Cumulative damage of welded joints

Cumulative damage of welded joints

Related titles: Fatigue in railway engineering (ISBN-13: 978-1-85573-740-2; ISBN-10: 1-85573-740-X) This authoritative new book includes every aspect of fatigue in the railway system with chapters on the following topics: fundamentals of fatigue, infrastructure – which predominantly concentrates on rail fatigue, rolling contact fatigue and sleepers and ballast. Chapters examine fatigue in vehicles, bridges, tunnels and viaducts. The book concludes with the various fatigue issues present in barriers and ticket machines. It is a valuable resource for all concerned with the rail industry, academics and researchers of railway infrastructure systems. Processes and mechanisms of welding residual stress and distortion (ISBN-13: 978-1-85573-771-6; ISBN-10: 1-85573-771-X) Measurement techniques for characterisation of residual stress and distortion have improved significantly. Through the collaboration of experts, this book provides a comprehensive treatment of the subject. It develops sufficient theoretical treatments on heat transfer, solid mechanics and materials behaviour that are essential for understanding and determining welding residual stress and distortion. The book will be useful for advanced analysis of the subject and provides examples and practical solutions for welding engineers. Fatigue strength of welded structures 3rd edition (ISBN-13: 978-1-85573-506-4; ISBN-10: 1-85573-506-7) Extensive research on the fatigue behaviour of welded structures has brought a greater understanding of the design methods that can be employed to reduce premature or progressive fatigue cracking. This book highlights the need for a design approach that incorporates an understanding of the fatigue problem and effects of welding. Specific recommendations for the design of welded joints in components or structures are discussed as is an account of the application of fracture mechanics to fatigue. This latest edition of Fatigue stength of welded structures has been revised to take into account recent advances in technology providing a practical approach with upto-date design rules. Details of these and other Woodhead Publishing materials books and journals, as well as materials books from Maney Publishing, can be obtained by: • visiting our website at www.woodheadpublishing.com • contacting Customer Services (e-mail: [email protected]; fax: +44 (0) 1223 893694; tel.: +44 (0) 1223 891358 ext. 30; address: Woodhead Publishing Limited, Abington Hall, Abington, Cambridge CB1 6AH, England) If you would like to receive information on forthcoming titles, please send your address details to: Francis Dodds (address, tel. and fax as above; email; [email protected]). Please confirm which subject areas you are interested in. Maney currently publishes 16 peer-reviewed materials science and engineering journals. For further information visit www.maney.co.uk/journals.

Cumulative damage of welded joints Tim Gurney

Woodhead Publishing and Maney Publishing on behalf of The Institute of Materials, Minerals & Mining

CRC Press Boca Raton Boston New York Washington, DC

WOODHEAD

PUBLISHING LIMITED

Cambridge England

Woodhead Publishing Limited and Maney Publishing Limited on behalf of The Institute of Materials, Minerals & Mining Published by Woodhead Publishing Limited, Abington Hall, Abington, Cambridge CB1 6AH, England www.woodheadpublishing.com Published in North America by CRC Press LLC, 6000 Broken Sound Parkway, NW, Suite 300, Boca Raton, FL 33487, USA First published 2006, Woodhead Publishing Limited and CRC Press LLC © 2006, Woodhead Publishing Limited The author has asserted his moral rights. This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. Reasonable efforts have been made to publish reliable data and information, but the author and the publishers cannot assume responsibility for the validity of all materials. Neither the author nor the publishers, nor anyone else associated with this publication, shall be liable for any loss, damage or liability directly or indirectly caused or alleged to be caused by this book. Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming and recording, or by any information storage or retrieval system, without permission in writing from Woodhead Publishing Limited. The consent of Woodhead Publishing Limited does not extend to copying for general distribution, for promotion, for creating new works, or for resale. Specific permission must be obtained in writing from Woodhead Publishing Limited for such copying. Trademark notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation, without intent to infringe. 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. Woodhead Publishing Limited ISBN-13: 978-1-85573-938-3 (book) Woodhead Publishing Limited ISBN-10: 1-85573-938-0 (book) Woodhead Publishing Limited ISBN-13: 978-1-84569-103-5 (e-book) Woodhead Publishing Limited ISBN-10: 1-84569-103-2 (e-book) CRC Press ISBN-13: 978-0-8493-3475-7 CRC Press ISBN-10: 0-8493-3475-6 CRC Press order number: WP3475 The publishers’ policy is to use permanent paper from mills that operate a sustainable forestry policy, and which has been manufactured from pulp which is processed using acid-free and elementary chlorine-free practices. Furthermore, the publishers ensure that the text paper and cover board used have met acceptable environmental accreditation standard Typeset by Replika Press Pvt Ltd, India Printed by T J International Limited, Padstow, Cornwall, England

Contents

Preface

ix

Nomenclature

xiii

1

Introduction

1

1.1 1.2 1.3 1.4 1.5

Background Characteristics of fatigue cracking Fatigue testing The S-N curve and fatigue strength Fracture mechanics assessment of constant amplitude fatigue behaviour

1 3 5 8 10

2

The constant amplitude database

19

2.1 2.2 2.3 2.4

Introduction Method of analysis and joint design classification Influence of plate thickness Influence of mean stress

19 21 65 70

3

Residual stresses

75

3.1 3.2 3.3 3.4 3.5

75 75 78 79

3.6

Introduction The formation of residual stresses Comparison between static and fatigue conditions Approximate theoretical analysis Tests on welded specimens under constant amplitude loading Prior overloading

4

Variable amplitude loading and testing

100

4.1 4.2 4.3 4.4

Introduction Variable amplitude loading Rainflow counting Reservoir counting

100 100 103 106

83 95

vi

Contents

4.5 4.6 4.7 4.8

Level-crossing counting Statistical interpretation of count data Miner’s rule Variable amplitude fatigue testing: a brief history

107 109 111 117

5

Tests under two and three level loading

125

5.1 5.2 5.3

125 130

5.4 5.5

Introduction Theoretical analysis Fatigue tests using stress sequences with excursions of two sizes Influence of stress ratio and residual stresses Summary of findings

134 134 144

6

The influence of spectrum shape and block length

147

6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8

Introduction Fatigue tests under concave upwards spectra Fatigue tests under Rayleigh and Laplace loading spectra Tests under Weibull stress spectra Influence of spectrum shape and clipping ratio combined Influence of block length and clipping ratio combined Influence of block length and spectrum shape combined Summary

147 148 150 171 177 180 183 189

7

The influence of narrow band, wide band and service loading

193

7.1 7.2 7.3 7.4 7.5 7.6

Introduction Comparing loading types Tests under narrow band loading Tests under wide band loading Tests under service loading spectra Summary

193 196 207 221 228 242

8

The influence of cycles of small stress range

244

8.1 8.2 8.3 8.4 8.5

Introduction Block testing of low stresses Comparative tests on stress relieved joints Predicting fatigue life Summary

244 247 251 258 261

9

Design for variable amplitude loading

263

9.1 9.2 9.3

Introduction Testing for different types of stress The area rule

263 265 273

Contents

vii

9.4 9.5

Possible modifications to Miner’s rule The fracture mechanics approach

274 275

10

More on the fracture mechanics approach – the effect of stress interaction

284

10.1 10.2 10.3 10.4

Introduction Summary of experimental evidence about stress interaction effects Discussion Concluding remarks

284 287 302 304

Appendix A Statistical analysis of constant amplitude test data

305

Appendix B Fatigue loading spectra

325

Appendix C Summary of test results obtained under variable amplitude loading

356

Appendix D Probability density curves

413

Appendix E Transformation of units in the Paris fatigue crack prorogation equation

416

References and further reading

418

Index

434

Preface

So far as welded joints are concerned, research on the problem of fatigue did not really start until the time of the Second World War. It was not, after all, until then that welding began to become the fabrication method of choice for many structures; up to that time the use of rivets had been almost universal. Indeed, if welding had not been invented, many structures, and even whole industries, such as nuclear power generation, simply could not exist. However, as welding started to become more widely used, fatigue soon made itself apparent and it was not long before fatigue failures in service began to become quite numerous. That was not particularly surprising because, in the early days of welding, structures were necessarily being designed, so far as fatigue was concerned, to very inadequate rules, but it took a few years for them to suffer enough loading cycles for the problem to become apparent. By the mid-1960s one could almost say that the problem had reached epidemic proportions and at that time it was estimated that, in Britain alone, the loss resulting from fatigue failures of welded structures was running into tens of millions of pounds each year. In retrospect, and speaking in general terms, it is fairly easy to define the general phases in which the research was carried out. In the beginning, when everyone was effectively starting from scratch, the main objective was to define, by testing, the basic constant amplitude fatigue strengths of the various types of joint that are commonly used, and there are many of them. This involved producing constant amplitude S-N curves for each type of joint, primarily under pulsating tension loading but also, to a rather limited extent, under other stress ratios. The results of that work, carried out in many countries, were subsequently used to define the basic design S-N curves that now exist in various design Standards around the world. In order to set the background to the remainder of this book, the results of that work are summarised in Chapter 2 and Appendix A. Rather surprisingly, again in retrospect, the work on defining constant amplitude S-N curves tended to start by looking at transverse butt welds. At the time they were considered to be the most important joints; obviously they are important, but it was not then realised that fillet welded joints would

x

Preface

give much lower fatigue strengths and that fillet welded joints were going to exist in most structures. However, the work on butt welds also highlighted another potential problem, namely the influence of weld defects. Thus, following the work on ‘good quality’ butt welds, research followed two main routes. The first involved looking at the effects of defects in butt welds (e.g. porosity, slag inclusions and lack of penetration) and ultimately led to the fatigue part of various defect acceptance Standards, such as the British PD6493. The second was concerned with obtaining basic fatigue data for a wide range of different types of fillet welded joints. It was at this stage that it was realised that residual stresses, an inevitable consequence of welding, could be an important variable in the overall fatigue problem; a brief summary of their significance is included in Chapter 3. It was also discovered that the fatigue strength of fillet welded joints could be very low indeed and attempts were made to see if any improvement could be obtained by such simple methods as using high tensile steel. Unfortunately, however, that produced no improvement whatever. It was at about this time that fracture mechanics was developed and shown to be applicable to the fatigue problem. More or less concurrently metallurgical investigation showed that fatigue cracks in welded joints starting at the weld toe were almost invariably associated with small crack-like intrusions. Thus for the first time it became possible to analyse, admittedly in relatively simple terms at that time, the growth of fatigue cracks in welded joints, particularly as crack propagation data for various parent materials was also then becoming available. As a result of that work it quickly became obvious why increasing the strength of the parent material did not lead to an increase in the fatigue strength of welded joints. At this stage the work on fillet welded joints also divided into two parts. The first was devoted to investigating other methods (such as spot heating, peening, toe grinding and TIG dressing) of improving the fatigue strength of such joints, but by then there was a reasonably sound knowledge of what the problems really were and consequently the work could be approached with specific aims in view, such as the elimination of toe defects, the introduction of compressive residual stresses, etc. The second branch of the work was more concerned with the exploitation of fracture mechanics principles to examine, theoretically, the influence of various aspects of joint geometry and then to confirm the findings by testing. It was as a result of this work that such features as thickness effect first became apparent, many of which are now incorporated in current design Standards. All the work summarised above was carried out under constant amplitude loading. In the beginning that was because it was the only type of test that could be performed, but more recently it was primarily because constant amplitude testing machines were fairly readily available and, in any case, it was desirable to obtain strength data for Standards which were consistent with what already existed.

Preface

xi

The fact remains, however, that many people realised that constant amplitude tests did not adequately represent service conditions which, in most instances, involved variable amplitude loading. In the relatively early days an attempt was made to ‘bridge the gap’ by means of tests under block programme loading, that being the only form of variable amplitude loading that was possible with the testing machines of the time. Unfortunately, however, as discussed in this book, that could give very wrong answers. It was not until the arrival of servo-controlled machines that it became possible to carry out realistic tests. In general, variable amplitude testing has involved two types of approach. The first has involved tests on various types of joints under loading spectra which were intended to simulate the loading that was expected to occur in service, mainly with a view to assessing whether or not the experimental lives obtained were better or worse than those predicted by Miner’s rule. Effectively, therefore, they were aimed at justifying the use of that rule for design purposes. The second approach has been to try to define what particular features of a loading spectrum are likely to lead to safe or unsafe predictions. The results obtained in both approaches are summarised in this book, although it is concentrated rather more on the second approach, since much of that work has probably been seen only by a relatively restricted audience. While it cannot be said that we yet know all the answers to the problem, we do at least know some of the situations where the current design rules ‘do not work’. It is to be hoped that, by bringing all the facts together in one place, it may help others to tidy up some of the loose ends. No apology is made for attempting to set out the experimental data results in considerable detail. In the preparation of this book I am particularly indebted to the Welding Institute, by whom I used to be employed, for their permission to make use of material which has previously appeared in the Institute’s research reports. I am also most grateful to the many other investigators and organisations who have so willingly and generously permitted me to make use of their results. So far as possible the source of all such information is referenced at the point of use.

Nomenclature

Note: In general, the following list does not include variables which occur in only one specific chapter; they are defined at the point of use. a B C, m E Ei I k K Kt L MK Mp Ms Mt N NB Nc NE NL Npred pi

( )

Crack length Plate thickness Material constants in crack propagation equation and in equation of S-N curve Young’s modulus No. of cycles per block in which stress equals or exceeds Pi σ Irregularity factor Constant defining spectrum shape of Weibull distribution (for Rayleigh k = 0.5, for Laplace k = 1.0) Stress intensity factor Stress concentration factor Attachment length (toe to toe) Correction factor for K allowing for crack being situated at position of stress concentration Correction factor for K allowing for crack tip plasticity Correction factor for K allowing for crack mouth being at a free surface Correction factor for K allowing for plate thickness Cycles to failure No. of blocks to failure Constant amplitude life under top limit stress in the spectrum Number of exceedences (cycles) Block length (cycles) Predicted life (cycles) Proportional stress (= Stress/Maximum stress)

P ∆s or P(S) Probability of load exceedence σ R Stress ratio (Smin / Smax)

xiv

Nomenclature

RG S Smin Smax Sm Sr SB T tB T′ vi y

Ratio of experimental life/life predicted by the Area rule Stress Minimum stress in cycle Maximum stress in cycle Mean stress Stress range Fatigue strength of joint using basic S-N curve Thickness Thickness corresponding to basic S-N curve Apparent thickness No. of cycles per block of stress range pi Correction factor for K (function of geometry) = M k MsMt M p φo

α

Non-dimensional crack length = a b

β ∆K ∆Kth σ σy φo n ΣN

( )

∆S max σ range of stress intensity factor Threshold value of ∆K for crack propagation to occur rms of process standard deviation of log N Yield stress Complete elliptic integral Clipping ratio =

