the stress at the joint due to the introduction of local bending stresses that have to be considered in both K-factor and yield load determination. In general, these local stresses will depend not only on the type and extent of the misalignment but also on the loading conditions (whether the plate is restrained or not) and on the geometrical
(a)
(b)
(c)
Figure 6.57: Examples of axial and angular misalignment – (a) Axial misalignment between flat plates; (b) Angular misalignment between flat plates; (c) Angular misalignment in a fillet welded joint.
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Fitness-for-Service Fracture Assessment of Structures Containing Cracks
complexity of the component, for example, with respect to stiffeners. Therefore, in the general case, a finite element analysis is required. For a number of simple cases, such as axial or angular misalignment between flat plates of identical or different thickness, longitudinal seam welds in tubes, pipes and vessels, ovality in pressurized pipes or vessels and axial or angular misalignment in butt and fillet welded joints, SINTAP/FITNET contains an appendix of analytical solutions [6.35] which was taken from other documents including [6.33] and [6.36]. These are based on a misalignment magnification factor km defined as km = 1 + s /m
(6.197)
In Eqn (6.197), s defines the maximum induced bending stress due to misalignment which has the same sign as the membrane stress m . For components loaded by combined tension and bending, the equation is used in conjunction with only the membrane stress. s always refers to the weld toe except for one case, that of a cruciform fillet weld. Note that the km solutions will be (sometimes unduly) conservative when applied to through thickness cracks.
6.12.3. Example of an Assessment Taking into Account Misalignment Example 6.12: A tension loaded plate with angular misalignment as shown in Fig. 6.58 with y = 5 mm and 2 = 300 mm. The plate dimensions are 2W = 200 mm and t = 10 mm, and the length of the through crack is 2a = 20 mm (Fig. 6.59). The tensile data are identical to Example 6.1, so Rp02 = 240 MPa, Rm = 617 MPa, E = 195 GPa and v = 03, and the toughness data are given by Jmat = 100 N/mm or Kmat = 1464 MPa · m1/2 . The misalignment analysis will be performed as an Option 1B analysis. The critical load is required.
t
α y 2l
Figure 6.58: Example 6.12: Definition of the misalignment parameters.
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227
B A
u
2h
2c
2W
t
Figure 6.59: Example 6.12: Plate and crack dimensions.
The misalignment-induced bending stress due to the straightening of the plate under tensile loading is obtained by ⎧ tanh 2 ⎪ 3 · 3y tanh 2 ⎪ ⎪ = · fixed ends ⎨ s t 2 t 2 2 = (6.198)
m ⎪ tanh 3 · · tanh 6y ⎪ ⎪ ⎩ = pinned ends t t with 2· = t
3m E
1/ 2 alpha in radians
and will be applied for fixed ends. The K-factor solution is √ KI = · a · m + fb · b
(6.199)
(6.200)
with fb being 1 at the front surface and −1 at the back surface [6.37]. Since this solution is obtained for an infinite plate, the plate under consideration here should be large compared to the crack size so that no edge effects influence the result. Limiting conditions are c/W < 015 and 2c/h < 03. The yield load is obtained from [6.38] b2 b + + m2 3 9 Lr = (6.201) Y 1 − c W
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Fitness-for-Service Fracture Assessment of Structures Containing Cracks
Table 6.26: Example 6.12: fLr function, FAD assessment path and CDF J-integral values of the analysis neglecting misalignment Membrane stress m in MPa 0 40 80 120 160 200 240 280 320 360 400 440 480
Lr
fLr
Kr (assessment pt.)
J in N/mm
0000 0167 0333 0500 0667 0833 1000 1167 1333 1500 1667 1833 2000
1000 0993 0973 0937 0872 0752 0559 0396 0294 0226 0179 0145 0119
0000 0048 0097 0145 0194 0242 0291 0339 0387 0436 0484 0533 0581
000 024 099 241 494 1038 2706 7320 17333 37074 73191 135413 237464
For comparison, the analysis was carried out without and with consideration of the misalignment effect. The first is summarised in Table 6.26 and the second in Table 6.27. Without misalignment the critical membrane stress is determined as mc = 294 MPa; with misalignment as mc = 172 MPa. There is a reduction in Table 6.27: Example 6.12: fLr function, FAD assessment path and CDF J-integral values of the analysis including misalignment Membrane stress m in MPa 0 40 80 120 160 200 240 280 320 360
Bending stress b in MPa
Lr (assessment pt.)
Kr (assessment pt.)
J in N/mm
0 57 110 159 204 246 285 322 357 390
0000 0264 0520 0767 1007 1242 1472 1697 1918 2137
0000 0118 0230 0337 0440 0540 0636 0729 0820 0908
0 144 611 1741 6415 24501 72421 179638 392333 778106
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229
(b)
(a)
1000 Option 1B
1.0
Option 1B 320
Misalignment
800 J-integral in N/mm
Kr
240
0.6 160
400
Membrane stress σm in MPa:
0.4
320
80
Misalignment
400
80
0.5
Membrane stress σm in MPa: 160 80
200
240
0
600
No misalignment
0.2
0
400
240
0.8
No misalignment 320 Jmat = 100 N/mm
160
240
1.0
Lr
0 1.5
2.0
0
0.5
1.0
1.5
2.0
Lr
Figure 6.60: Example 6.12: Comparison of the analysis with and without considering the misalignment effect. (a) FAD approach; (b) CDF approach.
the load carrying capacity of about 40%. It should be noted that, as mentioned above, the analysis could be significantly conservative for through wall crack configurations as in the present example. Both results are illustrated graphically in Fig. 6.60.
6.13. Reliability Aspects and Significance of the Results 6.13.1. General Aspects The failure assessment methodology described in Chapter 6, so far has been deterministic, that is, a single set of input parameters leads to a single result in terms of critical load, critical crack size or minimum required toughness. However, in reality, the input data will usually show some uncertainty and scatter that have to be taken into account since they make the result of the analysis less precise. The comments provided in this section describe the basic principles of how the significance of a SINTAP/FITNET analysis can be assessed and how it can be supported by statistical methods. For more detailed information the reader is referred to the more comprehensive presentations in [6.39, 6.40].
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Fitness-for-Service Fracture Assessment of Structures Containing Cracks
6.13.2. Reserve Factors and Sensitivity Analysis Reserve factors are margins against failure that can be defined for any input parameter such as load, crack size, toughness, etc. The load reserve factor FL is the ratio of the critical load to the assessment load, the crack size reserve factor Fa is the ratio of the critical crack size to the crack size at the assessment point and the toughness reserve factor FK is the ratio of the toughness at the assessment point to the toughness that would move this point to the FAD failure line. Similar reserve factors can be defined for J-integral and CTOD toughness parameters, the yield strength of the material, the magnitude of residual stresses etc. Reserve factors Fx (Fx > 1) can be applied in conjunction with safety factors Sx required in plant specific codes such that Fx ≥ Sx where the exponent x stands for the parameter to be assessed. Note, however, that the meaning of the reserve factor is limited if it is not complemented by a sensitivity analysis. The latter provides the user with information on how sensitive the result of the assessment is to variations in the input parameters. Even a large reserve factor can be meaningless if minor changes in the respective input parameter move the assessment point across the assessment line (FAD approach) or the crack driving force above the toughness limit (CDF approach). The sensitivity analysis on its part makes sense only in conjunction with the known variability (or uncertainty) of the parameter. Note that, although reserve factors and sensitivity analysis provide qualitative rather than quantitative information, they are of immense value for any application as they give the user an immediate indication of how trustworthy his results are and whether he should confirm them by statistical means or not. Therefore, sensitivity analysis is recommended for any application.
6.13.3. Reliability Analysis In order to guarantee reliable analyses that are on the safe side on the one hand, but are not overly conservative on the other, the users have various methods at their disposal: (1) A deterministic lower (or upper) bound analysis using lower (or upper) bound values to all input parameters such as toughness, crack size, applied loading, residual stresses etc. can be used. (2) A probabilistic analysis using statistical distributions of the input parameters can be carried out. This will give failure probabilities of the component under consideration that can then be checked against required target failure probabilities or can used in a subsequent risk analysis.
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231
(3) Statistically derived partial safety factors can be applied to all input parameters, to ensure that a given target failure probability will not be exceeded. The assessment itself is performed deterministically. In real applications these different possibilities will frequently be combined, that is, some input parameters will be lower bounds whereas others will be introduced as statistical distributions. A common probabilistic analysis is Monte Carlo simulation which is, however, frequently replaced by more efficient but approximate methods such as the First or Second Order Reliability Methods (FORM/SORM) [6.40, 6.41]. These are available as commercial and in-house computer codes. The variations in the input parameters can be described by different types of statistical distributions. Commonly, a normal, lognormal, or Weibull distribution is used for the lower shelf and ductile-to-brittle transition toughness. Within SINTAP/FITNET a Weibull distribution is used in the framework of the Master Curve approach. This is given in terms of scale, shape and shift parameters or in terms of a mean value and standard deviation in Section 4.4.5.2.2. On the upper shelf the lognormal distribution seems to be appropriate [6.42], but the normal distribution is applied as well [6.43]. The best fit of yield and ultimate strengths is provided by a lognormal distribution [6.44]. Only limited information is available on statistical variations of the crack size determined by NDI methods. The distribution most often used for this parameter is the normal distribution. A number of studies came to the conclusion that small crack sizes tend to be overestimated by NDI, whereas large cracks tend to be underestimated [6.41]. Where no empirical data are available, reference data for the coefficient of variation COV and the ratio of standard deviation to mean value are proposed for various input parameters in [6.41]. These and some additional information are summarised in Tables 6.28 and 6.29.
6.13.4. Example of a Simplified Reliability Analysis Example 6.13: A simplified reliability analysis will be performed with the fracture toughness being the only input parameter modelled by a statistical distribution. No special probabilistic software is needed in this case. A flat plate containing a surface crack as in Examples 5.7 and 6.1 is loaded by tension. The plate dimensions are 2W = 2000 mm and t = 50 mm and the crack dimensions are a = 30 mm and a/c = 02. The material is a ferritic steel showing a yield strength Y = ReL = 300 MPa, a tensile strength Rm = 450 MPa, a Young’s modulus E = 210 GPa and a Poisson’s ratio = 03. A yield plateau is expected. The toughness
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Fitness-for-Service Fracture Assessment of Structures Containing Cracks
Table 6.28: Reference data for probability analyses which can be used when no empirical data are available. The coefficient of variation is determined by COV = / (data according to [6.40] and [6.41] and supplemented by [6.45]) Parameter
Application range
Yield strength
Measured values1 Specified min values12
Tensile strength
Measured values Specified min values2 Welds
Young’s modulus Toughness
= 003 = Re + 70 MPa = 30 MPa = 005 = Rm + 70 MPa = 30 MPa = 001 = 005
Lower shelf /transition 34 ≈ 02 ! ! ! 03 Upper shelf 5 Upper shelf/ Charpy data
Crack size
Mean value and standard deviation
Remarks
[6.43] [6.43] [6.43] [6.43] [6.45] [6.45] refers to Kmed and to Kmat in Section 4.4.5.2.2
≈ 005 ! ! ! 01 < 01 Table 6.29
= 007 is provided in [6.45], however no distinction is made with respect to the application range. according to manufacturer’s specification (Re designates the standardised values of Y . in terms of Kmat . = 007 is provided in [6.45] but without distinction of application ranges. This value seems, however, unrealistically small in the ductile-to-brittle transition region of ferritic steels. 5) COV values between 0.076 and 0.165 are provided for various aluminium alloys (2024-T351, 7075-T651 and 7475-T7351) in [6.45]. 1) 2) 3) 4)
distribution of the rather brittle material is modelled by a three-parameter Weibull distribution (Eqn 4.25) with a scale parameter K0 = 50 MPa·m1/2 . The results of the analysis are summarised in Tables 6.30 and 6.31 and Figs 6.61 and 6.62. Note that the failure probability of the component is designated by Pf in order to distinguish it from the failure probability P of toughness test specimens (Section 4.4.5.2). Once the failure probability of the component has been determined it has to be checked against an allowable or target probability. This is provided in BS 7910 [6.47] as an application of the general principles summarised in [6.48]
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233
Table 6.29: Typical values of defect sizing error for ultrasonic inspection (according to [6.40] and [6.41]). The sizing error refers to the standard deviation whereas the mean value is given as 0.4 times the wall thickness Component type
Material
NDI technique
Sizing error (crack depth) in mm
Plate (wall thickness: >75 mm)
Ferritic steel
Advanced UT Good practice UT Low effectiveness UT
5 12 15
Pipe (wall thickness: 30–75 mm) (diameter > 250 mm)
Ferritic steel
Pipe (wall thickness: 10–30 mm) (diameter > 250 mm)
Ferritic steel
Advanced UT Good practice UT Low effectiveness UT Advanced UT Good practice UT Low effectiveness UT
5 15 15 3 5 10
Pipe (wall thickness: 5–30 mm) (diameter < 250 mm)
Ferritic steel
Pipe (wall thickness: >30 mm) (diameter > 250 mm)
Wrought austenitic steel
Advanced UT Good practice UT Low effectiveness UT Advanced UT Good practice UT Low effectiveness UT
3 5 10 5 5 7
Pipe (wall thickness: <30 mm) (diameter < 250 mm)
Wrought austenitic steel
Advanced UT Good practice UT Low effectiveness UT
2 3 5
Table 6.30: Example 6.13: Toughness Kmat and Jmat as a function of specimen failure probability P Specimen failure probability P 0.95 0.90 0.50 0.10 0.05
Kmat (Pf ) in MPa · m1/2
Jmat (Pf ) in N/mm
59.45 56.96 47.37 37.09 34.28
1532 1406 972 596 509
and adopted by SINTAP/FITNET (Table 6.32). The appropriate target failure probability depends on the situation, being very low where the consequences of failure are severe or very severe, for example, the loss of human life.