Miner summation

1 Introduction

1.1

Background

Fatigue can be defined as a mechanism of failure which involves the formation and growth of a crack or cracks under the action of repeated stresses. Ultimately, of course, the crack may propagate to such an extent that total fracture of the member may occur. In other words, fatigue failure can only occur in a structure in which the stresses vary with time in a repeated manner. Note, however, that this does not imply that the various stresses all have to be of the same magnitude; it is much more usual for them to have a wide range of magnitudes, but both types of loading can lead to fatigue failure. Fatigue has, in fact, been recognised as a problem, for more than a century, dating at the very least from the classic fatigue tests on railway axles carried out by Wohler, details of which were published in 1871 (Wohler, 1871). So far as welded joints are concerned, however, the problem really dates only from the Second World War, or even slightly after it, since it was not until then that welding became a major production process, and after that it took a little time for structures to amass sufficient load cycles for fatigue to rear its head. More recently, of course, welding has become the fabrication method of choice for almost all types of structure, so that scope for running into fatigue problems in welded structures is immense. Potentially there are many different sources of cyclic loading (often referred to as fatigue loading, for obvious reasons), any or all of which may be relevant in particular situations, and the following list includes some typical types of structure in which they may occur. The list is not by any means exhaustive but is merely intended to show examples. It includes, for example: 1. Fluctuating live loads Bridges, crane gantry girders, diesel engine frames, locomotive underframes and bogies, lorry chassis frames and axles, ships, cranes, earth moving equipment, farm machinery, rock crushers and presses. 2. Acceleration forces in, or due to, moving structures 1

2

Cumulative damage of welded joints

3. Pressure changes Pressure vessels and pipework. 4. Temperature fluctuations Process equipment involved with hot or cold materials, liquids or gases. 5. Mechanical vibrations Rotating machinery, grading equipment and conveyors. 6. Environmental loading (wind, currents and waves, particularly when vortex shedding is induced). Offshore platforms and rigs, ships, wind turbines and chimneys (wind). Even though, as stated previously, this list is by no means exhaustive, it does give some indication of the very wide range of types of structures in which the possibility of fatigue failure needs to be considered at the design stage. It should come as no surprise to learn that, so far as the structural types included above are concerned, the possibility has been proved to be real in all instances. Fatigue is, in fact, the most common cause of structural failure, although with the advent of modern design rules one senses than it is becoming less common than it was a few years ago. To avoid fatigue failures the main requirements are, firstly to recognise at the design stage that the loading is such as to make fatigue a possibility and, secondly, to design the structure with a high inherent fatigue strength. This depends, in particular, upon paying great attention to the detailed design of the joints; many well conceived structures have been spoilt by poor detail design and bad selection of joint types. As implied above, modern fatigue design Standards give plenty of guidance on the problem of joint selection and detailing to avoid fatigue problems. Essentially the need is to keep the severity of stress concentrations to a minimum, particularly in regions of high stress range. Unfortunately, however, the fact remains that fatigue strength is not a ‘constant’ material property, like yield stress or the modulus of elasticity; nor is it something like column buckling strength, which can be calculated if the basic material properties and shape of the section are known. In theory, for any given joint geometry, its fatigue strength can be assessed by fracture mechanics techniques, and significant advances in this direction have been made in recent years. However, such calculations necessarily involve having to make several assumptions, so that while, qualitatively, they provide a good description of the fatigue cracking process, quantitatively the end result may contain large errors. As a method of trying to define absolute strengths it is not, therefore, an attractive proposition at the moment, although fracture mechanics can be a useful tool in some circumstances, such as when it is required to make comparisons between geometries with everything else held constant. In current circumstances, therefore, the fatigue strength of a joint can really only be defined experimentally, and that is the method that was used to derive the various joint strengths which are specified in design Standards.

Introduction

1.2

3

Characteristics of fatigue cracking

When a ductile metal is loaded ‘statically’, so that the load is gradually increased from zero to a maximum, final rupture of the material is preceded by very large strains. Thus, in a tensile test on mild steel, the percentage elongation is usually about 25% and the ultimate tensile strength of the material is about 430 N/mm2. However, if the same material is subjected to repeated loads fatigue cracking may occur as a result of stresses much lower than the elastic limit and there will be no plastic deformation. Because of this lack of deformation fatigue cracks are exceedingly difficult to see, particularly in the initial stages of crack growth. This is true even in the ‘clean’ conditions of a laboratory, and when the likely location of any cracks can be predicted. Herein lies the particular danger of fatigue failure. Since no significant dimensional change occurs in a cracked member the crack may easily progress to a considerable extent before it is discovered. Although there may be some difficulty in deciding whether a crack in a structure is the result of fatigue until it has been opened up, examination of the fracture surfaces usually enables fatigue cracking to be recognised quite easily. A fatigue fracture surface normally has a characteristic appearance with two or three distinct and recognisable regions. In most cases fatigue cracks originate at the surface, usually at a stress concentration such as a weld toe, although in the context of welded structures an important minority start away from the surface, initiated at weld defects. In either case, both with structural steels and other materials, the region surrounding the origin has a smooth, silky appearance which extends to the limit of the fatigue fracture proper. There is often a tendency for this smooth region to grow slightly, but progressively, rougher in texture as the distance from the nucleus of the fatigue crack increases. In the immediate vicinity of the origin the surface may appear extremely smooth, a feature that is probably accentuated by rubbing of the surfaces together as the crack propagates. Careful examination of this smooth part of the fracture surface frequently reveals the existence both of concentric rings or beach markings around the fracture nucleus and of radial lines emanating from it, and these can often be helpful in locating the origin. Figure 1.1 shows a diagrammatic representation of a fatigue crack. The direction in which a fatigue crack propagates is always perpendicular to the line of action of the stresses causing it. As the crack progresses the stress on the residual cross-section increases so that there is a corresponding increase in the rate of crack propagation. Ultimately a stage is reached when the remaining area is insufficient to support the applied load and final rupture occurs. Thus, if the stresses are comparatively low the ‘fatigue’ area will be relatively large, but if the stresses are high it will be correspondingly smaller.

4

Cumulative damage of welded joints Origin

Area of final rupture

1.1 Diagrammatic representation of a typical fatigue fracture surface.

1.2 Fracture surface of a specimen in which fatigue cracking was followed by a brittle fracture.

The fracture surface of the final rupture area may be either crystalline or fibrous depending on whether the fracture was brittle or ductile. Figures 1.2 and 1.3 shows typical examples of specimens in which the initial cracking was due to fatigue but in which the final fracture were brittle and ductile respectively. Considerable information can be obtained from a careful study of service failures and examination of the fracture surfaces may be invaluable in diagnosing the cause of failure. In the case of fatigue cracks such an examination will normally not only indicate the point of initiation but will also show the direction of the critical stresses and give some idea of their magnitude. Fractures caused by low stresses usually have a very smooth fatigue area,

Introduction

5

1.3 Fracture surface of a specimen in which fatigue cracking was followed by ductile failure.

and the residual area at final fracture is small. High stresses, on the other hand, normally result in relatively small fatigue areas and tend to cause multiple initiation of cracking. This information can be most valuable in devising a cure, since it will indicate what details need to be modified and where the stresses are too high. It may even pinpoint an unsuspected mode of loading which was ignored at the design stage. It is therefore most important to ensure that as much information as is available is extracted from service failures and that, whenever possible, they are not hastily repaired before that has been done.

1.3

Fatigue testing

Although this book is concerned primarily with fatigue under variable amplitude loading, the fact remains that by far the larger proportion of fatigue tests have been carried out under constant amplitude conditions (i.e. with the minimum and maximum stresses in the cycle remaining constant throughout

6

Cumulative damage of welded joints

the test). That was, for example, the method used to derive virtually all the results used in the drafting of fatigue design rules, and it is against those rules that the results of tests under variable amplitude loading are almost always assessed. It is therefore necessary to start by considering, at least in summary, the type of constant amplitude data that exist, the way in which they were derived and the form of the results that emerge from constant amplitude tests. However, it may be helpful to start by considering the parameters that can be used in the definition of the stress cycle to which a constant amplitude fatigue test specimen may be subjected. The four basic parameters are: 1. 2. 3. 4.

The The The The

minimum stress in the cycle maximum stress in the cycle mean stress stress range

Smin Smax Sm = 1/2 (Smin + Smax) Sr = Smax – Smm

The cycle is fully defined provided that any two of these four quantities are known. In using certain types of fatigue-testing equipment it is convenient to define the loading cycle by means of the mean stress and the stress range, since the machine is set up by first of all applying the mean stress as a static load and then setting the machine to cycle about that mean stress with the required range. However, most design engineers find it more convenient to think in terms of the minimum stress and maximum stress in the cycle, these in many cases corresponding to the ‘dead load stress’ and ‘dead plus live load stress’ respectively. It will often be convenient, however, to refer to the loading cycle by the stress ratio, R, which is defined as the algebraic ratio Smin/Smax, tensile stresses being considered as positive and compressive stresses as negative. Figure 1.4 shows some typical types of stress cycle used in fatigue testing. A study of the literature shows that the most common type of cycle used in the testing of welded joints has been that shown in Fig. 1.4(a), in which the stress is varied between zero and tension and which is known as a ‘pulsating tension’ cycle. A fairly large number of results also exist for the case shown in Fig. 1.4(b), where the maximum and minimum stresses are equal in magnitude but opposite in sign – usually referred to as an ‘alternating’ cycle – and for that shown in Fig. 1.4(c), where both the maximum and minimum stresses in the cycle are tensile. The particular case in which Smin = 0.5 Smax is known as a ‘half tensile’ cycle. Other cycles involving unequal tensile and compressive stresses, or wholly compressive stresses, may of course be used. Any of these cycles may be produced either by axial loading or in bending, but in the latter case it is necessary to specify to what point the stress refers – e.g., the extreme fibre – since the stress distribution will not be uniform over the cross-section. It is usual to present results in terms of the nominal

Tension

Introduction

7

Stress

Smax

Time

0

Smax

Time

0

Compression

Stress

Tension

(a) Pulsating tension.

Smin = – Smax

Tension

(b) Alternating.

Stress

Smax

Smin =

0

1 2

Smax

Time (c) Half tensile.

1.4 Typical stress cycles that have been used in fatigue tests on welded joints.

stress on the cross-section in the case of specimens loaded axially and on a given plane in the case of specimens loaded in bending, ignoring the effect of any stress concentrations. In the early days of fatigue testing, up to approximately the 1950s, the testing machines available were capable of subjecting specimens only to constant amplitude loading. Initially such tests were usually carried out in rotating bending; this involved small cylindrical specimens, with or without notches, being loaded as cantilevers or as beams in four point bending. As the specimen rotated the stress at any point in it varied between limits that were equal in magnitude but opposite in sign (i.e. alternating loading, see Fig. 1.4(b), the plane and direction of loading remaining constant throughout. While this type of test was suitable for determining what might be called the

8

Cumulative damage of welded joints

basic fatigue strength of materials, it did little to help define fatigue strength in the engineering sense, i.e., taking account of real changes of shape, the presence of joints, etc. Such information had to await the development of somewhat more sophisticated testing machines which were capable of testing much larger specimens. In general these tests either involved axial loading or tests in bending, particularly on specimens in the form of beams. At this stage, however, they were still restricted to constant amplitude loading, although the type of stress cycle was usually pulsating tension (Fig. 1.4(a)) rather than alternating. Nevertheless, so far as welded joints are concerned, this type of testing has been used, and still is used, to build up a large database of information relating to the influence of joint geometry on fatigue strength, and also the influence of (nominally) small perturbations from it. These have included, for example, the influence of joint size, small variations to a basic geometry, residual stresses, etc. In the main this is the information that has been used to prepare the fatigue design rules that now exist in various National design Standards, and a brief review of the work is given in Chapter 2. At this point it is important to note that constant amplitude loading can be fully defined and is completely reproducible. In other words it should be possible, apart from experimental error, to carry out identical tests, or series of tests, in different laboratories. Consequently, the overall database of results for any particular type of specimen will often include results from several different sources, thereby giving a high degree of confidence that the mean overall result is credibly accurate. With the advent of newer types of (mainly hydraulic) testing machines it also became easier to obtain experimental data for a much wider range of mean stress conditions, even though most of the data have related to pulsating tension loading or alternating loading.

1.4

The S-N curve and fatigue strength

It would not normally be very useful to try to define the fatigue strength of a joint (or anything else) by carrying out a single constant amplitude fatigue test, if for no other reason than that the results of such tests are inevitably subject to scatter. In reality, to determine the fatigue strength of a particular joint under any particular conditions of (constant amplitude) loading it is necessary to test several similar specimens – about eight is a fairly typical number – but in some circumstances it may be necessary to test many more than that. The specimens in the test series are then subjected to a number of different stress ranges (Smax – Smin) while maintaining a constant relationship between Smax and Smin. For example, if the series were to be tested under pulsating tension loading each specimen would have Smin = 0 (or some very small positive value of stress), but the value of Smax would be different for

Introduction

9

320 300 280 260 240 220

Upper limit stress (N/mm2)

200 180 160 140 120

100

80

60

105

106 Endurance (cycles)

107

1.5 Typical set of fatigue test results, plotted on the basis of Log S against Log N, for a transverse butt weld in steel.

different specimens, although it might sometimes be decided to test more than one specimen at a given value of Smax. There are many other relationships between Smin and Smax that can be, and have been, used; the two most usual ones are (i) to have Smax and Smin equal in magnitude but opposite in sign (alternating loading) or (ii) to carry out the test with both Smax and Smin tensile but with Smin = 0.5 Smax (half tensile loading), but there are many other possibilities. Then, for each specimen, the number of loading cycles required to produce failure is noted, so that a relationship is obtained between the applied stress range (Sr) and the number of cycles to failure (N). This relationship can then be plotted as an S-N curve, although it is usually more useful to plot it in the form log S v log N. A typical example is shown in Fig. 1.5. In most situations the form of the relationship between log S and log N is usually a straight line, at least for endurances up to about 2 × 106 cycles and often for larger endurances than that. In passing it should be noted that there are also some situations, particularly under compressive loading, where the S-N curve is not straight, but in the context of this book they are of no great

10

Cumulative damage of welded joints

consequence. It should also be noted that, in much of the early work on fatigue, S-N curves were plotted on the basis of a linear (not a log) scale of stress against N or log N, and they are also not straight. In the light of this summary, and notably of the results shown in Fig. 1.5, two things are obvious. Firstly, there is no exact relationship between S and N; in other words, if you know the stress range you do not necessarily know the corresponding exact value of N, since there is inevitably some scatter. Secondly, there is no such thing as a single absolute value of fatigue strength; not only is it necessary to take account of the scatter but it is also necessary to define the life, in terms of N, in which you are interested. As will be discussed in more detail later, the situation is further confused by the fact that the S-N curve used for design purposes is deliberately set below the mean experimental S-N curve, so it also becomes necessary to define whether one is referring to the experimental curve or the design curve. However, provided one defines what S-N curve is being used, together with the relevant life (in cycles), the concept of an absolute value of fatigue strength for the joint under consideration is sometimes a useful one. Thus, referring to Fig. 1.5, it makes sense to say (for example) that the mean experimental fatigue strength for that particular test series was 130 N/mm2 at 2 × 106 cycles; it would not make sense if one omitted either the life or the fact that it referred to the mean experimental S-N curve. It is worth noting, in passing, that for historical reasons such results are very often quoted in terms of the strength at 2 × 106 cycles.

1.5

Fracture mechanics assessment of constant amplitude fatigue behaviour

The method of assessment of fatigue behaviour outlined above, involving the obtaining of an experimental S-N curve, only gives information relating to the failure condition of the joint. The alternative approach, using fracture mechanics, gives information about the progress of failure and involves an assessment of the rate of crack propagation. At this stage it is of interest to compare the form of the relationship between S and N predicted by this approach with that actually found experimentally. The fatigue failure process actually consists of two or three phases. The first part of the life is taken up in initiating a crack, while the second phase consists of progressive crack propagation and the third phase constitutes final rapid failure. Indeed the second phase can be subdivided into two stages, stage 1 consisting of crack growth in shear and stage 2 of growth perpendicular to the applied tensile stress, but stage 1 is normally of limited extent and does not really occur in welded joints anyway. In welded joints it so happens that the initiation stage is very often almost non-existent. Fatigue cracks in welded joints usually initiate at the weld toe

Introduction

11

where there are small very sharp defects which can really be regarded as small cracks. The other main initiation sites in welded joints are the weld root, in load-carrying transverse fillet welds, and weld defects, and both of these can also usually be regarded as crack-like. In other words, fatigue cracks in nearly all welded joints start, effectively, at pre-existing flaws, the presence of which can be taken as the equivalent of an initiated crack. In consequence the whole, or at least the major part, of the fatigue life is taken up by the propagation phase of crack growth. Over the years a large number of empirical laws have been proposed in attempts to characterise the rate of fatigue crack propagation. Unfortunately, most of these laws were derived from relatively small amounts of data and were found to apply only under specific conditions. It was not until fracture mechanics was applied to the problem that a more satisfactory law was derived, and one which could be applied to many design situations. This law relates the rate of crack propagation to the range of stress intensity factor (∆K) at the tip of the advancing crack. As is well known, the stress intensity factor, K, is a single term parameter which describes the stress conditions adjacent to the tip of a crack. For the simple situation of an infinite plate subjected to an in-plane uniform stress, σ, perpendicular to a central through-thickness crack of length 2a. K = σ √(π a)

[1.1]

It will be seen that the dimensions of K are stress × √(length), so that the same value of K can be obtained with a high stress and a short crack as with a lower stress and a longer crack. Equation [1.1] which is valid for both plane stress and plain strain conditions, can be regarded as the fundamental equation for K since, in general, K = Yσ √(π a)

[1.2]

where Y is a correction factor which is a function of the geometry of the crack and of the cracked body under consideration. In more detail this equation can be written as: K=

M k MsMt M p σ φ0

( π a)

[1.3]

where the various factors account for particular features of the geometry. The factor Mp is the correction to allow for crack tip plasticity; in most cases, it can be taken as being equal to 1.0. The factor Ms is the correction to allow for the fact that the mouth of the crack is situated at a free surface. (It will be recalled that the basic formula K = σ √ (π a) refers to a through-thickness crack in the centre of an infinite plate.) The value of Ms depends on the crack depth: width ratio a/2c. Several investigators have proposed values for it, and Maddox (1975) has shown that the equation Ms = 1 + 0.12 (1 – 0.75a/c) is

12

Cumulative damage of welded joints

within 1.5% of all of them. Clearly the maximum value, which applies in the case of a continuous surface crack (a/c) is Ms = 1.12. The factor Mt is a correction for plate thickness, to allow for the fact that there is a free surface ahead of the crack. Its value depends both on the shape of the crack, a/2c, and on the crack depth: plate thickness ratio, a/B. Maddox has computed the value of the combined correction factor MsMt/φ0, where φ0 is the complete elliptic integral, for a range of values of a/B and a/2c, and he found that, for most practical purposes, it was permissible to make the simplifying assumption that MsMt/φ0 = 1.0, or possibly 1.12 for safety. Consequently, if it is also assumed that Mp = 1.0, the stress intensity factor for a semi-elliptical crack at the toe of a weld can be taken to be approximately: K = Mk σ √(π a)

[1.4]

The factor Mk allows for the fact that the crack is situated at a position of stress concentration at the weld toe. For a crack of infinitely small depth, Mk is equal to the stress concentration factor Kt. However, as the crack increases in depth the crack tip gets steadily farther away from the stress concentration at the weld toe so that Mk decreases with increasing crack depth. The relationship can be defined by finite element analysis and values of Mk have been derived, and are available, for several basic joint geometries. Given that the value of K can be defined, it was suggested by Paris and Erdogan (1963) that, since fatigue crack propagation is clearly a phenomenon which must depend on the stresses around the crack tip, the stress intensity concept could also be used as a basis for defining quantitative values of crack propagation rate. As a result of a reanalysis of crack propagation data obtained by several investigators they suggested that the relevant parameter was the range of stress intensity factor, ∆K, where:

∆K = Kmax – Kmin

[1.5]

Kmax and Kmin being the values of K at the upper and lower limit stresses of the loading cycle. The particular relation which they proposed was of the form: da/dN = C (∆K)m

[1.6]

where C and m are assumed to be material constants. In reality the values of C and m for any particular material and set of conditions (stress ratio, environment, etc.), as obtained using standard fracture mechanics crack propagation tests, are not constant over the whole range of ∆K. Typically the form of the relationship between da/dN and ∆K is as shown in Fig. 1.6. It will be seen that this curve is sigmoidal in shape, although the central region is approximately linear and equivalent to eqn [1.6].

Introduction

13

log (rate of propagation, da/dN)

Kmax approaches Kc

Linear ‘Paris law’ region

Nonpropagating cracks

∆Kth log (range of stress intensity factor, ∆K)

1.6 Idealised relationship between da/dN and ∆K.

The lower point of inflection arises because of the need for ∆K to exceed a threshold value, ∆Kth, before any propagation can occur at all. It is because of the existence of this threshold that it is possible, given the right combination of crack size and applied stress, for some cracks not to propagate under fatigue loading. It has been suggested by Pook (1972) that the theoretical threshold corresponds to a propagation rate of one lattice spacing per cycle, which for metals is typically about 4 × 10–7 mm. However, even slower propagation rates have been recorded and this may be due to the crack only propagating on part of the crack front at any given time. From this it is clear that there is a need to test to very large numbers of cycles in order to ensure that no crack growth is occurring, which tends to make this type of testing expensive. The transition from the threshold to the central linear region of the curve of log (da/dN) against log ∆K is usually very steep; in other words a very small increase in ∆K above ∆Kth corresponds to a very large increase in da/dN. At the upper end of the curve the second point of inflection represents the approach of the maximum stress intensity in the cycle, Kmax, to the critical value (Kc or KIc) at which fast fracture would occur.

14

Cumulative damage of welded joints

Note: extrapolated plane strain curve is safe

D

Plane stress

log (da/dN)

C

Plane strain

B

∆Kth

log ∆K

1.7 Typical crack propagation relationship for thin specimens.

In fact the relation is slightly more complicated than the idealised one shown in Fig. 1.6, in that the ‘linear’ region can be composed of two linear regions with a further intermediate point of inflection, as shown in Fig. 1.7. The first linear region (BC) corresponds to crack propagation under predominantly plane strain conditions, while the second linear region (CD) corresponds to plane stress fracture. This type of relation is, therefore, obtained only if the specimen is sufficiently thin to permit relaxation to plane stress conditions before final rupture. In the practical situation, however, the plane stress region is normally ignored. In welded joints, at least, fatigue cracks are usually either partly or fully embedded and fracture in a plane strain mode for most of their lives. Indeed the percentage life which occurs under plane stress conditions will usually be extremely limited. In addition, as can be seen from Fig. 1.7, it is safe, from the calculation point of view, to assume that the plane strain relation also applies in the plane stress region, in that extrapolation of the plane strain relation (BC) leads to a faster rate of propagation for a given value of ∆K than the plane stress relation (CD). Equally it is safe to extrapolate the plane strain relation back into the threshold region of the curve. It is

Introduction

15

4.2

3.8

m

3.4 C = 1.315 × 10–4/895.4m

3.0

2.6 2.2 1.8

10–15

10–14

10–13

10–12

10–11

10–10

C

1.8 Relationship between m and C for steels tested in air at R = 䊊 (䊉 structural steels, 䊐 high strength steels, × weld metal, 䉱 HAZ).

apparent, therefore, that the Paris crack propagation law (eqn [1.2]), based on plane strain behaviour, can be applied with considerable confidence. This implies, however, that the values of C and m must be known. While it is not the purpose of this book to present a comprehensive review of available data on crack propagation rates it may, nevertheless, be useful briefly to indicate some of the general trends in the data. As far as tests in air under pulsating tension loading are concerned crack propagation data have now been obtained for a wide range of materials. For steels under plane strain conditions it has been shown (Gurney, 1979) that log C is linearly related to m (see Fig. 1.8), the relation being: C = (1.315 × 10–4)/895.4m

[1.7]

Inserting this into eqn [1.6] we get da = 1.315 × 10 –1  ∆K  dN  895.4 

m

[1.8]

which implies that the da/dN – ∆K relations for all steels pass through the point da/dN = 1.315 × 10–4 mm/cycle at ∆K = 895 N/mm3/2, as shown in Fig. 1.9. (To transform to other units, see Appendix C.) Unfortunately, it is not at the moment known what defines the value of m (and hence of C) for any particular material. It has been suggested by Barsom (1974) that m should be taken as 2.25 for martensitic steels, 3.0 for ferrite pearlite steels and 3.25 for austenitic steels. One thing that does seem to be clear is that there is a tendency for m to decrease as yield strength increases. For structural steels m usually lies in the range 2.4 to 3.6, the value m = 3.0 being one that is

16

Cumulative damage of welded joints 3 m = 3.6 2

Rate of crack propagation, da/dN (mm/cycle)

1.315 × 10–4 10–4

5 4 3 2

10–5

5 4 3

m=2 2.4 2.5

3 3.5

2

4 –6

10

100

2

3

4

5

1000

2

3

Range of stress intensity factor, ∆K (N/mm3/2)

1.9 Influence of m on propagation rate. For structural steels m usually lies between 2.4 and 3.6, so that the dashed line would be a conservative design curve.

frequently assumed for design purposes. From eqn [1.7] the corresponding value of C becomes 1.83 × 10–13. As can be seen from Fig. 1.9 the higher the value of m the better, since as m increases the rate of propagation decreases, at least in the all-important early part of the life. For materials other than steel, data are somewhat less numerous. A useful rule-of-thumb, however, is that if crack propagation data for different materials are plotted on the basis of ∆K/E, rather than ∆K, they tend to coincide. In this way a reasonable guess can be made of the propagation rate for another material by relating it to the propagation rate for steel. Although crack propagation data are most numerous for specimens tested in air under essentially pulsating tension loading, tests have also been carried out on many materials to study the influence of environment (e.g. temperature, corrosion, etc.) and mean stress. These tests have shown that both mean stress and environment can have a significant effect on propagation rate and it is important to use data relevant to the expected service conditions. In

Introduction

17

passing, it may be useful to note that in structural steels the rate of propagation does not seem to be strongly influenced by mean stress, at least when the cycle is fully tensile. With aluminium alloys, however, the rate of propagation can be very sensitive to mean stress, even under fully tensile loading, the sensitivity tending to increase with tensile strength. Hence, knowing that eqn [1.2] can be applied to the whole life and that the values of C and m can be defined, it becomes possible to calculate fatigue life. Thus, inserting the value of ∆K from eqn [1.2] in eqn [1.6], we get da/ dN = C (Y∆σ √ (πa))m which gives, on integration

∫

a2

a1

da = C ( ∆σ )) m N m (Y ( π a) )

[1.9]

where N is the number of cycles required for the crack to propagate from an initial size a1 to a final size a2. In some instances it may be more useful to express eqn [1.9] nondimensionally in terms of α = a/B where B is the plate thickness. It then becomes:

∫

α2

α1

m –1 dα = C ∆σ m B 2 N m (Y ( πα ) )

[1.10]

Examination of eqn [1.9] shows that, for a given type of joint (i.e. for a constant value of Y) and constant values of a1 and a2, the value of the integral is constant so that the equation can be written as: (∆σ)m N = constant

[1.11]

which is the equation of the S-N curve and is in good agreement with what has been found experimentally, as noted above. It implies that if the fatigue life of a specimen consists wholly of crack propagation, the S-N curve should be a straight line of slope – 1/m when plotted on the basis of log ∆σ against log N. As implied previously, welded joints fall effectively in this category, since fatigue cracks in many types of joint normally initiate at small sharp defects, the presence of which practically eliminates the need for a crack initiation period. It follows that the curves of log S against log N for these joints should be parallel to each other, and to a large extent they are. It is sometimes required to compare the behaviour of different sizes or geometries of specimen, in which the values of a1 and a2 may also be different. This can be done with the aid of eqn [1.10], which can be rewritten as: m

1/m   B m2 –1    1  ∆σ  I   N = C     

[1.12]

18

Cumulative damage of welded joints

where I is the value of the integral

∫

α2

α1

dα /(Y πα ) m . This is also the

equation of the S-N curve, and comparison with eqn [1.11] shows that our objective can be achieved, and the behaviour of different specimens reduced to a common basis, if we regard the stress parameter as:  m2 –1  ∆σ * = ∆σ  B   I 

1/m

[1.13]

Equation [1.12] then takes on the same form as eqn [1.11] and becomes (∆σ *)mN = constant

[1.14]

To make use of this approach it is of course necessary to be able to calculate Y for the particular type of joint and mode of cracking under consideration.

2 The constant amplitude database

2.1

Introduction

As noted in the previous chapter, it is the results that have been obtained in tests under constant amplitude loading that necessarily form the database from which the basic design S-N curves for the various types of joint have been derived. Design calculations relating to variable amplitude loading are then based on these curves. It is therefore of some interest to summarise how those design curves were derived and the type of information that now exists. It was noted previously that the use of welding really only ‘took off’ at the time of the Second World War. It was only after the war that it was recognised that there was a need to elucidate the fatigue performance of welded joints, particularly so that design rules could be compiled. At that stage the objective was therefore to define S-N curves, typically over the range of endurance from about 105 to 2 × 106 cycles, for a typical range of ‘basic’ joints. It was only considerably later that it was realised that geometry variations in a particular type of joint, such as changes in plate thickness or width and changes in weld size and shape, might have anything other than a relatively minor influence, so tests tended to be concentrated on specimens of fairly standard dimensions. Primarily as a result of what the available machines were capable of testing, both in terms of the maximum thickness that could be gripped in the machine jaws and also in terms of the maximum loads that could be applied, there was a great tendency for specimen dimensions to lie in the range 10–15 mm thick and 100–150 mm wide. By testing beams in bending it was possible to obtain test results for a slightly extended range of joint types, but the flange dimensions tended to be very similar to those of axially loaded specimens. Given that published data exist for several thousand test series it was clearly necessary to apply some fairly arbitrary restrictions to the data that were actually used in the analysis. The most recent review of the data, carried out to check the validity of the British fatigue design rules, was performed in 1987, and since none of the results obtained since then (except possibly for welded tubular joints) have 19

20

Cumulative damage of welded joints

given any very different strengths, it is convenient to base the results presented here on those used in that analysis. In other words the data considered were almost all obtained in the period 1959–1986. That effectively eliminates the very early test results, since it has to be recognised that considerable improvements in welding processes have occurred since then and it is therefore unlikely that the early results were obtained using joints representative of current welding practice. The data used were also subjected to other constraints. In the first place, since it has been shown in numerous investigations that the high cycle fatigue strength of welded joints in structural steel seems to be independent of the static strength of the parent material, that variable was ignored and results were accepted for joints in any steels with yield strengths in the range from approximately 225–850 N/mm2, although the great majority of the results related to normal structural steels. Secondly, the analyses were restricted to data emanating from specimens tested under uniaxial tensile loading or from tests on beams in bending, where the stress conditions in the flange are, in fact, similar to those in an axially loaded specimen. The stress ratio was required to be approximately zero (actually between –0.2 and +0.3, although few results exist for small negative stress ratios). Although many data exist for tests carried out at (larger) negative stress ratios (e.g. R = –1) it was felt that it might be unrealistic to make use of them. This was because, in a relatively small specimen, there might only have been small residual stresses, so that the stresses applied to the joint might actually have been partially compressive. In a real structure, high tensile residual stresses would be more likely to occur so that, in that case, the actual stress range under the same nominal loading would be fully tensile, and therefore more damaging. Hence, there is a danger that the test results obtained under compressive stress ratios may be somewhat optimistic. By the same token it would, theoretically, be safe to make use of the data obtained at higher (tensile) stress ratios, but simply for consistency they were also ignored. The main implication of the restriction on the range of stress ratios that were used, is that the proposed design stresses derived from this work refer primarily to as-welded (i.e. not stress-relieved) structures. However, for fully tensile loading the design stresses for stress-relieved joints would be expected to be very similar to those for as-welded joints. For stresses varying between tension and compression it is probable that an analysis of the relevant test results would suggest that there could be a considerable increase in design stress for stress-relieved structures. A brief summary of the influence of mean stress on the fatigue strength of welded joints under constant amplitude loading is included at the end of the chapter; residual stresses are considered in Chapter 3.