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Fitness-for-Service Fracture Assessment of Structures Containing Cracks
Table 6.31: Example 6.13: fLr function, J-integral and assessment paths Kr for different failure probabilities P m in MPa 20 40 60 80 100 120 140 160 180 200 220
Lr
fLr
Kr (P = 95%)
Kr (P = 50%)
Kr (P = 5%)
J in N/mm
0121 0242 0364 0485 0606 0727 0848 0970 1091 1212 1333
0996 0986 0968 0946 0919 0889 0858 0825 0153 0096 0062
0163 0325 0488 0650 0813 0975 1138 1300 1463 1625 1788
0204 0408 0612 0816 1020 1224 1428 1632 1836 2040 2244
0282 0564 0846 1128 1410 1692 1974 2256 2538 2820 3102
041 167 388 724 1198 1843 2698 3810 13923 44367 126585
(b)
(a)
30 Membrane stress in MPa:
2
1
80
1.2
3
60
Kr
1.0
120
0.8
40 100
0.6
1) Pf = 5%; Kmat = 34.28 MPa ⋅ m1/2 2) Pf = 50%; Kmat = 47.37 MPa ⋅ m1/2 3) Pf = 95%; Kmat = 59.47 MPa ⋅ m1/2
20 0.4
1) Pf = 5%; Jmat = 5.09 N/mm 2) Pf = 50%; Jmat = 9.72 N/mm 3) Pf = 95%; Jmat = 15.32 N/mm
20
Membrane stress in MPa:
100 Jmat
10
2 80
0.2
0.4
0.6
0.8
Lr
1.0
1.2
1.4
0
1
20 40
40 0
120 3
15
5
0.2 0
25
140
J-integral in N/mm
1.4
0
0.2
60
0.4
0.6
0.8
1.0
1.2
1.4
Lr
Figure 6.61: Example 6.13: Determination of the critical load for different failure probabilities P. (a) FAD approach; (b) CDF approach.
Frequently the failure probability Pf is substituted by a reliability index, , through the definition = −"−1 · Pf
(6.202)
Structural Assessment
235
100
Failure probability in %
80
2c
60
a
40
2W
t
20
0 60
70 80 90 100 110 Critical membrane stress σm in MPa
120
Figure 6.62: Example 6.13: Critical load as a function of the failure probability Pf of the component. Table 6.32: Target probability of failure (events/year) Failure consequences
Redundant component
Non-redundant component
23 × 10−1 10 × 10−3 70 × 10−5
10 × 10−3 70 × 10−5 10 × 10−5
Moderate Severe Very severe
Table 6.33: Relationship between Pf and (according to [6.48]) Pf
10−1
10−2
10−3
10−4
10−5
10−6
10−7
1.3
2.3
3.1
3.7
4.2
4.7
5.2
with "−1 being the inverse standardized normal distribution in [6.48]. Table 6.33 provides the conversion between Pf and .
6.13.5. Partial Safety Factors The concept of partial safety factors (PSF) comprises a deterministic analysis in conjunction with safety factors on all input parameters which have been
236
Fitness-for-Service Fracture Assessment of Structures Containing Cracks
determined beforehand by probabilistic analysis and which are available in tables now. The basic idea behind the concept is that, when applying the PSF to the input parameters, the failure probability of the component is guaranteed not to exceed the target probability. The analysis may, however, be overly conservative, so that imposing the PSFs leads to a failure probability much lower than the target probability. SINTAP/FITNET has adopted and supplemented the PSF tables of BS 7910 (Tables 6.34 to 6.38) [6.36, 6.39]. It should be noted that these factors were originally obtained for offshore structures and consequently, since they were derived for specific statistical distributions and specific geometries, care should be exercised in their general application. In particular they may be inappropriate for other statistical distributions. A basic problem is that the concept, when applied to multiple input parameters, is not unambiguous since various combinations of input parameters can yield identical failure probabilities. Where possible, a reliability analysis (Section 6.13.3) is preferred to a partial safety factor analysis.
Table 6.34: Recommended PSF for applied stress, Target failure probability Pf Reliability index Coefficient of variation COV 0.1 0.2 0.3
23×10−1
10×10−3
70×10−5
10×10−5
10×10−7
0.74
3.1
3.8
4.2
5.2
1.05 1.10 1.12
1.20 1,25 1.40
1.25 1.35 1.50
1.30 1.40 1.60
1.40 1.55 1.80
Table 6.35: Recommended PSF a for crack size, a Target failure probability Pf Reliability index Coefficient of variation COV 0.1 0.2 0.3 0.5
23×10−1
10×10−3
70×10−5
10×10−5
10×10−7
0.74
3.1
3.8
4.2
5.2
a
a
a
a
a
1.00 1.05 1.08 1.15
1.40 1.45 1.50 1.70
1.50 1.55 1.65 1.85
1.70 1.80 1.90 2.10
2.10 2.20 2.30 2.50
237
Structural Assessment
Table 6.36: Recommended PSF K for toughness, Kmat Target failure probability Pf Reliability index Coefficient of variation COV 0.1 0.2 0.3
23×10−1
10×10−3
70×10−5
10×10−5
10×10−7
0.74
3.1
3.8
4.2
5.2
K
K
K
K
K
1.00 1.00 1.00
1.30 1.80 2.85
1.50 2.60 NP
1.70 3.20 NP
2.0 2.5 NP
Table 6.37: Recommended PSF for toughness, mat Target failure probability Pf Reliability index Coefficient of variation COV 0.2 0.4 0.6
23×10−1
10×10−3
70×10−5
10×10−5
0.74
3.1
3.8
4.2
1.00 1.00 1.00
1.69 3.20 8.00
2.25 6.75 NP
2.89 10.0 NP
Table 6.38: Recommended PSF Y for yield strength, Y . Target failure probability Pf Reliability index Coefficient of variation COV 0.1
23×10−1 10×10−3 70×10−5 10×10−5 10×10−7 074
31
38
42
52
Y
Y
Y
Y
Y
100
105
110
120
150
The input parameters to the fracture assessment are obtained from the tabulated partial safety factors by the following rules: (1) Applied stress: is a multiplier to the mean stress of a normal distribution. (2) Crack size: a is a multiplier to the mean crack depth of a normal distribution. (3) Fracture toughness in terms of Kmat : K is a divider to the mean minus one standard deviation value of a Weibull distribution.
238
Fitness-for-Service Fracture Assessment of Structures Containing Cracks
(4) Fracture toughness in terms of mat : is a divider to the mean minus one standard deviation value of a Weibull distribution. (5) Yield strength: Y is a divider to the mean minus two standard deviations value of a lognormal distribution. Example 6.14: For the set of Kmat toughness values of Example 4.5 the scale parameter of the Weibull distribution was determined as Ko = 956 MPa ·m1/2 . This referred to a mean value Kmed = 885 MPa·m1/2 , a standard deviation Kmat = 192 MPa·m1/2 and a coefficient of variation COVK = 022 (Example 4.6). A component will be designed for redundancy and the consequences of failure classified as “severe”. Thus the target failure probability is given as 10×10−3 in Table 6.32. Based on this information, the partial factor on the fracture toughness is obtained as K = 197 by interpolation in Table 6.36. Finally, the toughness to be chosen for the assessment, Kmat is determined by Kmat =
1 · Kmed − Kmat 197
(6.203)
which gives an input value Kmat = 352 MPa · m1/2 .
6.14. Potential Benefit of Applying Advanced Options and Modules The conservatism of SINTAP/FITNET analysis can be reduced by choosing a higher order assessment option and/or by applying specific modules such as constraint corrections to fracture toughness (Section 6.7), using improved residual stress profiles (Section 6.6) or strength mismatch corrections (Section 6.11). It has, however, already been mentioned in Section 3.3 that none of these measures will necessarily provide a significant improvement of the assessment result. This is discussed in the final section of this chapter. Which parameter will be effective for improving a SINTAP/FITNET analysis primarily depends on the ligament yielding parameter Lr determined from the applied load and on the f(Lr )/Lr ratio as illustrated in Fig. 6.63. Constraint corrections are rather ineffective for Lr < 02 in the absence of secondary stresses and the same is true for the application of higher options and for strength mismatch corrections below Lr = 08. In contrast, the largest potential benefit of mismatch corrections is expected in the collapse-dominated
Structural Assessment
Potential advantage of applying constraint correction
239
Potential advantage of applying higher options or mismatch option f (Lr)/Lr = 1.1
1.2
Potential advantage of refinement of residual stresses
Fracture dominated
m
1.0
bi
ne
d
fa
ilu
re
Little benefit of refinment of tensile properties
0.8
f(Lr)
f (Lr)/Lr = 0.8 Co
m
od
e
0.6
f (Lr)/Lr = 0.4
0.4
Collapse dominated Little benefit of refinment of toughness properties
0.2 0
0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Lr
Figure 6.63: Effect of higher effort on the conservatism of a SINTAP/FITNET analysis depending on ligament yielding parameter Lr and f(Lr )/Lr ratio (according to [6.49]).
region, except for undermatched welds under plane stress conditions. Benefit from refining the tensile properties is expected above f(Lr )/Lr = 11, benefit from refining the toughness properties is of major value below f(Lr )/Lr = 04 and a refinement of residual stress profiles is of benefit in the range f(Lr )/Lr < 08.
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Chapter 7
Validation Examples 7.1. Introduction Since its publication in 1999, the SINTAP methodology has been subject to extended evaluation. A number of case studies are summarised in Volume II of FITNET [7.1]. Because the procedure contains elements of other methods, such as R6, BS 7910, ETM, etc., and because those methods have been revised with respect to SINTAP, the verification examples of these alternative approaches can also be used for validating SINTAP/FITNET to some extent. Considering investigations into all cases, hundreds of studies have been undertaken. It must be understood that only a limited number of validation examples can be reproduced within this chapter. These do, however, give an indication of the potential of the procedure.
7.2. Pipelines and Pressurised Tubes A total of 97 full-scale pipe tests, with through-wall and surface cracks subjected to internal pressure and four-point bending, were used for validating different assessment options in [7.2]. The examples comprise axial as well as circumferential cracks in six non-stainless grade and austenitic steels. All experimental data sets were taken from the literature. An example of an instability analysis is shown in Fig. 7.1. In that specific case, the predicted instability pressure, pc , was 16.9 MPa for Option 1 and 17.3 MPa for the modified Option 1, compared with the experimental value of 20.4 MPa. An Option 3 analysis was not possible because the stress– strain data available in the literature examples were not good enough. Note that the improved yield load solutions in Section 5.2.5.1, Example 5.11, were not available at the time the study was performed, which explains the relatively high conservatism. By applying these latter solutions, a noticeable reduction of conservatism is expected. This is also true for the configurations with a surface crack that are shown in Figs 7.2–7.4. Figure 7.2 shows Option 0 and 1 analyses of pipes subjected to bending. A comparison between Option 0, Option 1 and modified Option 1 analyses is provided in Fig. 7.3 for a number of cases. As defined above, modified Option 1 consists of Option 1 in conjunction with
242
Fitness-for-Service Fracture Assessment of Structures Containing Cracks
(a) 1200
J-integral in N/mm
1000
800
Pressure in MPa: 17.3 17.1 16.9
600
400
16.7 16.5
200
0
36.2
36.4
36.6
36.8
37
36.8
37
Crack depth in mm (b) 1200
J-integral in N/mm
1000
800
Pressure in MPa: 17.5 17.3 17.1 16.9
600
400
200
0
36.2
36.4
36.6
Crack depth in mm
Figure 7.1: Prediction of the instability load of a ferritic pipe with an axial surface crack subjected to internal pressure (according to [7.2]). (a) Option 1 analysis; (b) Option 1 analysis, but using an individually determined strain hardening exponent instead of Eqn (6.23).
individually determined strain hardening exponents N. In one case, data set J, two results are obtained as a result of the different R curves available in the literature. This demonstrates the effect of adequate input information on the degree of conservatism of the assessment result. Option 0 analyses of various pipes are shown in Fig. 7.4. As expected, the data sets show quite different but, in both cases, high conservatism of the analyses
Validation Examples
243
Predicted bending moment in kNm
50
40
30
20
Through wall cracks, ferritic steel, Option 0 Through wall cracks, ferritic steel, Option 1 Surface cracks, austenitic steel, Option 0 Surface cracks, austenitic steel, Option 1
10
0 0
10
20
30
40
50
Experimental bending moment in kNm
Predicted/experimental max load in %
Figure 7.2: Comparison with experimental data of Option 0 and Option 1 critical global bending moments for pipes containing circumferential cracks (according to [7.2]).
Option 0
A–D: Internal pressure J: Four-point bending
Option 1 Option 1*
Experimental failure
100 80 60 40 20 0
A
B
C
D/1
D/2
D/3
J
Figure 7.3: Comparison between Option 0 and Option 1 and modified Option 1 (Option 1∗ ) failure loads of six pipes subjected to internal pressure and on pipe subjected to bending with experimental data (according to [7.2]). The pipes were made of two non stainless grade steels.
(of the order of 20–70% with respect to the critical load). The differences are due to the empirical nature of the Charpy data-fracture toughness correlations, the conservatism of which varies from material to material.
244
Fitness-for-Service Fracture Assessment of Structures Containing Cracks
Predicted internal pressure at failure in MPa
15
Hollow cylinders with axial cracks subjected to internal pressure (5 data sets), steel
12
9
6
3
3 Data sets ( 2 Data sets (
): through wall cracks ) : surface cracks
0 0
3
6
9
12
15
Experimental internal pressure at failure in MPa
Figure 7.4: Comparison between Option 0 failure loads of five data sets of pipes subjected to internal pressure with experimental data (according to [7.2]). The pipes were made of three different steels.
Summarising, each analysis yielded conservative results, the experimental maximum load the component was able to sustain being underestimated. In addition, and as expected, the higher analysis levels yielded less conservative results than the lower levels.
7.3. Thin Wall Structures The SINTAP/FITNET module for thin walled structures has been validated for 61 thin wall plates made of four different aluminium alloys, one austenitic and two conventional steels in [7.3–7.5]. The plates were subjected to uniaxial tension, bending, biaxial tension with biaxial ratios of = Fx /Fy = −05 0 +05 and +1, and biaxial tension in conjunction with mixed mode. In each case, the fracture toughness in terms of a CTOD-5 -R-curve was obtained from C(T) specimens. The stress–strain curves and the 5 –a curves of the different materials are shown in Figs 7.5 and 7.6. Out of the complete data set only two specimens were assessed to be significantly non-conservative (in terms of instability load 11% and 27%). The reason was not fully clear in the first case, but was probably a result of material inhomogeneity. The second specimen was a 1000 mm wide, 1.6 mm-thick, plate subjected
Validation Examples
245
800
Engineering Stress, MPa
StE 550 600
AI 2024 T3 StE 460 AI 5083 H 321
400
200
T-Orientation 0 0
0.1
0.05
0.15
0.2
Engineering Strain
Figure 7.5: Thin wall validation exercise: the engineering stress-strain curves of the materials investigated in Figs 7.7 and 7.8. The Al 5083 H 321 curves show dynamic strain aging beyond a strain of 3%. 4
StE 460 M(T)
StE 550 M(T) StE 550 C(T)
CTOD-δ5, mm
3
StE 460 C(T)
2
A1 5083 H 321
1
A1 2024 T 3 0
0
5
10
15
20
25
Stable Crack Extension Δa, mm
Figure 7.6: Thin wall validation exercise: CTOD-5 R curves of the materials investigated in Figs 7.7 and 7.8.
246
Fitness-for-Service Fracture Assessment of Structures Containing Cracks
Predicted maximum force Fy in kN
1000
Fy
800
Fx
Fx
+20%
–20%
600
Fy λ = Fx / Fy
400
200
Option 3 Option 1 0 0
200
400
600
800
1000
Experimental maximum force Fy in kN
Figure 7.7: Predicted and experimentally determined failure loads of biaxially tension loaded plates (Option 1 and Option 3 analysis, according to [7.5]). The biaxiality ratios were = Fx /Fy = −05, 0 +05 and +1.
Predicted maximum force Fy in kN
1000
Fy +20%
800
Fx
Fx 45°
–20%
600
Fy λ = Fx/Fy
400
200
Option 3 Option 1 0 0
200
400
600
800
1000
Experimental maximum force Fy in kN
Figure 7.8: Predicted and experimentally determined failure loads of mixed mode loaded plates (Option 1 and Option 3 analysis, according to [7.5]). The biaxiality ratios were = Fx /Fy = −05 0 +05 and +1.
Validation Examples
247
to bending that was loaded with a buckling guide. In that case, the global yield load of the through thickness crack was simply not appropriate. All other cases showed conservative or “best estimate” results (max ± 6%, usually much better). Only the biaxially and mixed mode loaded cruciform plates are shown in Figs 7.7 and 7.8.
7.4. Strength Mismatched Configurations In [7.6], 12 tests on undermatched and overmatched bending plate tests carried out at TWI were used for case studies of the SINTAP procedure. The mismatch factor was approximately M = 148 for overmatching and M = 068 for undermatching. Two different crack lengths (a/W = 045 and 0.65) were introduced into the plates. The strength properties were: Material A: Y = 738 MPa, Rm = 849 MPa; Material B: Y = 497 MPa, Rm = 647 MPa. The plate sections were joined together by electron beam welding. The R curves (mean curves) were: Material A J = 0094 + 0095a078 Material B J = −0027 + 0311a040 (J in kN/mm and a in mm) The results are summarised in Fig. 7.9 and show that the analyses yielded moderately conservative results in all cases. Again, the higher analysis level showed less conservatism than the lower level. 70 a
Predicted failure load in kN
60
50
W Electron beam weld
2H
M = 1.48 a / W = 0.45 Option 2 M = 1.48 a / W = 0.45 Option 3 M = 1.48 a / W = 0.65 Option 2 M = 1.48 a / W = 0.65 Option 3
OverMatching E
Overmatching a
UnderMatching
40
W
2H
30
M = 0.68 a / W = 0.45 Option 2 M = 0.68 a / W = 0.45 Option 3 M = 0.68 a / W = 0.65 Option 2 M = 0.68 a / W = 0.65 Option 3
OverMatching
20
Material A
UnderMatching
10
0 0
Undermatching
10
20
Material B 30
40
50
60
70
Experimental failure load in kN
Figure 7.9: Predicted and experimentally determined failure loads of mismatched bend specimens (Option 1 and Option 3 analysis, according to [7.6]).
248
Fitness-for-Service Fracture Assessment of Structures Containing Cracks
7.5. Failure Investigation The application of SINTAP to a failure analysis was reported in [7.7]. A forklift fork broke at a hole, under a service load of 34.8 kN. Subsequent failure analysis
Figure 7.10: Fracture surface of the broken fork. The failure originated at two edge cracks left and right from the hole at the top side.
0.20
2c
Option 1
CTOD, δ[mm] Cracks
Lrmax
0.10
0.05
Critical condition δmat
0 1.4
Critical condition
Kr = KI /Kmat 1.0
Option 3
0.8 0.6
Option 1
0.4 0.2 0 0
0.2
0.4
0.6
0.8
1
1.2
1.4
Ligament yielding Lr
Figure 7.11: SINTAP CDF and FAD analysis of the broken fork.
Validation Examples
249
revealed that fracture originated from small edge cracks at the front face at either side of the hole (Fig. 7.10). The failure occurred unexpectedly because the original flaws were too small to be detected earlier. The crack lengths at the surface were 3 and 10 mm, leading to an overall crack size (including the hole) at the surface of 2c = 454 mm. The SINTAP analysis is illustrated in Fig. 7.11 for both CDF and FAD approaches. It revealed critical crack sizes (including the hole diameter) 2c = 332 mm (Option 1) and 2c = 356 mm (Option 2) which were of the same order (differences 27% and 22%, respectively) as the crack empirically found at the fracture surface. In conclusion, it was stated that the failure occurred as the consequence of inadequate design and not from inadmissible handling such as overloading. SINTAP was shown to be a suitable tool for the failure analysis.
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References
[1.1] Kocak, M., Webster, S., Janosch, J.J., Ainsworth, R.A. and Koers, R., Fitness for Service Procedure (FITNET), Final Draft 7, 2006. [1.2] R6, Revision 4, Assessment of the Integrity of Structures Containing Defects. British Energy Generation Ltd (BEGL), Barnwood, Gloucester, 2000. [1.3] BS 7910: Guide on Methods for Assessing the Acceptability of Flaws in Metallic Structures. British Standard Institution (BSI), London, 2005. [2.1] Zahoor, A., Ductile Fracture Handbook. Novotech, Cop & EPRI, Res. Proj. Vol. 1. 1989, pp. 1757–69, Vol. 2, 1990, Vol. 3, 1991. [2.2] Ainsworth, R.A., (Ed.), Special issue on Flaw Assessment Methods, Int. J. Press. Vess. Piping 77, 2000. [2.3] Ainsworth, R.A. and Schwalbe, K.-H., (Eds.), Practical Failure Assessment Methods, In: Milne, I., Ritchie, R.O. and Karihaloo, B., (Eds.), Comprehensive Structural Integrity (CSI), Elsevier, Amsterdam et al., Vol. 7, 2003. [2.4] Zerbst, U., Ainsworth, R.A. and Schwalbe, K.H., Basic principles of analytical flaw assessment methods, Int. J. Press Vess. Piping, 77, 2000, pp. 855–67. [2.5] Ainsworth, R.A., Failure assessment diagram methods, [2.3], Chapter 7.03, 2003, pp. 89–132. [2.6] Schwalbe, K.-H. and Zerbst, U., Crack driving force estimation methods. [2.3], Chapter 7.04, 2003, pp. 133–76. [2.7] Burdekin, F.M. and Dawes, M.G., Practical use of linear elastic and yielding fracture mechanics with particular reference to pressure vessels. In: Conference on Applications of Fracture Mechanics to Pressure Vessel Technology, Inst. Mech. Engineers, London, 1971, pp. 28–37. [2.8] Wells, A.A., Application of fracture mechanics at and beyond general yielding, British Welding Journal 10, 1963, pp. 563–70. [2.9] Dawes, M.G., TWI Report E/58/74, Published in summarised form in: Dawes, M.G. Fracture control in high strength weldments, Weld. J. Res, Suppl. 53, 1974, pp. 369–79. [2.10] PD 6493, Guidance on some methods for the derivation of acceptance levels for defects in fusion welded joints, British Standards Institution (BSI), London, 1980. [2.11] PD 6493, Guidance on methods for assessing the acceptability of flaws in fusion welded structures, British Standards Institution (BSI), London, 1991. [2.12] BS 7910, Guide on Methods for Assessing the Acceptability of Flaws in Metallic Structures, British Standard Institution (BSI), London, 2005. [2.13] API 1104, Welding of Pipelines and Related Facilities, App. A: Alternative acceptance standards for girth welds, American Petroleum Institute (API), 1994.