The constant amplitude database

2.2

21

Method of analysis and joint design classification

For each type of joint the individual test results were all assumed to form part of a single population. They were therefore plotted and analysed statistically, assuming that the whole set of results (often consisting of several individual test series) could be represented by a straight line on a plot of log stress (S) against log endurance (N), i.e. log N = log C – m log S

[2.1]

where m and C are constants for each individual population (i.e. joint type). This analysis was restricted to joints failing within the endurance range 5 × 104–5 × 106 cycles. In fact the analyses were carried out in two ways, firstly with the slope of the curve (m) not pre-defined, so as to obtain the actual best fit curve, and secondly with m pre-defined as (usually) 3.0, which (as discussed in Chapter 1) is a reasonable average value for structural steels and which is the value normally used for design purposes. For the higher strength ‘joints’, notably steel in the as-rolled condition and some types of continuous longitudinal welds, it was in fact found more realistic to use higher values of m (3.5 or 4.0) instead of 3.0. However, such joints are rarely critical because of the almost inevitable presence of lower strength joints, so it is not strictly relevant to pursue the matter here. For each set of data (i.e. each type of joint) the regression line and the standard deviation of log N (both for m = 3.0) were then calculated and the mean life and scatter band (±2 standard deviations) were plotted on the graph of the data. A summary of the results, showing the mean life and mean life minus two standard deviations of log N, corresponding to an applied stress of 124 N/mm2 is shown in Fig. 2.1. This includes only the lower strength joints, since it is those that are usually relevant in the design process. In general it would not have been practical to regard the lower limits of the various scatter bands as the design curves, since that would have meant that there would have been a different design curve for every joint, which would have been very unwieldy. It was therefore decided, since many joints have quite similar fatigue strengths, to limit the number of design S-N curves by considering all joints with similar strengths as belonging to a single ‘class’ and then to provide design S-N curves for each class. This is in fact the principle that has also been followed in design rules emanating from other countries, and although most are similar to the British rules there are, almost inevitably, some minor differences in the various joint classification systems that have been used. For convenience, at least in the British design rules, the various classes were designated by letters, with Classes A–G referring essentially to joints

22

Cumulative damage of welded joints Stiffener-to-web joints Submerged arc butt welds Class E Butt weld on backing bar K butt welds (cruciform) Intermittent longitudinal fillets Longitudinal NLC fillets Transverse NLC fillets Stud shear connectors Class F Transverse LC fillets (cruciform) Transverse LC fillets (lap joints) Longitudinal LC fillets (weld ends not on plate edge) Class F2 Gussets welded to plate edge Longitudinal LC fillets (weld end on plate edge) Beams with Ends not welded welded

Ends welded

cover plates

Wide plates, ends not welded

Class G 105

106 Endurance (cycles)

2

2.1 Comparison of mean life and mean life –2 standard deviations for various joints and Classes at 124 N/mm2 (note: the right-hand end of each bar represents the mean life and the left hand end the mean –2SD).

involving failure from the weld toe or end, while one (Class W) refers to failure through the weld throat from the weld root in load-carrying fillet welds. In fact one of the classes (Class A) is not really applicable in the context of welded joints, since it refers to parent material of uniform crosssection and with polished surfaces and thus represents an unattainable ideal; for that reason no design stresses are given for Class A. As far as Classes B, C and D are concerned the allocation of joints to classes was fairly obvious; for the lower classes (E, F, F2 and G) it became so when the results were plotted in the form shown in Fig. 2.1, although it proved necessary to make a few minor adjustments. Having decided upon the make-up of each class the results were re-analysed

The constant amplitude database

23

300

200

Stress range (N/mm2)

100 80

B C

60 50 D 40 E F 30 F2 G 20 W

10

105

106 107 Endurance (cycles)

108

2.2 Mean S-N curves for British joint classes.

so as to give the mean S-N curves and standard deviations for the classes. The results, expressed in the same terms as for the individual joints, are also shown in Fig. 2.1, while the S-N curves themselves are shown in Figs 2.2 and 2.3. The co-ordinates of these curves are given in Table 2.1. Since they are linear on a log stress against log N basis they can obviously be expressed as log N = log a – dσ – m log s, where σ is the standard deviation of log N and the curve is relevant to d standard deviations below the mean. The values of the relevant constants are given in Table 2.2. Obviously for any particular value of ‘a’ the equation of the S-N curve can be written as: Sm N = C

[2.2]

where C is a constant (different for each curve). For the set of mean curves C = a, while the relevant values of Sm N for the set of mean minus two standard deviation curves are also included in Table 2.2 for convenience.

24

Cumulative damage of welded joints 300

200

Stress range (N/mm2)

100 80 B 60 50

C

40 30

D E F

20 F2 G W

10

105

106 107 Endurance (cycles)

108

2.3 Mean –2 standard deviations S-N curves.

Reverting to the joint classification system (Fig. 2.1), by comparing the results for the classes and the results for individual joints it is easy to see that three joints appear to be in the wrong class. The reasons for these apparent anomalies are as follows. As far as transverse submerged arc butt welds are concerned the results suggest that they should be in Class F but have been placed in Class E. This was because most of the low test results are relatively old, and it is known that submerged arc butt welds made nowadays normally have a better weld shape and tend to give better strengths. The other two modifications both involved an apparent downgrading of class. The reason for putting stud shear connections in Class F rather than Class E was twofold; first the majority of the available test results refer to unloaded studs, and it seems possible that, with load applied to the studs, the fatigue strength may be lower; and secondly studs will normally be encased in concrete and therefore be uninspectable, so that it seemed prudent to apply a larger than normal factor of safety. Finally, the reason why transverse load-carrying fillet welds of the lap-plate type were downgraded from Class F to Class F2

The constant amplitude database

25

Table 2.1 Co-ordinates of S-N curves (N/mm2) Endurance, N (cycles) Curve

105

106

2 × 106

107(a)

2 × 107(b)

108

Class B Mean Mean – 1 S.D. Mean – 2 S.D.

391 352 317

220 198 178

185 167 150

124 111 100

104 94 84

70 63 56

Class C Mean Mean – 1 S.D. Mean – 2 S.D.

381 333 291

197 173 151

162 142 124

102 89 78

84 73 64

53 46 40

Class D Mean Mean – 1 S.D Mean – 2 S.D.

342 291 248

159 135 115

126 107 91

74 63 53

58 50 42

34 29 25

Class E Mean Mean – 1 S.D Mean – 2 S.D.

320 264 218

149 123 101

118 97 80

69 57 47

55 45 37

32 26 22

Class F Mean Mean – 1 S.D. Mean – 2 S.D.

258 219 185

120 101 86

95 80 68

56 47 40

44 37 32

26 22 18

Class F2 Mean Mean – 1 S.D. Mean – 2 S.D.

231 194 163

107 90 75

85 71 60

50 42 35

39 33 28

23 19 16

Class G Mean Mean – 1 S.D. Mean – 2 S.D.

178 155 135

83 72 63

66 57 50

38 33 29

30 27 23

18 15 13

Class W Mean Mean – 1 S.D. Mean – 2 S.D.

154 134 116

72 62 54

57 49 43

33 29 25

26 23 20

15 13 12

(a) Initial non-propagation stress S0. (b) Cut-off stress for cumulative damage calculations, but see further comments in Chapter 9.

 for joints in air, as given   in BS5400 and BS7608.

was that joints of the simple type which have normally been tested in the laboratory, (i.e. with a transverse weld alone) are virtually never used in practice; such a joint would always have welds along the sides as well and be more akin to a cover plate. A summary of the resulting joint classifications used in the British design rules, at least for the simple ‘basic’ joints, is shown in Table 2.3. As noted

12.6007

1.082 × 1014

3.988 × 1012

3.289 × 10

1.726 × 1012

1.231 × 10

0.566 × 1012

0.368 × 10

C

D

E

F

F2

G

W

12

12

11.5662

11.7525

12.0900

12.2370

12.5169

15.3697 14.0342

2.343 × 1015

B

12

log10a

a

Class

Table 2.2 Values of the relevant constants

26.6324

27.0614

27.8387

28.1770

28.8216

29.0144

32.3153

35.3900

logea

3.0

3.0

3.0

3.0

3.0

3.0

3.5

4.0

m

0.1846

0.1793

0.2279

0.2183

0.2509

0.2095

0.2041

0.1822

log10

0.4251

0.4129

0.5248

0.5027

0.5777

0.4824

0.4700

0.4194

loge

Standard deviation, σ

1.575 × 1011

2.50 × 1011

4.325 × 1011

6.31 × 1011

1.024 × 1012

1.516 × 1012

4.20 × 1013

Mean – 2SD

SmN

38

Cumulative damage of welded joints

previously there are some minor differences in other sets of design rules. At this stage it may be helpful to comment in a little more detail on the results for some of the individual joint types. Continuous longitudinal welds One of the more frequent uses of continuous longitudinal welds occurs in the web to flange joints of fabricated I beams and in general that is the type of specimen that has been used in obtaining fatigue test data. There are three main variants: 1. Joints made with full penetration (of the web) welds; 2. Fillet welds made automatically; 3. Fillet welds made manually. Given that the major stress concentration in welded joints is normally at the weld toe, or in some types of joint at the weld root, but that in continuous longitudinal welds both of those are parallel to the direction of stress, it would be anticipated that all these types of joint would give a relatively high fatigue strength, which has been shown to be the case. In most instances the mode of failure involves crack initiation from weld surface ripples or, in the case of fillet welds, from roughness at the weld root. Weld stop-start positions are particularly likely to be the source of failure since the resulting crater at that position tends to produce the most pronounced change in longitudinal profile. In service, however, this type of joint rarely leads to fatigue cracking since nearly all structures also contain details with much lower fatigue strengths, such as fillet welded stiffeners, etc. Intermittent longitudinal fillet welds In contrast to continuous longitudinal welds, intermittent welds inevitably result in multiple weld ends with every weld end acting as a stress concentration. As a result the fatigue strength is low, with a mean strength at 2 × 106 cycles of about 112 N/mm2. This is equivalent to a reduction in strength of about 35% compared with beams fabricated with continuous manual web-to-flange fillet welds. In other words, intermittent welds should not be used under fatigue loading conditions. Transverse butt welds In Chapter 1 it was noted that the behaviour of a structure under conditions likely to result in fatigue failure was determined primarily by the severity of the stress concentrations which it contained. With this in mind, the problem of joining together two plates may now be considered. Figure 2.4 shows four

The constant amplitude database

39

2.4 Various methods of making welded joints between two plates and the corresponding lines of stress flow.

possible methods by which such a joint could be made, one fabricated with a transverse butt weld, and three with fillet welds. The diagram also shows the influence of the form of the joint on the stress flow between the connected plates. It is quite obvious that the least disturbance to the stress flow occurs with the butt weld and it is therefore to be expected that this form of joint will give a better fatigue performance than would any of the fillet-welded joints. While this expectation is usually fulfilled, the fatigue strength of transverse butt welds can still vary between wide limits. In the absence of weld defects, the major stress concentration in a specimen containing a transverse butt weld with the weld reinforcement left in the aswelded condition occurs at the weld toes. In such a specimen it is therefore from the weld toe, either of the top or of the backing run, that fatigue failure invariably occurs (Fig. 2.5). Thus the propagating crack is initially located either just in weld metal or in the heat affected zone of the parent material, but subsequently it may spread either into the parent metal or into weld metal, depending on the type of joint. The latter mode of failure is associated in particular with butt welds made with a single-sided preparation failing from the toe of the penetration bead, as shown in Fig. 2.5b. The effect of weld defects is outside the scope of this book. However, even with so-called good-quality butt welds (i.e. welds without defects) a

40

Cumulative damage of welded joints (a)

HAZ

Fracture

(b)

Fracture

(c)

Fracture

2.5 Typical modes of fracture in specimens containing transverse butt welds: (a) in heat affected zone material and parent plate initiated at weld toe; (b) and (c) in weld metal initiated at the edge of the weld root.

large number of factors have at one time or another been thought to influence fatigue strength. These include, in addition to the normal variables such as type of loading and the applied stress ratio: • • • • • • • •

composition and mechanical properties of the parent material specimen width and thickness type and method of weld preparation welding process and type of electrode welding position weld shape post-weld machining post-weld heat treatment.

In reality, however, it appears that weld shape is the overriding factor determining the fatigue strength of transverse butt welds, and the influence of many of the other factors listed above depends upon their effects on the shape at the weld toe. Plate thickness also has an important influence, but since that is a factor in all types of transverse welds it is convenient to consider its influence separately later in this chapter. Influence of weld shape As far as is known the first investigators to appreciate the possible influence of weld profile on fatigue strength were Wilson et al. (1941) They found reasonable correlation between fatigue strength and a qualitative assessment of shape based upon the height of the reinforcement and the sharpness of the angle at its edge, with specimens in which the reinforcement flowed by a relatively smooth concave surface into the parent metal giving a higher fatigue strength than specimens in which the reinforcement was high and rough and which had a sharply re-entrant angle at the weld toe. Subsequently, Becker and Rieger (1954) also noted the importance of the shape at the weld toe;

The constant amplitude database

41

they noted that electrodes giving spray transfer produced a smooth gradual junction and higher fatigue strengths than electrodes giving globular transfer, for which the junctions were more abrupt. A more extensive study of the problem was carried out some years later by Newman and Gurney (1959). They tested several types of butt welds, made both by manual and automatic welding, and obtained a wide range of fatigue strengths, which varied from 100–178 N/mm2 at 2 × 106 cycles under pulsating tension loading. As a quantitative measure of reinforcement shape the (obtuse) angle θ between the plate surface and the tangent to the reinforcement at its point of contact with the plate surface was used. Examination of the specimens revealed that this ‘reinforcement angle’ varied along the length of a weld – particularly in manually welded joints – but that failure usually originated at the point of minimum angle. In order to try to make the measurements critical, a few specimens of each test series were selected from those which gave fatigue test results lying close to the relevant S-N curve; these were then sectioned at the point of crack initiation and the angle was measured with the aid of a projection microscope. The measured angles were then plotted against the fatigue strength at 2 × 106 cycles of the particular test series from which the specimen originated, as shown in Fig. 2.6. For the manually welded series the scatter was about 15°, but for the automatic welds it was somewhat less; but it can be seen that all the experimental points lie within a scatter band which can conveniently be located at its upper end by the strengths of plain plate with and without millscale. Figure 2.6 also contains, for comparison, comparable results obtained by other investigators, using reinforcement angles deduced from macrosections of the joints illustrated in the relevant reports. It will be seen that all these results lie within the scatter band. It is not suggested that this rough relationship adequately defines the fatigue strength of transverse butt welds. However, the fact that any relationship exists proves that reinforcement shape is important. It also shows that it may be possible partly to define ‘good’ and ‘bad’ shapes. The difference between the two, as found by Newman and Gurney (1959) is clearly demonstrated by the macrosections of four of the joints which were tested and which are shown in Fig. 2.7. Differences in reinforcement shape are almost certainly responsible for the different forms of fracture which are normally obtained in tests on manual and automatic welds respectively. Automatically welded joints tend to vary comparatively little in profile along their length, and fracture initiation has frequently been noted to occur over a considerable width of the specimen (Fig. 2.8(a)). This necessarily leads to the formation of a ‘continuous’ crack at the weld toe. In contrast, with manual welds, cracks are usually initiated only at isolated points along the toe of the weld (Fig. 2.8(b)) as might be expected in view of the greater variation in reinforcement angle. As a result the failure mode normally involves

42

Cumulative damage of welded joints 300

Fatigue strength at 2 × 106 cycles (N/mm2)

θ°

Plain plate (machined)

Fatigue crack Plain plate (with millscale)

200

100

0 100

110

120 130 140 150 160 Reinforcement angle, θ (deg.)

170

180

2.6 Relation between reinforcement angle and fatigue strength of transverse butt welds.

the propagation of one or more semi-elliptical cracks at the weld toe. It is the combination of the sharper reinforcement angle and the existence of a continuous toe crack that ultimately leads to the much lower fatigue strength of automatic as compared to manual welds. This was subsequently demonstrated by fracture mechanics methods combined with finite element stress analyses of a range of idealised joints. A comparison between the results of fatigue tests on manual and submerged arc welds is shown in Fig. 2.9 and this clearly demonstrates the difference in fatigue strength. It must, however, be noted that it is possible, with care, to adjust the welding conditions in submerged arc welding to give a much more favourable weld shape and consequently a higher strength. The obvious conclusion is, therefore, that if automatic transverse butt welds are to be used in a structure subjected to fatigue loading, then welding trials should be carried out to determine suitable conditions to produce favourable weld profiles. Transverse butt weld on a permanent backing bar From the point of view of fatigue strength, a transverse butt weld made on a permanent backing bar has a worse effective shape than is provided by a

The constant amplitude database

43

(a)

(b)

(c)

(d)

2.7 Macrosections of some of the joints tested by Newman and Gurney (1964). Note particularly the differences in reinforcement angle between the various specimens: (a) manual weld made with rutile electrodes giving a pulsating tension fatigue strength at 2 × 106 cycles of 154 N/mm2; (b) submerged arc (automatic) weld (100 N/ mm2); (c) close square butt manual weld made with deep penetration electrodes (108 N/mm2); (d) submerged arc weld made under controlled conditions to give good reinforcement shape (170 N/mm2).

normal weld reinforcement. Such welds invariably fail from the notch at the junction between the weld metal and the backing bar (Fig. 2.5(c)), and not from the toe of the reinforcement on the opposite side. The strength of such a joint is about 100 N/mm2 at 2 × 106 cycles under pulsating tension loading (Newman and Gurney, 1964) and is therefore about the same as the worst form of butt weld made without a backing bar. Cruciform joints In many respects cruciform joints can be regarded as another form of transverse butt weld. In all cases the form of the specimen that has been used to investigate the strength of this type of joint has been as shown in Fig. 2.10, in which the

44

Cumulative damage of welded joints

(a)

(b)

2.8 Typical fracture surfaces of transverse butt weld specimens, (a) automatic weld showing crack initiation over the major part of the specimen width; (b) manual weld with a few discrete points of initiation growing as semi-elliptical cracks.

load has been transferred from one longitudinal plate to another through a transverse plate by way of two welds placed directly opposite each other. Joints of this general type may in fact be made with full penetration welds, as shown in Fig. 2.10, with partial penetration welds or with fillet welds, and the external appearance may be very similar in each case. The behaviour under fatigue conditions may, however, be different, particularly in so far as the mode of failure of fillet welds and partial penetration butt welds may involve fatigue cracking through the weld, while full penetration welds seem to have resulted invariably in plate failure initiated at the weld toe. For this reason, the discussion here will be confined to full penetration welds; partial penetration welds will be considered in conjunction with fillet welds later in the chapter.

The constant amplitude database

45

99.8 99 98 95 90

Probability (%)

80 Manual welds

70 60 50 40 30 Submerged arc welds

20 10 5 2 1 0.5 0.2 60

80

100

120 140 160 180 200 220 Fatigue strength (N/mm2) at 2 × 106 cycles

240

260

2.9 Comparison of the fatigue strength at 2 × 106 cycles under pulsating tension loading of transverse butt welds made manually and automatically.

2.10 Type of specimen used to investigate the fatigue behaviour of K butt welds.

For this type of joint test results are most numerous for the alternating load condition. At 2 × 106 cycles strengths varying from ±69 to ±138 N/mm2 have been recorded. However, if one ignores the very high strengths obtained in Germany (Kloppel and Wiehermuller, 1957; Wintergest and Ruckerl, 1957), it seems more reasonable to accept a strength within the range of ±69 N/mm2 obtained by Ferguson (1942) to ±97 N/mm2 recorded by Höisveen and Persson (1963). Such figures are certainly more in accord with the corresponding pulsating tension fatigue strength of 100–108 N/mm2 found by Neumann (1960) and 117–128 N/mm2 found by Yamaguchi et al. (1966), Inoue (1974) and Donato et al. (1972).

46

Cumulative damage of welded joints

2.11 K butt weld with load causing bending of the transverse plate.

It is important to appreciate the type of specimen used in these investigations. In many cases in practice the load will be taken out of the longitudinal plate not by another longitudinal plate positioned immediately opposite the first but by bending of the transverse plate as shown in Fig. 2.11. Such a loading condition may easily produce a difference in fatigue strength, even with failure still occurring in the longitudinal plate from the weld toe, because of the more severe local stress conditions. Once again this may be visualised by considering the distribution of the hypothetical lines of stress flow, which must inevitably be more crowded in the region of the stress concentration than they would be if they were able to flow straight through the transverse plate. Few test results exist, however, for specimens loaded in this way. Stud shear connectors In composite construction there is an obvious need for attachments to be welded to the flange to act as shear connectors, and these often take the form of welded studs. In most applications the majority of the studs will be welded to the compression flange of the beam, so that even if the residual stresses induced by welding enable fatigue cracks to initiate and grow to an appreciable size, it is most unlikely that catastrophic failure would occur. However, the use of ‘continuous’ structures necessitates the welding of shear connectors to the tension flange in the regions where the beam passes over the supports. In these regions it would certainly be possible for a fatigue crack to propagate through the entire flange. What is more, such a crack would not be visible since it would be buried in concrete. Although few investigations have been made of the effect on fatigue strength of studs welded to a member, there is excellent agreement between the results that do exist. For the most part the specimens used have consisted of a flat plate, with one or more studs welded to it, subjected to axial loading. The results obtained with such specimens have varied from about 93–120 N/ mm2 at 2 × 106 cycles for arc welded studs tested at R = 0 and, in a single investigation (Selby et al., 1963), the strength at R = –1 was found to be 112

The constant amplitude database

47

N/mm2 (range). Friction welded studs gave 93 N/mm2 at R = 0. These strengths are very similar to those for non-load-carrying fillet-welded joints, which the stud closely resembles. Grinding the transition region of arc welded studs has given improved strengths ranging from 116–137 N/mm2. Unfortunately, this type of specimen does not fully represent a beam flange with stud shear connectors, since the stud is not subjected to a shear force at the same time as the member is stressed, as occurs in actual composite construction. A few comparative tests have, however, been carried out on beam specimens with and without a concrete slab (Selby et al., 1963). These showed some indication that the additional shear force imposed by the concrete did tend to reduce fatigue strength. Fillet welded joints Fillet welded joints can conveniently be divided into two types, those in which the welds are continuous and loaded parallel to their length and those in which the welds may be considered as discontinuous. The former class is typified by the fillet welds used to make web-to-flange joints in the construction of beams and the effect of such welds under conditions of repeated loading was considered above. This section will be restricted to a consideration of the effect of so-called ‘discontinuous’ welds, i.e., those that do not extend over the whole length of a stressed member. This group may be further sub-divided into welds that are either transverse or parallel to the direction of the applied stress, and in either of these cases the welds may be defined as load-carrying or non-load-carrying. In this context ‘load-carrying’ means that a major part of the load is transferred through the weld in shear while a ‘non-load-carrying’ weld may be defined as any weld on a stressed member, which does not carry the load, or an appreciable part of it, into the member. It should be realised, however, that a non-load-carrying weld, as defined here, is not entirely load free since it will be strained by the load in the stressed member on which it is laid. Typical examples of these four types of joint are shown in Figs 2.12 and 2.13, which also show, qualitatively, the stress distributions associated with each under axial tensile loading. Figure 2.12 refers to welds which are transverse, and Fig. 2.13 to welds that are parallel, to the direction of the applied stress. In the former case the stress distribution, although uniform along the length of the weld (i.e. across the width of the specimen) varies through the specimen thickness since the toes of the welds form stress concentrations. On the other hand, with longitudinal fillet welds it is the weld ends that form the stress concentrations, and the stress distribution is non-uniform both over the specimen width and through its thickness. In the case of the load-carrying welds it should also be noted that the stress distribution in the welds is not uniform. The transverse weld has a stress concentration

48

Cumulative damage of welded joints

(a) Non-load-carrying joint.

(b) Load-carrying joint.

2.12 Qualitative stress distributions in transverse fillet welded joints.

Shear stress in the weld

(a) Non-load-carrying joint.

(b) Load-carrying joint.

2.13 Qualitative stress distributions in longitudinal fillet welded joints.

at the weld root, while the longitudinal weld tends to transmit the majority of the load near its two ends and a lesser amount at the centre of its length. From a consideration of these stress distributions the possible modes of failure to be expected under repeated loading are obvious. If failure occurs in the plate, as will invariably happen with non-load-carrying fillet welds, the fatigue crack will be initiated at one of the weld ends in the case of a longitudinal weld or at the weld toe with a transverse weld. But in loadcarrying fillet-welded joints failure may either occur in the plate, with the cracks being initiated at the same points as in the case of non-load-carrying welds, or in the welds themselves. In the latter event failure will be initiated

The constant amplitude database

1

1

2

1

49

1

2

3

2.14 Modes of failure of fillet welded joints under fatigue loading. The black spots indicate points of crack initiation.

(a) Specimen with gussets welded on plate surfaces.

2.15 Typical specimens used to study the fatigue behaviour of joints involving longitudinal non-load-carrying fillet welds.

at the weld ends of a longitudinal weld or at the weld root of a transverse weld. All these modes of failure are shown diagrammatically in Fig. 2.14. Longitudinal non-load-carrying welds on the plate surface In the investigation of the fatigue strength of joints involving non-loadcarrying fillet welds on the plate surface the type of specimen which has usually been employed is as shown in Fig. 2.15. There have, however, been several variations of a comparatively minor nature, particularly with respect to the number of attachments on each surface. Nevertheless, the majority of tests have involved specimens with two gussets, one on each surface opposite each other, in order to maintain symmetry. A theoretical fracture mechanics study of this type of joint, Smith and Gurney (1986) suggested that, at least for specimens 13 mm thick and 152 mm wide, attachment length has a significant influence on fatigue life, particularly for attachments less than 150 mm long. It showed that fatigue life tended to increase as attachment length decreased, with a theoretical

50

Cumulative damage of welded joints

increase in life of about 70% on reducing length from 150 to 50 mm. For longer attachments the effect was much smaller with a reduction in length from 300 to 150 mm only giving a theoretical increase in life of about 10%. In general terms these findings are in agreement with available experimental results, which are summarised in Fig. 2.16. In the same study (Smith and Gurney, 1986) attachment height and thickness and weld leg length were all found only to have a relatively minor influence on fatigue life. However, rather surprisingly, an increase in main plate thickness was found to give a theoretical increase in fatigue life. This is in distinct contrast to the well proven finding for transverse attachments where an increase in thickness leads to a decrease in fatigue strength. (Part of the increase derives from the larger value of af in the larger joints, but the effect is still apparent even if a constant value of af is used.) Further work is still required to confirm this finding experimentally. Transverse non-load-carrying welds The type of specimen most often used in the study of transverse non-loadcarrying welds is shown in Figs 2.17(a) and 2.17(b) and consists merely of a gusset or gussets welded transversely to either one or both surfaces of the main plate. These types of specimen have been tested by a very large number of investigators and, in comparison with joints with longitudinal attachments, the results have been much more scattered. Based on a statistical analysis of the results of 454 individual tests under pulsating tension loading the mean fatigue strengths at 105 and 2 × 106 cycles were 315 and 105 N/mm2 respectively, with the limits of the scatter band, as represented by the mean ±2 standard deviations of log N, being 165 and 66 N/mm2 at 2 × 106 cycles. This may be compared with a scatter range from 114 to 68 N/mm2 at 2 × 106 cycles for specimens with longitudinal attachments. The influence of joint geometry on the fatigue strength of this type of joint has been studied in considerable detail, both theoretically, by means of fracture mechanics, and experimentally. As a result it has been found that the main plate thickness, the attachment thickness and the weld size can all have an important influence on fatigue strength. The influence of main plate thickness is considered in more detail later, since it is relevant to all types of joints involving welds transverse to the direction of stress (e.g. transverse butt welds, K butt welds, etc.). However, it may be helpful to summarise here the work relating to the influence of attachment thickness and weld size. It has in fact been found that these two variables can, to a very large extent, be considered as one, since the critical dimension, as far as attachment size is concerned, seems to be the overall distance from weld toe to weld toe, namely L (see Fig. 2.18). In general it is convenient to think in terms of the ratio L /T, where T is the main plate thickness.

Stress range, Nmm–2

40 104

50

60

70

80

90

100

150

200

300

350

400

3 4 5

6 7 8 9 105 2 3 4

5 6 7 8 9 106

Endurance, cycles

2

3

4

2.16 Results for longitudinal attachments, showing influence of length of stiffener.

2

50 mm long attachments 150 mm long attachments 305 mm long atttachments

305 mm long atttachments

5 6 7 8 9 107

150 mm long attachments

50 mm long attachments

52

Cumulative damage of welded joints

(a) (b) Specimens with welded gussets

(c) Specimens with welded pad plates.