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[2.34]
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[2.38]
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Structures, EFAM ETM 97, GKSS Report 98/E/6, GKSS Research Centre, Geesthacht, 1998. Schwalbe, K.-H., Zerbst, U., Brocks, W., Cornec, A., Heerens, J. and Amstutz, H., The ETM method for assessing the significance of crack-like defects in engineering structures, Fatigue Fract. Engng. Mat. Struct. 21, 1998, pp. 1215–31. Schwalbe, K.-H. and Zerbst, U., The Enginering Treatment Model, Int. J. Press. Vess. Piping 77, 2000, pp. 905–18. Schwalbe, K.-H., Kim Y.-J., Hao, S., Cornec, A., Koçak, M., The ETM Method for Assessing the Significance of Crack-Like Defects in Joints with Mechanical Heterogeneity (Strength Mismatch), EFAM ETM-MM 96, GKSS Report 97/E/9, GKSS Research Centre, Geesthacht, 1997. Ainsworth, R.A., Bannister, A. and Zerbst, U., An overview on the European flaw assessment procedure SINTAP and its validation, Int. J. Press. Vess. Piping, 77, 2000, pp. 869–76. SINTAP, Structural Integrity Assessment Procedure, Final Revision, EU-Project BE 95-1462, Brite Euram Programme, Brussels, 1999. Kocak, M., Webster, S., Janosch, J.J., Ainsworth, R.A. and Koers, R., Fitness for Service Procedure (FITNET), Final Draft 7, 2006. Ainsworth, R.A., Kim, Y.-J., Zerbst, U., Gutierrez-Solana, F. and Ruiz-Ocejo, J., Driving Force and Failure Assessment Diagram methods for defect assessment, 17th Int. Conf. on Offshore Mech. and Arctic Engng. (OMAE), Lisbon, Paper 98–2054, 1998. Kim, Y.-J., Ainsworth, R.A. and Kocak, M., Defect assessment procedure for strength mismatch structures – SINTAP, 12th European Conf. on Fracture (ECF 12), Sheffield, 1998, pp. 583–8.
[4.1] Anderson, T.L., Flaw characterization. In: Ainsworth, R.A. and Schwalbe, K.-H., (Eds.), Comprehensive Structural Integrity, Vol. 7: Practical Failure Assessment Methods, Elsevier, Amsterdam et al., 2003, 227–43. [4.2] Burdekin, F.M. and Sherry, A.H., On the recharacterisation of defects in structural integrity assessments due to snap-through, Int. J. Press. Vess. Piping, 2006. [4.3] R6, Revision 4, Assessment of the Integrity of Structures Containing Defects. British Energy Generation Ltd (BEGL), Barnwood, Gloucester, 2000. [4.4] API 579, Recommended Practice for Fitness for Service, American Petroleum Institute (API), Washington, 2000. [4.5] BS 7910, Guide on Methods for Assessing the Acceptability of Flaws in Metallic Structures, British Standard Institution, London, 2005. [4.6] Berger, C., Burr, A., Gugau, M., Habig, K.-H., Harsch, G., Kloos, K.H., Pyttel, B. and Speckhardt, H., Werkstofftechnik (Materials technology), Chapter E, Appendix E, In: Grote, K.-H. and Feldhusen, J., (Eds.), Dubbel-Taschenbuch für den Maschinenbau. Springer, Berlin, Heidelberg, New York, 2005.
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Appendix: “Fracture Toughness Test Standards”
This appendix summarises a number of test standards and recommended procedures that are of importance in the context of the SINTAP/FITNET fracture module. Standards Standard/guideline
Parameters
Remarks
ISO Standards and Draft Standards (International Standard Organisation) ISO 12135 (2003): Metallic materials – unified method of test for the determination of quasi-static fracture toughness
Fracture toughness based on K, J and , J and -R-curves
ISO CD 15653 (2006): Metallic materials – unified method of test for the determination of quasi-static fracture toughness of welds
Fracture toughness based on K, J and , J and -R-curves
Draft version
ISO Document FDIS 22889-N413.4 (2006): Metallic materials – method of test for the determination of resistance to stable crack extension using specimens of low constraint
Crack tip opening angle (CTOA), CTOD-5 , CTOA, plateau value, 5 -R-curve
Draft version
BSI (British Standards Institution) BS 7448, Part 1 (1991): Fracture mechanics toughness tests – method for determination of KIc , critical CTOD and critical J values of metallic materials
Fracture toughness based on K, J and , J and -R-curves
(Continued)
266
Appendix “Fracture Toughness Test Standards”
Standard/guideline
Parameters
Remarks
BS 7448, Part 2 (1997): Fracture mechanics toughness tests – method for determination of KIc , critical CTOD and critical J values of welds in metallic materials
Fracture toughness based on K, J and , J and -R-curves
Weldments
BS 7448, Part 3 (2005): Fracture mechanics toughness tests – method for determination of fracture toughness of metallic materials at rates of increase in stress intensity factor greater than √ 30 MPa m/s
Dynamic fracture toughness
ASTM Standards (American Society for Testing and Materials) ASTM E 1820-01 (2001): Standard Fracture toughness test method for fracture toughness based on K, J and , J and -R-curves Fracture toughness ASTM E 399-90 (reapproved KIc 1997): Standard test method for plane-strain fracture toughness of metallic materials ASTM E 1290-02 (2002): Standard test method for Crack-Tip Opening Displacement (CTOD) fracture toughness measurement
Crack tip opening displacement c
In particular, ductile-to-brittle transition
ASTM E 1221-96 (reapproved 2002): Standard test method for determining plane-strain crack-arrest fracture toughness, KIa , of ferritic steels
Crack arrest
Ferritic steels
ASTM E 1921-05 (2005): Standard Toughness scatter in test method for determination of the ductile-to-brittle reference temperature, T0 , for transition ferritic steels in the transition range
Ferritic steels, based on the VTT master curve approach (Continued)
Appendix “Fracture Toughness Test Standards”
267
Guidelines which are not standards Guideline
Parameters
Remarks
ESIS recommended procedure (European Structural Integrity Society) ESIS P3-03/06D (2006): Draft unified procedure for determining the fracture behaviour of materials. European Structural Integrity Society (ESIS)
Fracture toughness based on K, J and , J and -R-curves, CTOA, 5 -R-curve
Draft version, annexes for weldments, low constraint geometries, statistical issues, etc.
Dynamic fracture ESIS TC 5 (2005): Proposed standard methods for instrumented toughness pre-cracked Charpy impact testing of steels GKSS Research Centre
Submitted to ISO
EFAM GTP 02 – The GKSS test procedure for determining the fracture behaviour of materials
Annexes for weldments, low constraint geometries, statistical issues etc., basis of ESI P3/06D
Fracture toughness based on K, J and , J and -R-curves, CTOA, 5 -R-curve
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Glossary
Constraint and Stress Triaxiality Because of the large stress normal to the crack plane in mode I loading (y-direction), the material close to the crack tip should contract in the thickness and ligament directions (x and z in Fig. G1). Actually a type of necking occurs at the specimen surface. However, in the interior of the section deformation is prevented by the surrounding material. This phenomenon is designated as constraint. The hindered deformation corresponds to a high triaxial stress state, which is defined by a high ratio of the hydrostatic to the equivalent stresses. The constraint/triaxiality at the crack tip is affected by a number of features such as the specimen or component geometry and its dimensions in the thickness and ligament directions (out-of-plane and in-plane constraint), the loading geometry (bending causing a higher constraint than tension), the deformation behaviour of the material (namely its strain hardening exponent), and the magnitude of ligament yielding Lr . Note that strength mismatch also affects the constraint. Whereas the term constraint is used more qualitatively, the stress triaxiality is expressed quantitatively using various stress triaxiality parameters such as the triaxiality parameter h, the T-stress and the Q-stress. For a more detailed description see Section 4.4.6.2. The out-of-plane constraint refers to the terms plane strain and plane stress that are commonly used in fracture mechanics (→ “Plane Stress Versus Plane Strain Conditions”). Constraint affects the fracture toughness of the material with the consequence that there can be a transferability problem from test specimens to components. The larger the constraint and, in conjunction with this, the higher the stress triaxiality, the lower the fracture toughness. According to common test standards, lower bound fracture toughness values are determined by using high constraint specimens, that is, deeply cracked bend specimens with an almost square ligament (B ≈ b) and a thickness B not smaller than the component thickness. An exception to this is the toughness determination for thin walled plates (see Section 6.10). For cases where the constraint in the test specimens is much higher than in the component, an overconservative SINTAP/FITNET analysis will be the result. In order to reduce this effect, the procedure offers a methodology for describing and correcting toughness values for constraint (Section 4.4.6).
y
B
Surface
Centre line Thickness B
(centre line)
A a
Necking Crack depth (surface) Crack depth
b
Ligament length b
Surface
A-A
Surface
A
Surface
Glossary
Centre line
270
x
z
Figure G1: Deformation at a crack tip. The high mode I tensile stress causes necking at the surfaces. However, in the interior deformation is prevented, the material is constrained.
Crack Driving Force The term designates the crack tip loading in terms of K-factor, J-integral or crack tip opening displacement (CTOD), . Moreover, in SINTAP/FITNET, the terminology is used for an assessment philosophy different to the failure assessment diagram (FAD). In the CDF approach, the determination of the CDF and its comparison with fracture toughness are two separate steps.
Crack Idealisation The term crack idealisation describes the modelling of a real crack or crack-like flaw using a crack for which fracture mechanics solutions are available. This includes the shape and dimensions as well as the orientation of the crack and interaction criteria for coplanar cracks and cracks and surfaces. Idealization is applied to the modelling so that the resulting crack driving force is overestimated. Guidance is provided in Section 4.2.
Glossary
271
Crack Tip Opening Displacement The crack tip opening displacement, CTOD, is a crack tip parameter applicable to linear elastic as well as elastic–plastic deformation behaviour (small scale, contained and net-section yielding; → “Crack Tip Plasticity”). It is usually designated . When loaded, the crack tip blunts. In general terms, the magnitude of this blunting normal to the crack plane (y-direction; Fig. G1) is the CTOD. This simple and clear definition is, however, not easy to realise in practical applications as the whole deformed crack profile has to be described by a simple parameter only, which obviously is a matter of definition. That is the reason why, for the CTOD, in contrast to the J-integral, a number of definitions exist side-by-side, yielding slightly different values. (a) CTOD Definitions for Elastic–Plastic Deformation Behaviour One obvious definition is the displacement in the y-direction at the original crack tip (Fig. G2a). Since the initially sharp crack blunts in the x- as well as in the y-direction can immediately be obtained from the corresponding points on the crack profile. However, the experimental determination of this parameter is rather difficult, although possible, for example, by replica techniques. As an alternative approach, engineering definitions can be used such as 5 , which will be explained below. A further definition is based on the HRR field (designated according to its authors Hutchinson, Rice and Rosengren; Fig. G2b; → “J-Integral”) [G1, G2]. Based on this, the crack face displacements ux and uy are obtained by
(a)
(b)
σ
Original crack tip
σ
45°
y
uy
δ
δ
ux
x
45° r
σ
σ
Figure G2: Definitions of the crack tip opening displacement, CTOD. (a) As the finite displacement normal to the crack plane at the original tip of a blunted crack; (b) As the displacement at the intersection of 45 lines with the crack flanks.