2.17 Typical specimens used to study the fatigue behaviour of joints involving transverse non-load-carrying fillet welds. L

45°

a T

b

w

2.18 Joint geometry used in theoretical analysis of transverse nonload-carrying fillet welded joints.

By means of fracture mechanics analysis it has been found that the relationship between the fatigue strength for a particular plate thickness and ratio L /T is of the form shown in Fig. 2.19. This shows the results both for a semi-elliptical toe crack and a continuous toe crack. It should be noted that the two relationships are for slightly different thicknesses, namely 20 and 22 mm respectively, because they are the thicknesses at which the rate of decrease of fatigue strength of geometrically similar joints (constant L/T) changes; the reason for this change is not yet understood. Although the two relationships are, in principle, very similar to each other, the strength of the joint with a continuous toe crack is significantly lower than that with a semi-elliptical crack. It will be seen that, in both cases, fatigue strength seems to reach a minimum value at L / T = 2–2.7. In other words small attachments (i.e. those having L / T 10 mm

Type 3. Transverse butt welds (i.e. essentially perpendicular to the direction of applied stress) conditions), at the weld root. Unless made approximate method of allowing for eccentricity Notes on potential modes of failure With the weld ends machined flush with the on a permanent backing (type 3.3) welds made in the thickness direction is to multiply the plate edges, fatigue cracks in the as-welded entirely from one side are not classified for nominal stress by (1 + 3e/t), where e is the condition normally initiate at the weld toe, so fatigue purposes, since adequate control distance between centres of thickness of the that the fatigue strength depends largely upon cannot be exercised over the profile of the two abutting members (if one of the members the shape of the weld overfill. If this is dressed root bead which is where fatigue cracks would is tapered, the centre of the untapered thickness flush the stress concentration caused by it is be likely to initiate. must be used); and t is the thickness of the removed and failure is then associated with thinner member. With connections which are weld defects. In welds made on a permanent Design stresses supported laterally, e.g. flanges of a beam which backing strip, fatigue cracks initiate at the weld In the design of butt welds of types 3.1 or are supported by the web, eccentricity may be metal/strip junction, and in partial penetration 3.2 which are not aligned the stresses must neglected. welds (which should not be used under fatigue include the effect of any eccentricity. An 3.1 Parent metal adjacent to, or weld metal in, full penetration butt joints welded from both sides between plates of equal width and thickness or where differences in width and thickness are machined to a smooth transition not steeper than 1 in 4.

Note that this includes butt welds which do not completely traverse the member, such as circular welds used for inserting infilling plates into temporary holes.

Table 2.3 Continued Type number, description and notes on mode of failure

Class Explanatory comments

(a) With the weld overfill dressed flush with the surface and with the weld proved free from significant defects by nondestructive examination.

C

The significance of defects should be determined with the aid of specialist advice and/or by the use of fracture mechanics analysis. The nondestructive testing technique must be selected with a view to ensuring the detection of such significant defects.

(b) With the welds made in the shop, either manually or by an automatic process other than submerged arc, provided all runs are made in the flat position.

D

In general welds made positionally, or on site, or by the submerged arc process tend to have a poor reinforcement shape, from the point of view of fatigue strength. Hence such welds are downgraded from D to E.

(c) Welds made other than in (a) or (b).

E

In both (b) and (c) the corners of the cross-section of the stressed element at the weld toes should be dressed to a smooth profile. Note that step changes in thickness are, in general, not permitted under fatigue conditions, but that where the thickness of the thicker member is not greater than 1.15 × the thickness of the thinner member, the change can be accommodated in the weld profile without any machining. Step changes in width lead to large reductions in strength (see joint type 3.3).

Examples, including failure modes

t

e

Table 2.3 Continued Type number, description and notes on mode of failure

Class Explanatory comments

3.2 Parent metal adjacent to, or weld metal in, full penetration butt joints made on a permanent backing strip between plates of equal width and thickness or with differences in width and thickness machined to a smooth transition not steeper than 1 in 4.

F

Examples, including failure modes

Note that if the backing strip is fillet welded or tack welded to the member the joint could be reduced to class G (joint type 4.2).

No tack welds

3.3 Parent metal adjacent to, or weld metal in, full penetration butt-welded joints made from both sides between plates of unequal width, with the weld ends ground to a radius not less than 1.25 × the thickness t.

F2

Step changes in width can often be avoided by the use of shaped transition plates, arranged so as to enable butt welds to be made between plates of equal width. Note that for this detail the stress concentration has been taken into account in the joint classification.

Type 4. Welded attachments on the surface or edge of a stressed member Notes on potential modes of failure attachments involving a single as opposed When the weld is parallel to the direction of to a double weld, cracks may also initiate at the applied stress fatigue cracks normally the weld root. The cracks then propagate into initiate at the weld ends, but when it is the stressed member. When the welds are transverse to the direction of stressing they on or adjacent to the edge of the stressed usually initiate at the weld toe; for member the stress concentration is increased

t

r ≥ 1.25t

and the fatigue strength is reduced; this is the reason for specifying an ‘edge distance’ in some of these joints (see also note on edge distance in joint type 2)

Table 2.3 Continued Type number, description and notes on mode of failure

Class Explanatory comments

4.1 Parent metal (of the stressed member) adjacent to toes or ends of bevel-butt or filletwelded attachments, regardless of the orientation of the weld to the direction of applied stress, and whether or not the welds are continuous round the attachment.

Butt-welded joints should be made with an additional reinforcing fillet so as to provide a toe profile similar to that which would exist in a fillet-welded joint.

(a) With attachment length (parallel to the direction of the applied stress) ≤150 mm and with edge distance ≥10 mm.

F

(b) With attachment length (parallel to the direction of the applied stress) >150 mm and with edge distance ≥10 mm.

F2

4.2 Parent metal (of the stressed member) at the toes or the ends of butt- or fillet-welded attachments on or within 10 mm of the edges or corners of a stressed member, and regardless of the shape of the attachment.

G

The decrease in fatigue strength with increasing attachment length is because more load is transferred into the longer gusset, giving an increase in stress concentration.

Note that the classification aplies to all sizes of attachment. It would therefore include, for example, the junction of two flanges at right angles. In such situations a low fatigue classification can often be avoided by the use of a transition plate (see also joint type 3.3).

Examples, including failure modes

Edge distance

Edge distance

Table 2.3 Continued Type number, description and notes on mode of failure

Class Explanatory comments

4.3 Parent metal (of the stressed member) at the toe of a butt weld connecting the stressed member to another member slotted through it.

Note that this classification does not apply to fillet-welded joints (see joint type 5.1b). However, it does apply to loading in either direction (L or T in the sketch).

(a) With the length of the slotted-through member parallel to the direction of the applied stress ≤150 mm and with edge distance ≥10 mm.

F

(b) With the length of the slotted-through member parallel to the direction of the applied stress >150 mm and with edge distance ≥10 mm.

F2

(c) With edge distance 3.0

n 4.19 Histograms showing distributions of: (a) mean values of Σ in N n in each test series. For each test series (b) lower limit values of Σ N results published up to 1986 (146 test series).

Fig. 4.19 are, if anything, larger than the values which would be derived using the real or design S-N curves. In terms of individual test results (approx 1300) the values of Σ n ranged N from 0.16 to 9.12, after ignoring three test series for which the mean values of Σ n were more than 10.0. These were so out of line with the other results N that it was assumed that some error must have occurred in reporting the results. In terms of test series rather than individual results about 15% gave

124

Cumulative damage of welded joints

n mean values of Σ N which were less than 0.8 and 25% gave lower limit values which were less than 0.6. In other words, those results were clearly not being forecast correctly by Miner’s rule. A summary of the results is shown in the form of a histogram in Fig. 4.19(a). It is also obvious from Fig. 4.19 that test results obtained under block programme loading, which is clearly not typical of real service conditions, n tended to give relatively high values of Σ N . In other words, if those values n were ignored, the average values of Σ N would be smaller, i.e., less safe. It was as a result of this analysis that it was felt necessary to try to define what particular features of the loading history tended to lead to low values of n ΣN (i.e. to produce unsafe predictions of life). It is the results of this work that form the basis of Chapters 5–8. However, other investigators have also continued to test various types of joints under a range of loading spectra with n the objective of determining the particular values of Σ N relevant to those particular conditions.

5 Tests under two and three level loading

5.1

Introduction

Given that the survey of test results referred to in Chapter 4 clearly indicated that variable amplitude fatigue tests could give values of ∑ (n/N) well outside the expected range of values (either above or below it), there was an urgent need to try to discover why that was happening and possibly to define a more accurate rule. The obvious starting point seemed to be to try to keep the loading spectrum as simple as possible with only two magnitudes of stress range. That was achieved by applying stress histories consisting essentially of constant amplitude loading with one or more excursions applied on each stress cycle. With this type of loading there is no problem over cycle counting but there is certainly the potential for stress interaction. It should be noted that this type of loading is different from the simple two-step loading used in the early days of variable amplitude testing, in that in these new tests the loading effectively involved ‘complex cycles’ (i.e. with the small and large cycles combined), whereas the early work had involved the two different magnitudes of cycle being applied quite separately. Initially, the tests were restricted to wholly tensile loading with the peaks of the smaller cycles never greater than that of the main constant amplitude cycle, so it seemed reasonable to assume that the stress interaction effect should consistently give an increase in the expected life; in other words the values of ∑(n/N) were expected always to be greater than 1.0. The initial work was aimed at defining an empirical relationship for predicting the increase in life. In order to simplify the work all the tests in the first part of the investigation (Gurney, 1982, 1983a, b) involved specimens of the form shown in Fig. 5.1, since it was known that that type of specimen tended to give very little scatter in fatigue test results. They were subjected to stress cycles of the form shown in Fig. 5.2 and it is convenient to consider the results in terms of the number of ‘complex cycles’ to cause failure (where each complex cycle consists of a single cycle of magnitude σ and an associated number of small cycles of magnitude pσ, with 0 < p < 1.0). The main objectives of the initial 125

126

Cumulative damage of welded joints

12.5

150

150

38 12.5

5.1 Details of test specimen.

x y

y

z x

σ

z x

z

z

x x

Type 1

Type 2

Type 3

x y

z x

z z x

Type 4

x Type 5 (z = 0.285σ )

5.2 Stress sequences with a single size of excursion (z = pσ).

set of tests were to define the influence of the number of small cycles in each complex cycle and to define the magnitude of the small cycle which had no influence at all. In addition, it was anticipated that comparison of the lives obtained using cycles of Types 3 and 4 (see Fig. 5.2) should indicate whether a large uninterrupted rising range was or was not more damaging than a large falling range. At this stage it is relevant to note that it is easy to define upper and lower bounds of the fatigue life to be expected under the various stress histories. Thus, considering for example the cycle of Type 1 shown in Fig. 5.2, if

Tests under two and three level loading

127

p = 0 the whole cycle reduces to constant amplitude loading, giving an upper bound on life. Alternatively, if p = 1.0 the ‘cycle’ again becomes constant amplitude loading but with three cycles instead of one; this is obviously a lower bound situation. Hence, remembering that in the present context we are taking each of the stress histories shown in Fig. 5.2 to represent ‘a cycle’, the life for Type 1 will lie between Nc and 1/3 Nc, where Nc represents the life under constant amplitude loading of stress range σ. The results obtained in tests at R = 0 for each of the loading sequences of Types 1–4 with a constant maximum stress of 185 N/mm2 and a number of different magnitudes of stress excursion (z = pσ) are shown in Fig. 5.3. It is clear that, in each instance, a good linear correlation was obtained by plotting 1.0

0.9

0.8

0.7

Value of p

0.6

0.5

0.4

0.3

0.2

0.1

0 104

Sequence type 1 2 3 4 2

3

4

5 105 Endurance, cycles

2

3

4

5

Nc

5.3 Test results for pulsating tension stress sequences with peak stress = 185 N/mm2.

128

Cumulative damage of welded joints 1.0

0.9

0.8

Σ

n = 1.0 N

0.7

Value of p

0.6

0.5

0.4

0.3

0.2

0.1

0 104

2

3

4

5 105 Endurance, cycles

2

3

4

5

5.4 Comparison of test results obtained with stress sequence Z with those predicted by Miner’s rule.

p against ln N with the end points of the lines at p = 0 and p = 1.0 being defined as indicated above. If the results were to comply with Miner’s rule they would have to lie on a curve, as indicated in Fig. 5.4, where the results for Sequence 2 are replotted as an example. It is obvious that the straight-line interpretation is considerably more realistic. In fact, the calculated Miner summations for the tests for Sequences 1–4, lay in the range 0.64–1.02. In order to check that these results were not related solely to the choice of the magnitude of the peak stress range, some check tests were carried out using loading Sequence 4A (see Fig. 5.5) with peak ranges of 160 and 80 N/ mm2. For all practical purposes this sequence was the same as Sequence 4 (Fig. 5.2) but it had two subsidiary cycles rather than one on the falling

Tests under two and three level loading

129 p.σ

p.σ

σ

p.σ Sequence 4A

Sequence 4B (a)

z1 = p1.σ

z2 σ = p2.σ

Sequence 4C

z2 σ = p 2. σ

z1 = p1.σ

Type 7

Type 6

z2 σ = p2 . σ

z1 = p1.σ

Type 8 (b)

5.5 Stress sequences with excursions of two sizes or with a single size of excursion at different mean stresses.

major half-cycle. The results were virtually identical to those for Sequence 4, with values of ∑ (n/N) ranging from 0.71–1.00 for σmax = 160 and from 0.70 to 1.07 for σmax = 80 and the results again giving a good linear relationship when plotted on the basis of p against ln N. In other words, at least under tensile loading, the magnitude of the peak stress appears to be irrelevant. It is also apparent from Fig. 5.3 that there was no significant difference between the results for sequences Types 3 and 4. In other words it does not matter whether a ‘break’ in a cycle occurs on the rising or falling part of the cycle.

130

5.2

Cumulative damage of welded joints

Theoretical analysis

Stress sequences with a single size of excursion Based upon the experimental results shown in Fig. 5.3, it seems reasonable to deduce that there is a linear relation between p and ln N, where p is the ratio between the magnitude of the stress excursions and the constant amplitude stress range σ (i.e. z = pσ). Also, as noted previously, upper and lower bounds on the life can be deduced from the fact that the loading degenerates to constant amplitude loading at p = 0 and p = 1.0. Thus with p = 0 the life necessarily becomes Nc, since each block reduces to a single constant amplitude cycle, while with p = 1.0 the life becomes Nc / (ν + 1). Hence, the relation between p and NB (the number of ‘blocks’, or in this case the number of complex cycles, to failure) becomes: NB = Nc(ν + 1)–p

[5.1]

Alternatively, assuming that the constant amplitude S-N curve is a straight line of slope m, where for fillet welded joints m is approximately 3.0, the corresponding Miner summation for the same situation is: m n = n 1 + vn 1 p = n 1 (1 + vp m ) ΣN Nc Nc Nc

[5.2]

where n1 is the number of blocks to failure. This arises because the constant N amplitude lives for the two stress ranges, S and pS, are Nc and mc , respectively. p If the predicted life (in blocks) deduced from eqn [5.1] is written as n2, the ratio between that and the corresponding life predicted by Miner’s rule, n1, becomes, from eqns [5.1] and [5.2]: n1/n2 = (1 + ν)p/(1 + νp m )

[5.3]

With p equal either to 0 or 1.0 we clearly have n1 = n2 but n1 and n2 are different at intermediate values of p. It is of interest to define the values of p and ν giving the maximum value of n1/n2. Since n1/n2 > 0 the maximum will occur under the same conditions as for ln (n1/n2). Thus, taking logs of eqn [5.3] F = ln (n1/n2) = p ln (1 + ν) – ln (1 + νpm). Differentiation gives: m–1  ∂F = ln (1 + ν ) –  mνp  m  ∂p (1 + p) ν  

[5.4]

pm ∂F = p – ∂ν 1 + ν (1 + νp) m

[5.5]

To find the maximum value of F put ∂ F/∂ p = ∂ F/∂ ν = 0 and solve for p and ν. Dividing eqn [5.4] by eqn [5.5] we have:

Tests under two and three level loading

(1 + ν) ln (1 + ν) = mν

131

[5.6]

and from eqn [5.5] 1 + νpm – (1 + ν) pm–1 = 0

[5.7]

There is only one non-zero solution to eqn [5.6]. For m = 3 it is ν ≈ 16. With m = 3 eqn [5.7] can be rewritten as: (p – 1)(νp2 – p – 1) = 0

[5.8]

so that p = 1 or (1 ± (1 + 4ν) /2ν) 1/2

Hence, on inserting ν = 16, the positive non-trivial root is given by p = 0.285, and the corresponding value of n1/n2 is 1.636. This represents the maximum difference between the life predicted by eqn [5.1] and by Miner’s rule, eqn [5.1] giving a life of 1/1.636 = 0.61 of the life predicted by Miner’s rule. Stress sequences with excursions of two sizes Consider now a sequence consisting of a constant amplitude cycle σ with ν1 excursions of magnitude p1σ and ν2 excursions of p2σ (for typical examples see Fig. 5.5). Clearly there are various special cases which can be defined. Firstly, if p2 = 0 the sequence degenerates to one with a single size of excursion. In this case the life is given by: NB = Nc /(1 + ν1)p1[p2 = 0]

[5.9]

which is shown as the line XY in Fig. 5.6. NC 1 + v1

NC 1 + v1 + v 2 1.0

Y

Z B

p

0

p1′ p1 = p2

p2 = 0

A NC NC p1 p1 (1 + v 1 + v 2) (1 + v1 ) Log endurance

X NC

5.6 Derivation of possible theoretical S-N curve for stress history with excursions of two sizes.

132

Cumulative damage of welded joints

Secondly, in the situation where p1 = p2 the sequence again degenerates to the earlier case, but with (ν1 + ν2) small cycles. In this case the life can be represented by the line XZ, given by: NB =Nc/(1 + ν1 + ν2)p1 [p1 = p2]

[5.10]

Thus, in the particular case p1 = p 1′, the life corresponding to p2 = 0 is given by point A in Fig. 5.6 and the life corresponding to p1 = p2 = p 1′ is given by point B (the vertical axis of the graph now representing p2). It seems reasonable to assume that the straight line AB will represent the life corresponding to p1 = p 1′ with p1 ≥ p2 ≥ 0. It is easy to show that the equation of AB is: NB /Nc = (1 + ν1) p2/{(1 + ν1 + ν2) p2(1 + ν1) p1}

[5.11]

Alternatively, if it is assumed that Miner’s rule is correct, the predicted life is given as before (see eqn [5.2]) and is:

n 1 = N c /(1 + ν1 p 1m + ν 2 p 2m )

[5.12]

Hence, the ratio between the life n2 deduced from eqn [5.11] and the life n1 based on Miner’s rule becomes:

(1 + ν1 + ν 2 ) p2 (1 + ν1 ) p1 n1 = n2 (1 + ν1 p 1m + ν 2 p 2m )(1 + ν1 ) p2

[5.13]

General case: the area rule The theoretical S-N curve relating to stress sequences with minor cycles of more than two sizes can be derived in the same way as indicated above, by locating the end points of the curve which refer to particular special situations. If E i is the number of cycles per block during which the stress range equals, or exceeds, p iσ, we have: i

E i = Σ (νi)

[5.14]

Eo = νo = 1.0

[5.15]

o

and

By the method indicated above it is easy to show that, for a stress history consisting of the constant amplitude cycle and n sizes of minor cycle, n E NB = Π  i–1  1  Ei  Nc pi

[5.16]

It will be noted that eqn [5.16] includes eqns [5.1] and [5.11] for n = 1 and 2 respectively. Now, by definition, E i > E i–1 and Eo = 1, so that, taking logarithms of eqn [5.16], we get:

Tests under two and three level loading

133

1.0

Proportionate stress (p)

p1

p2

p3

N E0 = 1 NE 1 NE 2 Log (number of exceedences (NE))

NE 3

5.7 Diagrammatic representation of the Area rule. n NB = – Σ [p i (ln E i – ln E i–1 )] [5.17] 1 Nc Now consider, for example, a spectrum with three discrete levels of subsidiary stress, p1, p2 and p3. As can be seen from Fig. 5.7 eqn [5.17] represents the summation of the areas of the vertical strips in the diagram and is consequently equal to the total area. Clearly, as the steps between successive values of p tend to zero the exceedence curve tends to a ‘continuous’ rather than a ‘stepped’ curve. Hence, in general, we have:

ln

ln (NB /Nc ) = – area under exceedence curve p against ln E i [5.18] which can be re-written as: NB = Nc e–Area

[5.19]

where NB is the predicted life in blocks. Hence the predicted life in cycles becomes: N = NB × block length = Nc e–Area × block length

[5.20]

134

Cumulative damage of welded joints

Clearly, if correct, this ‘area rule’ is just as easy to apply as Miner’s rule, and it is interesting to see that, in contrast to Miner’s rule, it is independent of the slope of the S-N curve; it only requires knowledge of Nc, the constant amplitude life under the peak stress range in the spectrum, and of the form of the loading spectrum.

5.3

Fatigue tests using stress sequences with excursions of two sizes

Having defined a predicted life for stress sequences with several different magnitudes of cycle, some check tests were carried out using two simple sequences each consisting of a constant amplitude cycle with two associated minor cycles (see Fig. 5.5). The results obtained, together with the corresponding S-N curves predicted by eqn [5.18] are shown in Figs 5.8 and 5.9. It will be seen that the agreement between the predicted S-N curves and the experimental points was quite reasonable. If the results for both sequences are taken together, the ratio of actual life to predicted life ranged from about 0.75 to 1.26. This may, at first sight, appear to be a large variation, but it is very similar to the scatter normally obtained under constant amplitude loading. If all the constant amplitude results are taken together, the range of life between ±2 standard deviations is in fact about 0.5 to 2.0. In this context, therefore, the agreement was excellent. For comparison it may be noted that the corresponding values of ∑ (n/N) ranged from 0.61 to 1.0. This also represents a relatively small variation, but it is interesting that all but one specimen gave ∑ (n/N) less than 1.0 whereas, in the light of the anticipated stress interaction effect, it might be expected that all results would give ∑ (n/N) greater than 1.0. To that extent, the results in terms of Miner’s rule are ‘wrong’ on the unsafe side, whereas the relation predicted by eqn [5.18] is more nearly correct, the results lying within about ±25% of it.

5.4

Influence of stress ratio and residual stresses

All the tests considered to date related to as-welded specimens subjected to pulsating tension (R = 0) loading with a maximum stress range of 185 N/ mm. However, as was shown in Chapter 3, under constant amplitude loading, although the applied stress ratio does not usually have much effect on the fatigue strength of as-welded specimens containing high tensile residual stresses, it can certainly show an effect in stress relieved joints. In simple terms the reason for this is that, in the presence of high tensile residual stresses, the top limit stress in a loading cycle is inevitably equal to yield stress tension, regardless of the nominal stress ratio of the applied loading.

Tests under two and three level loading

135

NC = 4.8 × 10lL 1+2+2 1.0

0.9

0.8

0.7

p1 . σ

p 2. σ

p1

p2

1.0 0.75 0.5

Varies Varies Varies

Value of p2

0.6

0.5 S–N curve for p1 = p2

Predicted S–N curve for p1 = 1.0 0.4

Predicted S–N 0.3 curve for p1 = 0.75

0.2 Predicted S–N curve for p1 = 0.5 0.1

0 104

2

3

4

5

105

Endurance, cycles

2

3

4

5

NC

5.8 Fatigue test results for sequence type 6 (maximum stress = 185 N/mm2) and predicted S-N curves for P1 = 1.0, 0.75 and 0.5.

In other words all cycles tend to pulsate downwards from tensile yield (cf. stalactites) whatever their nominal stress ratio (see Fig. 3.5), so that little, if any, of the loading cycle will be in compression. In contrast, with stress relieved joints (assuming, for simplicity, that stress relief completely eliminates residual stresses) the actual stress ratio is the same as the nominal stress ratio of the applied loading. In this situation tests at R = –1 will involve the compressive part of the stress cycle actually being

136

Cumulative damage of welded joints

1.0

0.9

0.8

0.7 Predicted S-N curve for p1 = 1.0

Value of p2

0.6

72, 870

0.5 Predicted S-N curve for p1 = 0.75

S-N curve for p1 = p2

0.4

p 2. σ

p1 . σ

0.3

0.2 p1 0.1

0 104

Predicted S-N curve for p1 = 0.5

p2

1.0 Varies 0.75 Varies 0.5 Varies 2

3

4

5 105 Endurance, cycles

2

3

4

5

5.9 Fatigue test results for sequence type 7 (maximum stress = 185 N/mm2) and predicted S-N curves for p1 = 1.0, 0.75 and 0.5.

in compression, and therefore being less damaging. Indeed, as noted in Chapter 3, there is some evidence to suggest that, in stress relieved joints, only about 40% of the compressive half-cycle is damaging. In view of this situation it was of interest to determine to what extent the results of tests on stress-relieved joints and under alternating loading could be predicted by means of the ‘area rule’ (eqn [5.18]). Considering first of all the influence of stress ratio on as-welded joints, tests were made both at R = 0 and at R = –1 on specimens subjected to loading Sequence 1. As was the case with the tests at R = 0 (see Fig. 5.3) the tests at R = –1 also gave results which followed the linear area rule prediction, with all values of ∑ (n/N) in the

Tests under two and three level loading

137

range 0.68–0.88. In other words the area rule was again a better predictor of life than Miner’s rule for this particular type of loading. So far as the influence of stress relief is concerned, three pairs of test series are directly relevant to the issue. In the first place both as-welded and stress relieved specimens were tested under Sequence 1 both at R = 0 and at R = –1. Secondly, tests were also carried out both on as-welded and stress relieved joints using loading Sequence 3A (Fig. 5.5). In all these six test series only one specimen gave a value of ∑(n/N) greater than 1.0, and remembering that the values of ∑(n/N) were calculated assuming that the constant amplitude ln S v. ln N line could be extrapolated linearly downwards ad infinitum all the quoted values of ∑(n/N) will actually be too high. It is therefore clear that, in these particular instances, Miner’s rule was unsafe. In contrast, the area rule was a much better predictor of life. A summary of the relevant results is shown in Table 5.1. It is important to note, however, that the various values of ∑(n/N) were calculated using as a basis the relevant constant amplitude S-N curve for the particular conditions under consideration. Similarly the values of RG were calculated using eqn [5.18] to calculate the predicted life, again using the relevant value of Nc for the particular condition. Thus, although the values of ∑(n/N) or RG were the same for different test series, this does not imply that the fatigue lives were also the same. The remainder of this particular part of the test programme involved a different type of specimen (Fig. 5.10), for which the relevant constant amplitude S-N curves are shown in Figs 5.11 and 5.12. The loading sequences (Fig. 5.5) were also slightly different from those used previously. In general terms they were similar to Sequences 3 and 4 (Fig. 5.2) but instead of there being a single subsidiary cycle attached to each main cycle there were two subsidiary cycles. In this part of the work the primary intention was to investigate the

Table 5.1 Influence of stress relief on the results obtained under Sequences 1 and 3A Load

Specimen

sequence condition

Stress

No. of

Values of ∑(n/N)

Values of RG

ratio, R

results

Range of values

Mean

Range of values

Mean

1

As-welded Stress relieved

0 0

6 7

0.70–0.97 0.65–1.06

0.79 0.82

0.77–1.06 0.93–1.31

0.98 1.10

1

As-welded –1 Stress –1 relieved

4 10

0.68–0.88 0.58–0.91

0.82 0.74

0.92–1.15 0.78–1.31

1.03 1.02

3A

As-welded –1 Stress –1 relieved

5 5

0.66–0.94 0.64–0.89

0.82 0.77

0.87–1.15 0.94–1.18

1.05 1.05

138

Cumulative damage of welded joints 150

12

125

8 mm leg length fillet welds on each side of main plate

12 90 Dimensions in mm

5.10 Details of test specimen with longitudinal non-load-carrying edge attachments.

influence of the stress ratio of the subsidiary cycles, with the main constant amplitude cycle held constant at R = –1. The three cases considered were: 1. With the subsidiary cycles having the same mean stress as the main cycle (Sequence 4A). 2. With the subsidiary cycles having the same bottom limit stress as the main cycle (Sequence 4B). 3. With the subsidiary cycles having the same top limit stress as the main cycle (Sequence 4C). The first step, however, was to check whether there was any significant difference, for stress relieved specimens, between the effect of Sequences 3A and 4A, Sequence 3A having the subsidiary cycles following the compressive peak and Sequence 4A having them following the tensile peak. In effect this pair of tests represents a repeat of those using Sequences 3 and 4 (considered previously), but using stress relieved joints. The results obtained with these two sequences, together with the theoretical relationships derived by means of the area rule, are shown in Figs 5.13 and 5.14; they are also summarised in Table 5.2. In general Miner’s rule tended, apart from one specimen, to give unsafe predictions while the area rule gave quite good life predictions, particularly for Sequence 3A. With Sequence 4A it tended to give safe predictions. In qualitative terms that is not particularly surprising, since, with stress relieved joints, one would expect the order of the stresses in Sequence 4A to be beneficial. This is because the peak tensile stress of the main cycle would tend to create compressive residual stresses at

Stress, N/mm2

300

60 104

70

80

90

100

120

140

160

180

200

220

240

260

280

2 3 4 5

105 1.5 2 3

4

5

Endurance, cycles

106

1.5

5.11 Contact amplitude test results (for specimen shown in Fig. 5.10) at R = 0.

1.5

Mean S-N curve for all results

Rutile Low hydrogen

2

3

4

5

This investigation

Previous investigations

107

Range of stress, N/mm2

300

60 104

70

80

90

100

120

140

160

180

200

220

240

260

280

3 4 5

105 2 3

4

5

Endurance, cycles

106

1.5

2

R = – 1, rutile

3

5.12 Contact amplitude test results for stress relieved specimens at R = 0 and R = –1.

2

Symbol Stress ratio R=0 R=–1

R = 0, rutile

4

5

107

Tests under two and three level loading

141

1.0

0.9 100 100p 0.8

0

100p

Stressrelieved

–100 0.7 Aswelded 0.6

p

0.5

0.4 Predicted results (Eq. 2) 0.3

0.2

Constant amplitude lives: As-welded 250,000 Stress-relieved 650,000

0.1

0 104

105 Cycles to failure

5.13 Fatigue test results for specimens tested at R = –1 using sequence type 3A (peak stress = ± 100 N/mm2).

the crack tip that would tend to ‘blanket’ the smaller subsidiary stress cycles. The same would not be expected to occur in as-welded joints; nor would it be expected to occur with Sequence 3A, where the subsidiary cycles followed a compressive peak stress rather than a tensile one. Turning now to the tests aimed at establishing the influence of the stress ratio of the subsidiary cycles, the three series, subjected respectively to loading Sequences 4A, 4B and 4C (Fig. 5.5), all involved stress relieved specimens. It is interesting to start by considering what the damaging part of the main and subsidiary cycles might be for each of these loading sequences, assuming that only part of the compressive half-cycle, equal to 40% of the corresponding tensile half-cycle, is damaging. For simplicity let us assume

142

Cumulative damage of welded joints 1.0

0.8

0.6 100 p

100p 0

–100p

–100 0.4

0.2 Constant amplitude life = 650,000

0 105

106 Cycles to failure

5.14 Fatigue test results for stress relieved specimens tested at R = –1 using sequence type 4A (peak stress = ± 100 N/mm2).