272
Glossary
n/1+n E·J ˜ x ux 1/1+n u = · 0 · ·r u˜ y uy · 0 · 0 In
(G1)
The CTOD is then defined as = 2uy at the intersection points of two symmetric 45 lines, the origin of which is at the crack tip [G3]. This CTOD is commonly designated as 45 or t (with t meaning “tip”). The relation to the J-integral is given by 45 =
dn ·J Y
(G2)
with dn being a function of the stress state (plane strain or plane stress), the Ramberg–Osgood strain hardening exponent n n = +· (G3) 0 0 0 and the Y /E ratio of the material. Closed form solutions for Eqn (G2) are available [G4–G6]. Note that the 45 is defined up to stable crack initiation, rather than to stable crack extension, since the profile of the growing crack deviates from that plotted in Fig. G2b. (b) CTOD Definitions for Linear Elastic Deformation Behaviour For linear elastic material behaviour, is usually correlated with the K-factor. One correlation is based on the definition of the plastic zone corrected K-factor (→ “Crack Tip Plasticity”) [G7]. Based on the K-field solutions for the displacements ux and uy
KI cos /2 · − 1 + 2 · sin2 /2 ux
(G4) = · r/2 · uy 2G sin /2 · + 1 − 2 · cos2 /2 with G being the shear modulus and 3 − 4 for plane strain = 3 − /1 + for plane stress
(G5)
the crack tip opening displacement can be obtained as = 2uy at the real or physical crack tip when the tip of the plastic zone corrected crack is assumed to be at a + rY . Inserting the radius of the plastic zone for plane stress given by 2 1 KI rY = (G6) 2 Y
Glossary
(a)
273
(b) σ σyy
K
σY
σY
Keff δ
δ
uy σY
uy r a
rY
Plastic zone
a
σ
d
Figure G3: Definitions of the crack tip opening displacement, CTOD. (a) Based on plastic zone corrected K strain field; (b) based on Dugdale strip yield model.
into Eqn (G4) gives for r = rY , = 180 (Fig. G3a) and G=
E 2 · 1 +
(G7)
a plane stress CTOD = 2uy =
4 KI2 · E · Y
(G8)
A further approach is based on the Dugdale strip yield model. As above, the crack is formally extended by the plastic zone size and is then modelled as the crack opening at the physical crack tip. The plastic zone geometry is simplified as a yielded strip with no extension in the y-direction. For elastic–perfectly plastic deformation behaviour and tension loading the local mode I stress acting along the plastic strip is identical to a uniform remote stress yy of yield strength magnitude (Fig. G3b). It is then postulated that the effect of yielding is an imaginary increase in the crack length and that the opening of this crack portion is restrained by yy = Y . The problem has been solved by superposition of the solution for the uncracked sheet loaded by uniform tension = yy with the solution for the cracked sheet with no remote loading but with crack face pressure of yield load magnitude. From the demand that the yield strength must not be exceeded at any point along the strip length of it is obtained as · −1 d = a · sec 2Y
(G9)
274
Glossary
This solution was found to be in satisfactory agreement with plastic zone size measurements up to about 0.9 Y [G8]. The CTOD is determined as [G9] 8 a · Y · = · · ln sec (G10) E 2 · Y Applying a series expansion to the ln sec term gives an approximation of Eqn (G10): 2 4 8 · Y · a 1 1 = + +··· (G11) · · · · · ·E 2 2 Y 12 2 Y which can be transformed to 2 KI2 1 = +··· · 1+ · · Y · E 6 2 Y
(G12)
2 by reducing the lowest common denominator by 21 · 2 · and replacing 2 a Y by KI 2 . Neglecting the higher order terms for small-scale yielding conditions ( Y ) simplifies Eqn (G12) to =
KI2 Y · E
(G13)
which is close to Eqn (G8) above. Note that the use of the strip yield model is restricted by a number of inherent assumptions such as an infinite tension geometry, non-hardening deformation behaviour and plane stress. In order to obtain a more general expression both Eqns (G8) and (G13) were re-written as =
KI2 m · Y · E
(G14)
with m being an additional dimensionless constraint factor usually being assumed as m = 2 for plane strain and m = 1 for plane stress and E/1 − 2 for plane strain, E = (G15) E for plane stress.
Glossary
275
Note that the constant m has been replaced in SINTAP/FITNET by a value m = 1 5 based on 25% percentile values of an empirical correlation of a large data set of structural steels with strain hardening N ≥ 0 05. (c) Further CTOD definitions for material testing The CTOD definitions discussed so far are widely restricted to the description of the crack driving force in components. Laboratory measurements are usually performed on edge cracked three-point bending specimens using a plastic hinge approach. The plastic displacement V is measured at the crack mouth opening. For bending geometries a plastic hinge forms in the ligament that separates the regions of tension from those of compression. Assuming straight crack fronts, the plastic CTOD, p , can simply be determined by the theorem of intersecting lines as p =
r · W − a ·V a + r · W − a
(G16)
using the information provided in Fig. G4a. The only unknown in Eqn (G16) is the rotational factor r which defines the plastic hinge at a distance of rW − a ahead of the crack tip. This is specified as r = 0 4 or 0.44 [G10, G11] for typical materials and test specimens in the common test standards. Adding the elastic component the complete CTOD is then obtained by = e + p =
KI2 r · W − a + ·V m · Y · E a + r · W − a p
(a)
(G17)
(b) V
a r (W-a)
Δa ao
Plastic hinge
bend specimen
δ
δ5 + 5 mm
Figure G4: Definitions of the crack tip opening displacement, CTOD. (a) Based on a plastic hinge in bending specimens; (b) 5 concept.
276
Glossary
with m and E as discussed under point (b). Note that no plastic hinge develops within the ligament of high strain hardening materials with the consequence of an overestimation of in experimental determination. An alternative definition is provided by the 5 parameter [G12]. This is measured at the surfaces at two gauge points located 5 mm apart on a straight line going through the original crack tip (Fig. G4b). In contrast to the plastic hinge CTOD, 5 can be determined both on laboratory specimens and on components with any geometry. Within SINTAP/FITNET, the 5 definition is applied to thin walled structures (Section 6.10) where the direct experimental determination of the crack driving force without any calibration functions is an essential advantage and this is more so as the standard CTOD definition based on the plastic hinge is usually not applicable for such structures. Note that 5 is one of two parameters used by an ISO draft standard for toughness determination in low constraint geometries [G13].
Crack Tip Plasticity With respect to ligament plasticity, the terms “small-scale yielding,” “contained yielding,” “net-section yielding” and “gross-section yielding” can be distinguished. The term small-scale yielding is used when the size of the plastic zone is small (of the order of a few percent) compared to the K controlled zone and characteristic dimensions of the crack and the component such as wall thickness, crack and ligament length. The radius of the plastic zone, rY is usually estimated by [G14] 2 KI for plane strain, 1 6 (G18) rY = · for plane stress. 1 2 Y In Eqn (G18) the cross section of the plastic zone is assumed to be circular. However, in reality it shows a shape comparable to the schematic drawing in Fig. G5.
Plastic zone Plane strain
Crack
Plane stress 2rp
Figure G5: Schematic drawing of the mode I plastic zone varying from plane stress conditions at the surface to plane strain conditions in the interior.
Glossary
277
Note that the dimension rY is a first approximation rather than an exact description. For example, it does not take into account material hardening. Various improvements of Eqn (G18) have been proposed to overcome this shortcoming. For example, it is extended by 2 n−1 KI 1 6 · rY = · 1 2 n+1 Y
for plane strain, for plane stress,
(G19)
with n being the Ramberg–Osgood strain hardening exponent (see Eqn (G3)) [G15]. In SINTAP/FITNET terminology, the small-scale yielding regime can be limited to a ligament yielding parameter Lr < 0 5 to 0.6. At Lr values higher than these, the ligament is subjected to contained yielding and, at and above Lr = 1, to net-section yielding. Net-section yielding means that the whole ligament is plastified. Alternative terms sometimes used for this are “general yielding” or “fully–plastic yielding.” Net-section yielding usually does not cause failure of the ligament because the material is still strain hardening. When yielding reaches the region of the body volume remote from the crack plane, it is designated as gross-section yielding. In fundamental fracture mechanics, the K concept can only be applied when small-scale yielding conditions exist. However, within SINTAP/FITNET a formal extension is provided by a plasticity correction performed by the fLr functions.
Ductile-to-Brittle Transition Materials with body-centred cubic and hexagonal lattices, for example, ferritic steels, show a distinct transition behaviour from brittle cleavage fracture at low temperatures (→ “Lower Shelf”) to micro-ductile fracture at higher temperatures (→ “Upper Shelf”). The ductile-to-brittle transition in between these two ranges is characterised by initially stable micro-ductile crack extension that is suddenly terminated by cleavage. The measured toughness values show a large scatter which can be explained by the so-called Weakest Link model [G16] (see Section 4.4.5.1) and which requires statistical description. In the framework of SINTAP/FITNET the latter is provided by the Master Curve concept (see Sections 4.4.5.2 and 4.4.5.3).
Failure Assessment Diagram Approach The FAD approach is one of two possible assessment philosophies followed in SINTAP/FITNET. The failure assessment diagram is a failure line that
278
Glossary
interpolates between the failure modes of elastic fracture and plastic collapse. The ordinate of the diagram is derived as the crack driving force normalised by the fracture toughness of the material, the abscissa is the applied load relative to the yield load of the cracked structure. The FAD line is almost geometryindependent but does depend on the stress–strain curve of the material. In order to perform a failure assessment, the relative position of an assessment point (which takes into account the component and crack geometry as well as the deformation and fracture properties of the material) with respect to the FAD line has to be determined. A detailed description is given in Section 6.2.
Global and Local Failure Global failure is the failure of the component, or at least of the whole cross section containing the crack, whereas local failure refers to the failure of a limited area in the ligament, e.g., leakage while the component or cross section stays intact. Local failure occurs earlier and at lower loads than global failure and, therefore, local analyses tend to be the more conservative. In SINTAP/FITNET this is mainly considered using global and local yield loads (Section 5.2.2).
J-Integral Although derived for non-linear elastic behaviour, the J-integral is applicable to elastic–plastic deformation behaviour (or net-section yielding) comparable to the crack tip opening displacement, CTOD. The difference in the material laws is illustrated in Fig. G6. Although the loading behaviour is identical for both material laws, its unloading behaviour is significantly different. The non-linear elastic Nonlinear elastic
Loading
Loading Stress
Elastic-plastic
Unloading
Unloading
Strain
Figure G6: Deformation behaviour of non-linear elastic and elastic–plastic materials.
Glossary
279
material unloads along the same path but the elastic–plastic material follows a linear path, leaving irreversible deformation after unloading. As a consequence, the J-integral is only applicable to elastic–plastic deformation behaviour if no unloading occurs. The J-integral can be interpreted in different ways: (a) Path-Independent Line Integral For an arbitrary path around the tip of a crack (Fig. G7) the J-integral is obtained by ui w · dy − Ti ds (G20) J= x
with w being the strain energy density w=
ij ij dij
(G21)
0
Ti designates the components of the traction vector normal to , ui the displacement vector components and ds the length increment along . With the exception of contours very close to the crack tip, the resulting J is independent of the chosen contour and is equal to zero if no crack exists [G3]. Equation (G20) has to be modified by an area correction when gradients of secondary stress exist within the contour . (b) Non-linear Energy Release Rate The J-integral can also be interpreted as a generalised energy release rate [G3] J=−
dU dA
(G22)
y
x Contour Γ ds
Figure G7: J-integral: arbitrary contour around the crack tip.
280
Glossary
In Eqn (G22), U is the potential energy and A is the crack area. For unit thickness, A is usually replaced by the crack depth a. For linear elastic deformation behaviour, J is identical to the linear elastic energy release rate which permits correlation with the K-factor: 2 K /E for plane stress, (G23) J= K2 1 − 2 /E for plane strain. (c) Non-linear “Stress Intensity Factor” in the HRR Field In conjunction with the HRR field, J can be interpreted as the equivalent of a “stress intensity factor” for non-linear elastic deformation behaviour [G1, G2]. The HRR stress field is expressed as
J ij = 0 · · 0 · 0 · In · r
1/1+n · ˜ ij n
(G24)
with zz =
0 5 · xx + yy 0
for plane strain, for plane stress,
(G25)
and iz = 0. In describes an integration constant dependent on whether plane strain or plane stress conditions are given and on strain hardening. ˜ ij n is a dimensionless geometry function and 0 , 0 , and n are parameters of the Ramberg–Osgood description of the stress–strain curve (Eqn (G3)). The information provided by Eqn (G23) can be summarised as follows: • The J-integral describes the asymptotic field as r → 0 of the stress distribution in the ligament comparable to the K-factor, but with a (1/r) type singularity. Note that for linear elastic material behaviour, i.e., n√= 1, the exponent in Eqn (G24) becomes 1/2 and the singularity becomes 1 r which is consistent with the definition of K (→ “Stress Intensity Factor”). • In contrast to K, the J-integral, as a measure of the crack driving force, is not only dependent on the component and crack geometry and dimensions and on the loading type but, in addition, on the deformation characteristics of the material, i.e., its stress–strain curve. With the exception of Option 5 in SINTAP/FITNET the elastic–plastic J-integral is determined by none of the above-mentioned definitions. It is, instead, based on a formal plasticity correction of the elastic J-integral (obtained from K) provided by the fLr functions.
Glossary
281
Ligament The ligament is the intact cross section ahead of a crack.
Multi-optional Approach The SINTAP/FITNET procedure comprises various assessment options with respect to complexity and accuracy. The lowest option (0) is the simplest and most conservative. Only the yield strength has to be available as a strength parameter. The fracture toughness may be estimated from Charpy data. The higher options (1 and 3) require the yield and tensile strength or the complete stress–strain curve respectively, and fracture toughness data. For certain applications, the conservatism can be further reduced by stable tearing analyses, constraint corrections of the fracture toughness (Option 5) or by finite element analysis to provide input parameters (Option 4). A special option (2) refers to strength mismatched components consisting of two materials, e.g., weldments. If a component is shown not to be in a critical state at a certain option no further analysis is necessary. Otherwise, the analysis should be repeated at a higher level.
Plane Strain Versus Plane Stress Conditions Classical fracture mechanics is based on two-dimensional (2D) considerations of cracked structures. Three-dimensional effects on the stress–strain state are modelled by two boundary conditions: plane strain, in which the strain component in the thickness direction, z , is set to zero, and plane stress, in which the stress component in the thickness direction, zz , is set to zero (for nomenclature see Figs 4.23 and G1). In reality, the conditions ahead of a crack are neither plane strain nor plane stress but three-dimensional. There exist, however, limit cases. The central portions of a specimen or component will be subjected almost to plane strain, whereas the surfaces are closer to plane stress. Also, thick walled components can be described approximately by plane strain and thin walled components by plane stress. Whether plane strain or plane stress conditions prevail also depends on the development of ligament yielding (a predominantly plane strain state at low loads and, associated with this, small ligament yielding Lr may change to predominant plane stress at higher loads and increased ligament yielding Lr . The terms plane stress and plane strain refer to the out-of-plane constraint (→“Constraint and Stress Triaxiality”). Commonly, the fracture toughness is determined for plane strain conditions since these should generate lower-bound values (one exception to this is thin
282
Glossary
walled plates). On the other hand, for determining the crack driving force, plane stress conditions are usually conservative since they yield lower bound yield loads and therefore underestimate the critical loads or critical crack sizes.
Primary and Secondary Stresses In a SINTAP/FITNET analysis, the stresses have to be categorised as primary and secondary. Primary stresses are the result of the applied mechanical loads including any dead weight or inertia effects. In contrast, secondary stresses are due to suppressed local distortions or to thermal gradients and are self-equilibrating across the structure. Residual and thermal stresses are typical secondary stresses, although thermal loading can also produce primary stresses. The SINTAP/FITNET specification deviates from the general use of the terms insofar as primary stresses are defined as stresses that contribute to plastic collapse and secondary stresses are defined as stresses that do not. Note, however, that even stresses that are self-equilibrating over the entire component may sometimes contribute to plastic collapse in the net section containing the crack. They then have to be categorised as primary rather than as secondary. If it is not clear which kind of classification is appropriate, the treatment as primary stress is conservative. In SINTAP/FITNET secondary stresses are considered in the determination of the K-factor but not in the determination of the yield load or its corresponding parameters (Section 6.6).
Proportional Loading Proportional loading means that all stress components at all locations increase in proportion to the applied load. This excludes local effects such as unloading in the wake of a growing crack. Crack driving force solutions such as used in SINTAP/FITNET usually presuppose proportional loading.
Reliability and Failure Probability Input parameters, such as the crack size detected by non-destructive inspection and the fracture toughness, may be subject to considerable uncertainty or to scatter. Applying conservative lower bounds can lead to overconservative predictions of SINTAP/FITNET analyses. In such cases, suitable alternatives include statistical methods such as Monte Carlo simulations or First- and Second-Order Reliability
Glossary
283
approaches. The result of such an analysis will be a failure probability of the component, a statistical distribution of the critical load, the critical crack size, etc. Note that the failure probability of the component differs from the failure probability of specimens in the Master Curve concept (Section 4.4.5.2). An inverse reliability approach is provided by the concept of partial safety factors. The starting point is a target failure probability of the component. Which target probability will be acceptable depends on the consequences of the failure and whether or not the component is designed as redundant. Once the target probability is given, partial safety factors are determined for all relevant input parameters that are dependent on the coefficient of variation (the quotient of standard deviation and mean value) of the input distributions. Reliability aspects are discussed in Section 6.13.
Stable and Unstable Crack Extension These terms are used with respect to the state of the component. A stable crack only extends when the load is increased. In contrast to this, once a crack becomes unstable it will grow even if the load is decreased or removed. If it is not arrested at a later stage it will cause the component to fail. Stable crack extension usually occurs in displacement-controlled loading. Note that the plastic collapse criterion included in the SINTAP/FITNET fLr functions refers to an unstable failure mode. The terms stable and unstable have to be distinguished from the terms brittle and ductile as well as from the terms cleavage and micro-ductile.
Strain Hardening Exponent Within SINTAP/FITNET the strain hardening exponent N is the slope of the plastic branch of the stress–strain curve in double-logarithmic scales. The conservatively estimated strain hardening exponent in analysis Options 1 and 2 is also based on this original definition, although it represents an empirical lower bound to experimental material data. Further definitions are in use in the literature and competing documents, an example of this being the Ramberg–Osgood exponent (Eqn (G3)). Because the various definitions of N (or n) yield different numerical values these are not interchangeable. Only strain hardening exponents, the definition of which is in accordance with SINTAP/FITNET, are permitted for use here.
284
Glossary
Strength Mismatch Material compounds, such as bi-materials or welds, may consist of materials of different strength. In the case of welds it is necessary to distinguish between overmatching (the weld metal is of higher strength than the base plate) and undermatching (the weld metal is of lower strength than the base plate). Commonly, mismatch is defined by the yield strength ratio of the materials involved. However, a SINTAP/FITNET analysis differs in that the strain hardening coefficients or the complete stress–strain curves also have to be taken into account. The mismatch effect leads to local strain concentration zones and, consequently, to the potential for increased crack driving force that would not exist in homogeneous components. In SINTAP/FITNET, mismatch effects are explicitly treated for differences in the yield strengths of 10% or more. Two assessment options (Option 2 and a special module of Option 3) can be applied (Section 6.11).
Stress Intensity Factor, Linear Elastic (K-Factor) For linear elastic deformation behaviour, the stress (and strain) field ahead of a crack tip can be described by the stress intensity or K-factor [G17] in the following equation 1 ij = √ K · fij + higher order terms r
(G26)
with r and being polar coordinates describing the location of arbitrary material points (in Fig. G8 displayed as a square) in the ligament.
σyy
y
τxy τxy σxx
σxx τ τxy xy
r Crack
θ
σyy x
Figure G8: Stresses acting at an arbitrary material point in the ligament ahead of a crack.
Glossary
285
If one neglects the “higher order terms” in Eqn (G26), it can be written as KI
3
· 1 − sin · sin (G27) xx = √ · cos 2 2 2 2r KI
3
· 1 + sin · sin (G28) yy = √ · cos 2 2 2 2r xx + yy plane strain (G29) zz = 0 plane stress K
3
· sin · cos (G30) xy = √ I · cos 2 2 2 2r xz = yz = 0
(G31)
for crack opening mode I. Similar K solutions, which will not be reproduced here, are obtained for mode II and III crack opening. The information given by Eqns (G26)–(G31) is as follows: (a) The first term of Eqn (G26) describes a stress singularity, the amplitude of which, or, in fracture mechanics terminology, its intensity, is defined by the linear elastic K-factor.
√ (b) The stresses near the crack tip vary with 1 r and become infinite for r → 0, i.e., immediately at the crack tip. Note that the same is true with respect to strains near the crack tip. (c) K describes the distribution of all stress and strain components in the region ahead of the crack tip. (d) Since Eqns (G27)–(G31) indicate that the stresses become infinite at the crack tip itself, it is immediately clear that the equations cannot be applied there. The same statement has to be made for the stresses away from the crack tip such as demonstrated schematically in Fig. (G9). The region ahead of the crack tip, where Eqns (G27)–(G31) describe the real stress field in a satisfactory way, is designated as the K singularity dominated, or simply the K controlled, zone. (e) The higher order terms in Eqn (G26) which have been neglected in Eqns (G27)–(G31) depend on the component geometry. One example is the T-stress term that is briefly discussed in Section 4.4.6.2. The crack parallel stress, which is not included in the K solution, is used as a second parameter for describing constraint. (f) Equations (G26)–(G31) are restricted to linear elastic deformation behaviour. With increasing ligament yielding parameter Lr , the K controlled zone
286
Glossary
σyy Real stress distribution K field stress distribution
θ=0
Crack depth a
r K controlled region
Figure G9: Schematic view of the K controlled zone ahead of a crack tip (hatched region).
becomes smaller. Above Lr = 0.5–0.6, the K-factor loses its meaning as a crack driving force parameter and has to be replaced by elastic–plastic parameters such as the J-integral or the crack tip opening displacement, . Note that when K is used as an input parameter in SINTAP/FITNET analysis, it is formally defined as a parameter describing the loading at the crack tip. In order to apply the concept to real geometries, K has to be determined as a function of the applied load in terms of remote loads or stress distributions in the cross section bearing the crack. The general expression for this follows a format √ K = a · Y
(G32)
with being the applied stress, a the crack size and Y a geometry function characterising the component and loading geometry under consideration. Different types of K solutions have been described in detail in Section 5.1.2. Note that the applied load is given for the uncracked structure. Care has to be exercised for multi-path loading when the applied load in the uncracked member is affected by the damage of another member of the structure.
Sub-critical Crack Extension This term is used in contrast to stable crack extension although the phenomena have in common that the component is not immediately at risk. Whereas “stable” is used in conjunction with end-of-life criteria, sub-critical refers to crack extension due to fatigue, creep, stress corrosion or combined mechanisms. Cracks extend sub-critically at loads significantly lower than the failure loads.
Glossary
287
True Stress–Strain Curve The SINTAP/FITNET procedure is based on the true stress–strain curve to describe the deformation behaviour of the material. Up to the maximum point of the uniaxial engineering stress–strain curve, the true stress is obtained as t = 1 + and the true or logarithmic strain as t = ln1 + with and being the engineering stress and strain.
Upper and Lower Shelf These terms refer to the different failure mechanisms, which materials with body centred cubic and hexagonal lattices can experience. On the upper shelf those materials fail by micro-ductile fracture. The final unstable ductile failure is preceded by a certain amount of stable ductile crack extension, which is described by a crack resistance or R-curve. With increased stress triaxiality (→ “Constraint and Stress Triaxiality”) because of, for example, the presence of notches, geometrical transitions or thicker sections, the failure mechanism will occur at higher temperatures. Faster loading rates can increase not only the apparant “ductile” toughness but also the ductile-to-brittle transition temperature or even induce fully brittle behaviour. (→ “Ductile-to-Brittle Transition”). On the lower shelf the failure occurs by cleavage, perhaps following marginal stable crack extension. In contrast to the ductile-to-brittle transition region, scatter in the measured toughness values is rather moderate on the upper and lower shelves.
Yield Load The term yield load or net section yield load as used in this book is frequently designated as the limit load. In common mechanics terminology the term marks the maximum load that a cracked structure of perfectly plastic material can sustain. However, strain-hardening materials display two limit states: one at which yielding spreads over the whole ligament, and a second at which the maximum load is reached and the component fails. Following the terminology of this book, the first limit state is associated with the term “yield load” and the second with the term “collapse load”, although it should be noted that that the definition for the yield load follows an ideal concept and is often equal to the limit load for an ideally plastic material with a strength equal to the material yield strength. However, the yield loads available in the SINTAP/FITNET compendium have been obtained by a variety of different methods. The yield load is often the most important model parameter affecting the accuracy of a SINTAP/FITNET analysis.
288
Glossary
References [G1] Hutchinson, J.W., Singular behaviour at the end of a tensile crack tip in a hardening material, J. Mech. Phys. Solids, 1968, 16, 13–31. [G2] Rice, J.R. and Rosengren, G.F., Plain strain deformation near a crack tip in a power-law hardening material, J. Mech. Phys. Solids, 1968, 16, 1–12. [G3] Rice, J.R., A path independent integral and the approximate analysis of strain concentration by notches and cracks, J. Appl. Mech., 1968, 35, 379–386. [G4] Shih, C.F., Relationship between the J integral and the crack opening displacement for stationary and extending cracks, J. Mech. Phys. Solids, 1981, 29, 305–326. [G5] Schwalbe, K.-H., A modification of the COD concept and its tentative application to the residual strength of center cracked panels, ASTM STP 700 (pp. 500–512), American Society for Testing and Materials, 1981. [G6] Schwalbe, K.-H., Some aspects of the crack tip opening displacement concept, J. Aeronaut. Soc. India, Special issue on fracture mechanics dedicated to Prof. G.R. Irwin, 1986, 271–285. [G7] Wells, A.A., Unstable crack propagation in metals: Cleavage and fast fracture, in: Proc. Crack Propagation Symposium, Cranfield, UK, Band 1, Paper 84. [G8] Dugdale, D.S., Yielding of steel sheets containing slits, J. Mech. Phys. Solids, 1960, 8, 100–108. [G9] Burdekin, F.M. and Stone, D.E.W., The crack opening displacement approach to fracture mechanics in yielding materials, J. Strain Analysis, 1966, 1, 145–153. [G10] BS 5762 (1979): Methods for Crack Opening Displacement (COD) Testing, British Standards Institution (BSI), London (last time revised in 1991). [G11] ASTM E 1290 (1993): Standard Test Method for Crack-Tip Opening Displacement (CTOD) Fracture Toughness Measurement, American Society for Testing and Materials. Note that the plastic hinge based approach is replaced by an p1 based approach (comparable to J determination in the more recent version of the standard E 1290-02. [G12] Schwalbe, K.-H., Introduction of 5 as an operational definition of the CTOD and its practical use, ASTM STP 1256 (pp. 763–778), American Society for Testing and Materials, 1995. [G13] Schwalbe, K.-H, Newman, J.C., Jr. and Shannon, J.L., Jr., Fracture mechanics testing on specimens with low constraint—standardisation activities within ISO and ASTM, Eng. Fract. Mech., 2005, 72, 557–576. [G14] Irwin, G.R., Plastic zone near a crack and fracture toughness, Proc. Sagamore Research Conf., Vol. 4, 1961. [G15] Kumar, V. and Lee, Y.J., Fully plastic solutions for a single-edge cracked plate subjected to combined tension and bending, in: Kumar, V., German, M.D., Wilkening, W.W., Andrews, W.R., deLorenzi, H.G. and Mowbray, D.F. (Eds.), Advances in Elastic–Plastic Fracture Analysis, General Electric Company, Schenectady, New York, Section 2, 1984. [G16] Landes, J.D. and Shaffer, D.H. Statistical characterization of fracture in the transition regime, ASTM STP 700 (pp. 368–382), American Society for Testing and Materials, Philadelphia. [G17] Irwin, G.R., Analysis of stresses and strains near the end of a crack traversing a plate, J. Appl. Mech., 1957, 24, 361–364.
Subject Index
acceptable conditions of a component see structural integrity analyses aluminium alloys see metallic materials analytical yield load solutions see net section yield load analytical stress intensity factor (K factor) solutions, types of see stress intensity factor American Society for Testing and Materials (ASTM) ASTM E 1921 standard 57, 60, 62, 266 further ASTM test standards and standard activities 198, 199, 266 anisotropy of stress-strain behaviour see stress-strain curve assessment path see Failure Assessment Diagram (FAD) assessment point see Failure Assessment Diagram (FAD) ASTM see American Society for Testing and Materials austenitic steel see metallic materials bainitic steels see metallic materials basic crack types see crack geometry Basic Option see multi-optional concept BEGL see British Energy Generation Ltd bend specimens see fracture toughness test specimens bending stress see also stress profile through wall thickness(stress linearization) due to misalignment 26, 225–27 general 48, 89, 125 global bending stress 32, 95–97 input parameter for K factor determination 32–33, 36 input parameter for net section yield load determination 35–36 bi-modal Master Curve approach see Master Curve approach blunting, crack tip see fracture toughness
British Energy Generation Ltd (BEGL, former Central Electricity Generating Board, CEGB) general 1–3, 10, 12–13, 89 R6 Routine, general aspects 114, 241 R6 Routine, older versions 10, 12 R6 Routine, Revision 3 3, 13–15, 17 R6 Routine, Revision 4 3, 55–56, 81, 84, 89, 115–16, 120–22, 165, 198, 211, 216 British Standards Institution (BSI) BS 7910 standard 3, 9, 15, 39, 53, 89, 107, 116, 232, 236–38, 241 BSI test standards 265–66 brittle fracture 51–52, see also fracture mechanisms BS 7910 standard see British Standards Institution (BSI) ß constraint parameter see constraint CDF philosophy see Crack Driving Force (CDF) CEGB see British Energy Generation Ltd CEN see Comité Européen de Normalisation (European Committee for Standardisation) census criterion see Master Curve approach Charpy energy see Charpy test Charpy test see also fracture toughness (reference toughness) Charpy energy 85–87 sub-sized Charpy specimens 88 transition temperature 87–88 cleavage fracture see fracture mechanisms coefficients of variation (COV) see also reliability analysis crack dimensions 232, 236 fracture toughness data 64, 232, 237–38 reliability analysis 231–35 stress-strain data 232, 237 collapse, plastic 8, 10–11, 14, 19, 25, 29, 112, 137, 141, 144, 148–51, 162, 211–13, 219, 238–39, 278, 282–83, 287
290
Subject Index
Comité Européen de Normalisation (European Committee for Standardisation) FITNET document 3–5, 18, 26–27, 69, 84, 89, 115, 130, 194–98, 211, 216, 241 compact tension specimens C(T) see fracture mechanics test specimens conservatism of analysis see structural integrity analyses constraint see also fracture toughness (geometry dependency) along crack front of semi-elliptical surface cracks 76, 78, 80, 189 constraint parameters 76–85 general 2–3, 20, 22–23, 26, 57, 59, 74–76, 238–39, 269–70 in-plane and out-of-plane constraint 75, 199, 269–70, 281–82 necking of specimen surface 269–70 Q stress 79–80 stress triaxiality 74, 76–78 T stress 78–79, 83–85, 189–90 thin-walled structures 199, 266–67 treatment in Crack Driving Force (CDF) approach 188–91 treatment in Failure Assessment Diagram (FAD) approach 188–91 ß parameter 79–84, 190 contained yielding see ligament yielding continuous and discontinuous yielding see stress-strain curve (Lüders’s plateau) corner cracks see crack geometry, see also worked examples COV see coefficient of variation crack arrest analyses see structural integrity analyses Crack Driving Force (CDF) see also structural integrity analyses CDF philosophy 12, 24–25, 143–46, 270 constraint, treatment of 188–91 determination of critical conditions 137–38, 144–46, 153–62 misalignment, treatment of 225–29 primary and secondary stresses, treatment of 162–65, 172–85 stable crack initiation, treatment of 144 strength mismatch, treatment of 208–20 thin walled structures, treatment of 198–209, 247 unstable ductile fracture, ductile tearing, treatment of 144–46, 159–62, 202–08, 241–44 crack geometry basic crack types 36–37 corner cracks 37, 105 crack front length 57, 60–61, 87
crack plane 41–43 embedded cracks 37 surface cracks, semi-elliptical 37, 91, 95, 100, 108, 117, 120, 153, 172, 189, 241–44 through wall cracks 37, 132, 202, 206, 214, 217, 224, 227, 243–44, 246–248 crack front length see crack geometry crack idealisation crack orientation 41–43, 191, 206 crack plane 41–43 crack re-characterisation 40–41 crack shape idealisation 36–37 projected crack depth 41–43 crack interaction between multiple cracks or with surfaces 38–40 crack-like defects planar and volumetric flaws 36 crack opening modes 110, 192 crack orientation see crack idealisation crack plane see crack idealisation crack re-characterisation see crack idealisation crack shape idealisation see crack idealisation crack tip blunting see fracture toughness crack tip opening displacement; CTOD CTOD-5 199–200, 203, 244–47, 275–76 definitions of 271–76 determination in SINTAP/FITNET 143, 200–02 use in other methods: Design Curves 9–10; ETM 15–16 crack types see crack geometry critical conditions of a component see structural integrity analyses C(T) specimen see fracture toughness test specimens CTOD see crack tip opening displacement CTOD-5 see crack tip opening displacement Design Curve approach see The Welding Institute ductile fracture see fracture mechanisms (micro-ductile fracture) ductile tearing see also fracture mechanisms treatment in Crack Driving Force (CDF) approach 144–46 treatment in Failure Assessment Diagram (FAD) approach 140–42 ductile-to-brittle transition see fracture mechanisms embedded cracks see crack geometry energy release rate see J integral (definitions) engineering stress-strain curve see stress-strain curve Engineering Treatment Model (ETM) see GKSS Research Centre
Subject Index
EPRI (Electric Power Generation Institute) approach 11–12, 14, 90, 116, equivalent K factor for mixed mode loading see stress intensity factor ESIS see European Structural Integrity Society ETM see Engineering Treatment Model European Structural Integrity Society (ESIS) ESIS test procedures 55, 223, 267 FAD philosophy see Failure Assessment Diagram (FAD) Failure Assessment Diagram (FAD) see also structural integrity analyses assessment path 10–11, 141–43 assessment point 10–11, 24–25, 139–42 constraint, treatment of 188–91 determination of critical conditions 137–38, 141–42, 153–62 FAD philosophy 10–11, 13–15, 24–25, 139–43, 277–78 failure line, FAD line 10–11, 24–25, 139–41, 277–78 misalignment, treatment of 225–29 non-unique solutions 142–43 primary and secondary stresses, treatment of 162–65, 172–85 stable crack initiation, treatment of 144 strength mismatch, treatment of 207–20, 247 unstable ductile fracture, ductile tearing, treatment of 141–42, 159–62, 241–44 failure investigation see structural integrity analyes see also validation examples failure line see Failure Assessment Diagram (FAD) failure probability of components 231–32, 233–35 scatter of fracture resistance 57, 62, 233 FEM see structural integrity analyses ferritic steels see metallic materials FFS see structural integrity analyses (fitness-for-service) finite element method (FEM) see structural integrity analyses First Order Reliability Method, FORM see reliability analysis fitness-for-service analyses see structural integrity analyses flaw assessment see structural integrity analyses flow stress see stress-strain curve FORM see reliability analysis fracture assessment see structural integrity analyses
291
fracture mechanisms cleavage fracture 19, 25, 51–53, 55, 83, 137, 277, 287 ductile tearing 51, 54–56, 80 ductile-to-brittle transition 51–52, 56–57 micro-ductile fracture 19, 25, 51–52, 137, 277, 287 fracture modes stable crack initiation (and extension) 19, 51–54, 80, 137, 198–99, 283 unstable crack extension 19, 51–52, 137, 144–46, 283 fracture resistance curve (R curve) see fracture toughness fracture toughness crack tip blunting 52–56, 271 ductile-to-brittle transition 51–53, 56–74, 80, 87, 195–96, 231–32 266, 277 geometry dependency (constraint) 74–75 geometry dependency (statistical weakest link effect) 57 inhomogeneity 63, 68–74 lower shelf fracture toughness 51–54, 80, 85, 195–06, 231–32, 287 R curve, fracture resistance curve 50–51, 54–56, 80, 86, 140–42, 144–46, 160, 195, 198–202, 242, 244, 265–67, 287 reference toughness based on Charpy data 2, 21–23, 27, 85–88, 232, 242–244 scatter 51–52, 54–57, 68, 140, 266, 277, 287 transition curve 51–52 upper shelf fracture toughness 51–53, 54–56, 68, 80, 86, 195–96, 231–32, 287 fracture toughness test procedures low constraint (thin walled) geometries 199, 265, 267, 276 plane strain fracture toughness 53, 140, 265–67 required number of tests 53–54, 67–68 test procedures and standards 55, 57, 60, 62, 198–99, 223, 265–67, 276 weldments 265–267 fracture toughness test specimens bend specimens 54, 60, 74–76, 80, 140, 223, 247, 269 compact tension specimens; C(T) 61, 189, 199, 201, 244 cruciform specimens 202–08, 246 tension specimens, M(T) specimens 54, 74–76, 132, 199, 223 fully-plastic yielding see ligament yielding fusion line see weldments
292
Subject Index
general yielding see ligament yielding geometry factor solutions in table format see stress intensity factor GKSS Research Centre Geesthacht Engineering Treatment Model (ETM) 15–17, 90, 116, 148–149, 215 GKSS fracture toughness test procedure 223, 267 global bending stress see bending stress component global yield load see net section yield load HAZ see weldments heat affected zone (HAZ) see weldments HRR field solutions 11–12, 75, 79, 271–72, 280 impact loading see rapid loading in-plane constraint see constraint instability load see structural integrity analyses interaction effects of cracks with surfaces see crack interaction interaction effects of multiple cracks see crack interaction International Organisation for Standardisation ISO test standards 199, 265, 276 ISO see International Organisation for Standardisation J-integral definitions 278–80 determination according to the SINTAP/FITNET approach 143 K-factor see stress intensity factor leak-before-break analyses see structural integrity analyses ligament yielding contained yielding 8, 15, 25, 59, 161, 164, 271, 276–77 fully-plastic yielding 276–77 general yielding 276–77 gross-section yielding 39–40, 276–77 net-section yielding 8–9, 25, 39–40, 59, 163, 271, 276–77 plastic collapse, plastic collapse limit 8, 10, 14, 25, 112, 141, 148–51, 210–13 plastic zone 276–77 small-scale yielding 25, 59, 163, 271, 274, 276–277 ligament yielding parameter Lr see net-section yield load limit load see net-section yield load
loading rate effect on stress-strain data 50, 194–96 effect on fracture toughness 194–97 local yield load see net-section yield load lower shelf fracture toughness see fracture toughness lower yield strength see stress-strain curve Lüders’ plateau see stress-strain curve Lüders’ strain see stress-strain curve Master Curve approach see also fracture mechanisms (ductile-to-brittle transition) application to inhomogeneous materials (stage 1–3 analyses) 68–74 ASTM E 1921 standard 57, 60, 62, 266 bi-modal Master Curve approach 69, 73–74 census criterion 58–59, 63, 69 transition temperature To 64–68, 71–73, 84–85, 87, 197, 287 membrane stress see also stress profile through wall thickness (stress linearization) input parameter for K-factor determination 30–34 input parameter for yield load determination 35–36 metallic materials aluminium alloys 45, 47, 153, 168, 170, 198–99, 201–202, 209, 232, 244–46 austenitic steels 14, 17, 45–46, 52, 152–53, 168–71, 172, 233, 241, 244 bainitic steels 50–52, 55–56, 80 C-Mn steels 14 ferritic steels 17, 45, 50–52, 55–57, 80, 85, 152–53, 168, 170–71, 186, 231, 232–33, 241–44, 266, 277 micro-ductile fracture see fracture mechanisms minimum toughness, required see structural integrity analyses see also worked examples misalignment see weld misalignment treatment in Crack Driving Force (CDF) approach 225–29 treatment in Failure Assessment Diagram (FAD) approach 225–29 mismatch factor, mismatch ratio see strength mismatch Mismatch Option see multi-optional concept mixed mode loading see also crack opening modes treatment of in SINTAP/FITNET 191–94, 201–02, 205–08 modulus of elasticity (Young’s modulus) see stress-strain curve Monte Carlo simulation see reliability analysis
Subject Index
M(T) specimens see fracture mechanics test specimens multi-optional concept of SINTAP/FITNET see also structural integrity analyses (conservatism of analysis) higher assessment options/specific modules 21–23 Option 0 analyses (“Basic Option”) 21–22, 24, 147–148, 152, 242–44 Option 1 analyses (“Standard Option”) 21–22, 24, 148–150, 152–162, 172–85, 188–91, 226–29, 231–35, 242–44, 248–49 Option 2 analyses (“Mismatch Option”) 21–22, 24, 210–18, 247 Option 3 analyses (“Stress-Strain Defined Option) 21–23, 24, 151–162, 172–85, 202–08, 219–20, 244–49 R6 Options 1 and 2 14–15, 17 necking see constraint net-section reference stress ref see net-section yield load net-section yielding see ligament yielding net-section yield load analytical solutions 112–13, 115–16 conservatism 114–15 definition 8, 112–14 determination 112–13 example, types of solutions 116–35 ligament yielding parameter Lr 112–13 local and global yield load 113–14 net-section reference stress ref 112–13 solutions for strength mismatch components 129–35, 214–16 sources of analytical solutions 115–16 non-unique solutions see Failure Assessment Diagram (FAD) Option 0, 1, 2, 3 analyses see multi-optional concept out-of-plane constraint see constraint overload test see structural integrity analyses partial safety factors see reliability analysis plastic collapse see ligament yielding plastic zone see ligament yielding plastic zone corrected stress intensity factor see also stress intensity factor determination in the EPRI approach 11 determination in the ETM approach 15–16 in conjunction with primary-secondary stress interaction 164–65 Poisson’s ratio see stress-strain curve
293
post-weld heat treatment (PWHT) see welding residual stresses profiles post weld mechanical treatment see welding residual stresses profiles primary stresses 29–30, 162–65, 172–85 projected crack depth see crack idealisation proof strength see stress-strain curve proof test see structural integrity analyses proportional loading 282 PWHT see welding residual stresses profiles Q-stress see constraint Quasi-static versus dynamic analyses see rapid loading R6 routine see British Energy Generation Ltd R-curve see fracture toughness R-curve analyses see structural integrity analyses, also worked examples rapid loading 194–97 reference stress see net-section yield load Reference Stress approach 12–13 reference toughness based on Charpy data see fracture toughness reliability analysis see also coefficient of variation (COV) First Order Reliability method (FORM) 231, 282–283 Monte Carlo simulation 231, 282–283 partial safety factors 231, 235–38, 282–283 Second Order Reliability method (SORM) 231, 282–283 residual stresses see welding residual stress profiles, also secondary stresses Second Order Reliability Method, SORM see reliability analysis secondary stresses 29–30, 162–65, 172–85 section modulus 32, 35 SEN(B) specimens see fracture toughness test specimens (bend specimens) sensitivity analysis 71, 80, 114, 230 SIF see stress intensity factor small-scale yielding see ligament yielding SORM see reliability analysis stable crack initiation see fracture modes Stage 1, 2, 3 analyses see Master Curve approach Standard Option see multi-optional concept steel see metallic materials stepwise graded conservatism see multi-optional concept
294
Subject Index
strain hardening exponent see also stress-strain curve general 8, 195–96, 269, 283 mismatch corrected 211–13 power law definition of the strain hardening exponent 15, 82 Ramberg-Osgood definition of the strain hardening exponent 12, 272, 277 SINTAP/FITNET definition of the strain hardening exponent 46–47, 283 strength mismatch see also weldments effect on constraint and fracture toughness 222–23 effect on the yield load 130–35 mismatch factor/mismatch ratio 40, 46, 50, 130, 209, 220, 247, 284 treatment in Crack Driving Force (CDF) approach 214–18 treatment in Failure Assessment Diagram (FAD) approach 214–18 stress intensity factor (K-factor) definition 284–86 determination by code solutions (based on linearised stress profiles) 107–10 determination by handbook solutions 89–90 determination by weight function solutions 92–104 equivalent K factor for mixed mode loading 193–94 individual determination using finite elements 111 superposition of 110 treatment of geometry factor solutions available in table format 110–11 types of analytical solutions 90 stress linearization see stress profile through wall thickness stress profile through wall thickness polynomial fit 30–32 stress linearization 32–36 Stress-Strain Defined Option see multi-optional concept stress-strain curve anisotropy 200–01 engineering and true stress-strain curve 44–45 flow stress 46–47 Lüders’ plateau, Lüders’strain 47–49, 148–49, 210–13 Poisson’s ratio 44–45 proof strength 45, 147, 150, 169 strain hardening see strain hardening exponent scatter of stress-strain properties in welds 50, 221–22 yield strength, lower and upper 45–46 Young’s modulus 44–45
stress-strain field ahead of a crack 77, 284–85 see also HRR field solutions stress triaxiality see constraint structural assessment see structural integrity analyses structural integrity analyses acceptable conditions of a component 137–39 conservatism of analysis 15, 23, 45–47, 112–15, 125, 147, 171–72, 185–86, 200–01, 238–39, 241, 281 crack arrest analyses 41, 195, 197–98 Crack Driving Force (CDF) versus Failure Assessment Diagram (FAD) philosophy 24–25 critical conditions of a component 20–21, 137–39 critical crack dimensions 137–39, 157–59 critical load 137–39, 153–57 failure investigation 248–49 finite element method (FEM) 23, 30, 34, 50, 78–79, 92, 111, 113, 128–29, 226, 281 fitness-for-service analyses (FFS), also fitness-for-purpose 19–20 instability load 141–42, 144–46, 159–62, 202–07, 241–47 leak-before-break analyses 2, 4, 41–42, 46, 76, 114 minimum toughness, required 137–39, 158–59 stepwise graded conservatism see multi-optional concept proof test, overload test 46–48, 51, 76, 114, 186–87 R curve analyses see instability load sub-critical crack extension (fatigue or creep crack extension etc.) 3–4, 19, 286 sub-critical crack extension see structural integrity analyses substitute component geometry 34, 125 superposition of K-factors see stress intensity factor surface cracks see crack geometry T-stress see constraint target probability for component failure see reliability analysis tension specimens see fracture toughness test specimens thermal loading 29, 95–97, 105, 162, 282 The Welding Institute Design Curve approach 9–10 thin wall structures definition 198–99 treatment in SINTAP/FITNET 200–07 Three parameter Weibull distribution see Master Curve approach
Subject Index
through wall cracks see crack geometry, also worked examples transition temperature see Charpy test, also Master Curve approach true stress-strain curve see stress-strain curve TWI see The Welding Institute Two parameter Weibull distribution see Master Curve approach unstable crack extension see fracture modes upper shelf fracture toughness see fracture toughness upper yield strength see stress-strain curve V factor (for primary and secondary stresses interaction) 164–65 validation examples failure investigation 248–49 forklift 248–49 pipelines 241–44 pressurised tubes 241–44 strength mismatch configurations 247 thin wall structures 244–47 VTT see Master Curve approach weakest link model see fracture toughness (ductile-to-brittle transition) Weibull distribution see Master Curve approach welding residual stress profiles at surface 168–70 longitudinal (parallel to fusion line) and transverse (normal to fusion line) 168–72, 178 post-weld heat treatment (PWHT) 187 post weld mechanical treatment 186–87 residual stress profile compendium 2, 26, 46, 165–72, 185–86 through thickness 170–72 treatment in structural assessment 159–65, 172–85 weldments cracks at the fusion line 131–35, 172–85, 216–18, 223 cracks in the weld metal 131–35, 214–16, 223 heat affected zone (HAZ) strength and toughness properties 69, 71, 197, 221, 223 misalignment see weld misalignment mismatch see strength mismatch residual stresses see welding residual stress profiles weld misalignment definition 225 treatment in structural assessment 225–29
295
worked examples axial (longitudinal) cracks in hollow cylinders 92–95, 118–24 bending and membrane stress components 33, 36, 107–10 biaxial tension loading 202–08 circumferential cracks in hollow cylinders 95–97 constraint 81–83, 188–92 corner cracks 105–07 crack instability, R-curve analysis 162–63, 202–07 critical crack size 157–58 critical load 153–57, 159–62, 172–85, 188–92, 202–08, 214–08, 226–29, 231–35 cylinders, hollow 92–97, 118–24 fracture toughness distribution at the upper shelf, R curve 55–56 fracture toughness distribution in the ductile-to-brittle transition 61–62, 64–67, 238 Master Curve analysis 61–62, 64–67, 238 minimum required fracture toughness 158–59 mixed mode loading 205–08 net section yield load 116–28, 131–35, 202–07, 214–15, 216–18, 226–29 nozzle section 105–07 partial safety factor 238 plates 107–10, 116–18, 125–28, 131–35, 153–62, 172–85, 188–92, 202–08, 214–08, 226–29, 231–35 polynomial stress profile 30–31 reliability analysis 231–35 secondary stresses 172–85 semi-elliptical surface cracks 92–95, 97–103, 107–10, 116–18, 118–24, 125–28, 153–62, 172–85, 188–92, 231–35 strength mismatch 131–35, 214–18 stress intensity factor 90–110, 154–57, 172–85, 204–05, 214–15, 226–29 stress linearization 33, 36, 107–10 thin wall structures 202–08 through thickness cracks 131–35, 202–08, 214–08, 226–29 weld misalignment 226–29 welding residual stresses 172–85 yield load see net section yield load yield strain see stress-strain curve yield strength see stress-strain curve Young’s modulus see stress-strain curve
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