Table 5.2 Summary of results obtained using Sequences 3A and 4A with stress relieved joints Value of ∑(n/N)

Value of RG

Load sequence

Range of value

Mean

Range of values

Mean

3A 4A

0.64–0.89 0.76–1.27

0.77 0.95

0.94–1.18 1.09–1.57

1.05 1.30

Tests under two and three level loading

143

that the main cycle is nominally ± 100 N/mm2 (actually it was ± 75 N/mm2). This would be expected to give a ‘damaging range’ of 140 N/mm2 (i.e. 100 N/mm2 + 40% of 100 N/mm2), and that would be relevant to all three sequences. However the damaging ranges of the subsidiary cycles will be quite different. For Sequence 4A, in which the main and subsidiary cycles have the same mean stress, the nominal range of the subsidiary cycles will be ±100 p and the corresponding damaging range will be +100 p to –40 p, giving 140 p. Clearly, this will increase linearly from zero to 140 N/mm2 as p increases from 0 to 1.0, as the test results would be predicted by the area rule to follow the same form of linear relationship as has been shown to exist earlier in this chapter for as-welded specimens. As can be seen from Fig. 5.14 that was, to a close approximation, the case, although to some extent the area rule tended to underestimate the actual life. Turning now to Sequence 4B, we now have the situation that the main and subsidiary cycles have the same bottom limit stress rather than the same mean stress. In this situation, at small values of p, the subsidiary cycles will be wholly compressive and therefore do no damage. It is not until p exceeds 0.5 that the subsidiary cycles will acquire a tensile component and start to do damage. In other words, for p ≤ 0.5 the subsidiary cycles would be expected to do no damage and therefore for there to be no reduction in life. However, as p increases from 0.5 to 1.0 there will be a linear increase in the tensile component, and therefore also in the damaging effect of the subsidiary cycles. Ultimately, at p = 1.0, the subsidiary cycles will each be as damaging as the main cycle. Thus the test results would be expected to follow the relationship shown in Fig. 5.15, and it will be seen that the correlation between experiment and theory is excellent. For comparison Fig. 5.15 also shows the corresponding results for as-welded specimens. In that case, due to the presence of high tensile residual stresses, all cycles would be expected to be damaging so that the linear relationship would hold, which it does. So far as Sequence 4C is concerned, the main and subsidiary cycles have the same top limit stress. Consequently, for p ≤ 0.5 the whole of the subsidiary cycle will be tensile, and therefore damaging. For 0.5 ≤ p ≤ 0.7 part of the subsidiary cycle will be compressive, but since it will be less than 40% of the tensile half cycle it will still be damaging. However, for p > 0.7 the compressive half-cycle will be outside the 40% limit and therefore any proportion of the subsidiary cycle exceeding that will be inactive. In other words, the subsidiary cycles reach their most damaging state at p = 0.7, so that cycles with p > 0.7 would not be expected to be any more damaging than p = 0.7. Consequently, the expected relationship between p and endurance becomes as shown in Fig. 5.16. In this instance the correlation between the actual and theoretical results is not as good as with Sequence 4B, but the fit is not unreasonable, even if scattered.

144

Cumulative damage of welded joints

1.0 1/3 Nc

1N C 3

0.8 Aswelded Results predicted by the area rule

Value of p

0.6

Stress relieved 0.4 Results predicted by the area rule assuming that the whole of the minor stress cycles are damaging

0.2

Nc for as welded specimens = 24400

Rutile electrodes Low hydrogen electrodes 0 104

2

3

4 5

105 2 3 Endurance, blocks

Nc = 525 000 (from Fig. 5.12) 4 5

106

2

5.15 Test results for specimens subjected to loading using sequence 4B with peak stress = ± 75 N/mm2.

5.5

Summary of findings

It will be recalled that the work discussed in this chapter started with the confident prediction that all the loading sequences used (except Sequence 4C) would involve a beneficial stress interaction effect, since the tensile peak of the main constant amplitude cycle was greater than that of the subsidiary cycles. It was therefore anticipated that the subsidiary cycles would, to some extent, be ‘blanketed’ by compressive residual stresses introduced by the main tensile peaks. In other words it was anticipated that the values of ∑(n/N) would consistently be greater than 1.0. That, however, turned out to be false. On the other hand, even though it is not consistent with a simple fracture mechanics approach (which Miner’s rule is), the ‘area rule’ gave very good correlation with the experimentally determined lives. A summary of both sets of results is shown in the form of histograms in Fig. 5.17. This shows that only just over 10% of the results were ‘safe’ by Miner’s rule, the 50% probability level being at ∑(n/N) = 0.82. In contrast Fig. 5.17(b), from which the results obtained in tests on stress relieved specimens at R = –1 have been

Tests under two and three level loading

145

1.0 1/3 Nc

0.8

Value of p

0.6

0.4

75 150.p 0.2

0 –75

1 block Nc = 525000 (from Fig. 5)

0 104

2

3

4

5

105 Endurance, blocks

2

3

4

5

5.16 Results for stress relieved specimens made with rutile electrodes using loading sequence 4 C (peak stress = ± 75 N/mm2).

excluded, shows that the mean value of RG was 1.03, with 90% of the results lying within the range 0.79–1.32. This is very comparable to the scatter that one would expect to obtain in constant amplitude tests on the same type of specimen. Obviously, there are potential problems in trying to apply the area rule in situations involving stress relieved specimens subjected to loading cycles with a compressive component. Even so, it has to be said that, apart from loading Sequence 4C, the values of RG were well within acceptable limits,

146

Cumulative damage of welded joints 25

Number of test results

(a) 20 15 10

5

Number of test results

0

Value of ∑n/N (b)

15 10

5 0 0.4

Value of

0.8

1.2 1.6 Actual life Life predicted by the area rule

5.17 Histograms summarising the values of (a) in tests under 2 and 3 level loading.

Σn

N

(b) RG obtained

ranging from 0.79–1.57. For comparison Miner’s rule was much the same, values of ∑(n/N) being 0.62–1.49. With Sequence 4C, however, there was the occasional low value of RG, the range of values being from 0.43–1.16 while the corresponding values of ∑(n/N) were all less than 1.0, ranging from 0.32–0.89. In the light of these results it is tempting to suggest that the ‘area rule’ tends to give a more accurate prediction of life than Miner’s rule, but it has to be remembered that the tests considered in this chapter all involved very short loading ‘blocks’, typically in the range 2–5 cycles. There was therefore an obvious need to carry out further testing with longer block lengths in order to establish whether the same held true in that situation. The work is considered in the following chapters.

6 The influence of spectrum shape and block length

6.1

Introduction

In Chapter 5 it was demonstrated that, with very short block lengths, the area rule tended to give very much better life predictions than Miner’s rule. The problem which then arises is whether the same is also true with longer block lengths. Let us start by considering what life is predicted by Miner’s rule where the result is expressed in terms of the various proportional stresses, pi  where p = stress range under consideration  . Thus if the spectrum contains, i maximum range in spectrum   in each block, v1 cycles of Smax (where in this instance v1 = 1) and vi cycles of each of the lower proportional stresses pi, we get k  m  Σ (v p ) N k i i  v n =N 1 B ΣN + 2 (v i p im ) = N Σ B Nc Nc c 1    

[6.1]

where NB is the number of blocks to failure (each block containing v1 = 1 application of the top limit stress, so that p1 = 1.0 and v1 = 1 (see Fig. 6.1)) and Nc is the expected life to failure under constant amplitude loading with stress equal to the top limit stress range in the spectrum (Smax). Hence, for n = 1.0, the number of cycles to failure predicted by Miner’s rule becomes ΣN Nc ⋅ NL Predicted life = NL · NB = [6.2] Σ (v i p im ) where NL is the number of cycles per block. Meanwhile, the corresponding prediction derived from the area rule is (see eqn [5.20]). Predicted life = e–Area · Nc · NL

[6.3]

Given these two life predictions, which are clearly different, it is not difficult 147

148

Cumulative damage of welded joints

Stress

Smax

Si = pi × Smax

0 Time

Relative stress range, p

1 block (a)

1.0

0.5

0 1

Block length Number of exceedences (log scale), NE per block (b)

6.1 Schematic illustration of variable amplitude loading. (a) Definition of block length, (b) exceedence diagram.

to deduce the types of spectrum that would be expected to give the greatest differences between them. These were computed, for a variety of block lengths ranging from 16 to 512 cycles, and all were found to lie on a single, concave upwards, exceedence curve, as shown in Fig. 6.2. This shows, for example, that with a block length of 16 cycles the minimum range in the spectrum needs to be p = 0.27, for 64 cycles it is p = 0.15 and for 512 cycles it is p = 0.06. The important point to note about concave upwards spectra is that, compared with convex spectra, they have very few cycles with high stress ranges combined with a relatively large number of cycles with low stress ranges.

6.2

Fatigue tests under concave upwards spectra

Clearly, it was of interest to define the relevant experimental values of Σ n N

The influence of spectrum shape and block length

149

1.0 Number of stress levels in the spectrum 3 4 5 6 0.5 0.4

Relative stress, p

0.3

0.2

0.1

0.05 0.04

0.03 1

10

100 Number of exceedences

1000 2000

6.2 Details of ‘most damaging’ spectra defined by theoretical analysis (see Table 6.1).

and of RG for these ‘most damaging’ spectra and the results are summarised in Table 6.1. From this it will be seen that the values both of Σ n and of RG N were, except possibly for the results for the shortest block length, remarkably insensitive to block length. The mean value of Σ n for the whole set of tests N was 0.85 while the corresponding value of RG was 1.63. For these tests,

150

Cumulative damage of welded joints Table 6.1 Summary of test results for the ‘most damaging’ spectra Block length

Minimum stress range (p)

No. of levels

16 64 106 128 256 512

0.27 0.15 0.12 0.10 0.08 0.06

5 7 3 8 9 10

Σn

N ————————— Range Mean

RG ————————— Range Mean

0.65–0.88 0.83–0.94 0.78–0.85 0.77–0.95 0.83–1.04 0.85–0.91

1.13–1.53 1.61–1.79 1.41–1.55 1.55–1.89 1.69–2.11 1.75–1.89

0.76 0.90 0.80 0.86 0.93 0.88

1.31 1.74 1.45 1.73 1.91 1.82

therefore, Miner’s rule was consistently unsafe while the area rule overpredicted life by a factor ranging from 1.13 to 2.11. For comparison some tests were also carried out using four other concave upwards spectra with block lengths ranging from 8 to 512 cycles. These did not comply with the ‘most damaging’ criterion, and the overall mean values of Σ n and of RG were 0.89 and 1.27 respectively. In other words, the value N of Σ n was very similar to those found for the ‘most damaging’ spectra but N the value of RG was somewhat reduced. In summary, therefore, this part of the work on concave spectra with relatively short block lengths produced values of Σ n which were very similar to the values found previously N (Chapter 5) using ‘complex cycles’ consisting of a single constant amplitude cycle with a few associated supplementary cycles; in general the block lengths in those tests were mainly less than five cycles. In contrast, the values of RG were somewhat greater than in the previous tests. In other words, Miner’s rule was still giving unsafe life predictions but the area rule was consistently giving over-safe predictions. It is therefore of considerable interest to investigate whether this behaviour is merely a function of the fact that the spectrum shape was concave upwards or whether it is also a function of block length. Some enlightenment on this question can be derived from the results of tests carried out subsequently under convex and Laplace loading spectra.

6.3

Fatigue tests under Rayleigh and Laplace loading spectra

In view of the results obtained under concave loading spectra some tests were carried out to determine whether similar results are obtained under convex and linear (Laplace) loading. A commonly occurring load spectrum of the convex type is the Rayleigh distribution, which is what has been used in several investigations, and one of the larger of these was particularly concerned with the study of the influence of block length; some similar tests have involved the Laplace distribution.

The influence of spectrum shape and block length

151

For both of these types of distribution the probability of load exceedence P(∆s/σ) can be expressed as: 1

P(∆S/σ) = exp {– k ∆S/ σ ) k }

[6.4]

where ∆S is the stress range σ is the r.m.s of the process k = 0.5 for the Rayleigh distribution of 1.0 for Laplace. In other words, in a block of NL cycles, the number of cycles which exceed a stress ∆S is given by P(∆s/σ) × NL. σ When plotted as exceedence diagrams (see Fig. 6.3) the Rayleigh distribution is of the convex upwards type, while the Laplace distribution plots as a straight line. However, this Rayleigh distribution is equally well known as a probability density curve which, in effect, gives the frequency of occurrence of the various stress ranges in the loading spectrum. By way of example it has been shown by Schilling et al. (1975) in the USA that highway bridge loading tends to follow a Rayleigh distribution, but with a constant finite minimum stress representing the dead load stress (i.e. with no traffic) in the structure. As shown in Fig. 6.4 the frequency of occurrence of the various stress ranges can be defined by a histogram in which the height of each bar represents the percentage of stress ranges within some interval represented by the width of the bar. Such frequency of occurrence data can then be normalised by dividing the height of each bar by the width of the bar to obtain a probability density curve, as illustrated at the bottom of Fig. 6.4. 1.0

Relative stress

0.8 Rayleigh, k = 0.5

0.6

0.4 Laplace, k =1.0 0.2

0 1

10 100 Number of exceedences

1000

6.3 Comparison of exceedence diagrams for Rayleigh and Laplace spectra for block length = 1000 cycles.

152

Cumulative damage of welded joints 14 13 12

Idealised vehicle loading S max

11 9

Sr

Stress, ksi

10 8 7 6

S min.

4 3 2 1 0

Deod lood

5

20 16 12 8 4 0 Occurrence, %

Time

Probability density, l/ksi

0.25 0.20 0.15 0.10

0.05 Conversion factor: 0 1 ksi = 6.895 MPa 0

1

2

3 4 5 6 7 8 Stress range, ksi

9 10

6.4 Method of deriving Rayleigh load distributions for tests relating to highway bridges in the USA.

Thus, the proportion of the area under the probability density curve between any two values of stress range (Sr) represents the fraction of stress ranges within that interval, and since the whole curve represents the complete spectrum the total area under the curve is equal to 1.0. Now, reverting to eqn [6.4], if the block length is restricted to some value, NL, the maximum peak stress range in the spectrum, ∆Smax, which occurs only once in each block, will be related to NL. Thus, writing β = ∆Smax/σ (where β is the clipping ratio) we get from eqn [6.4]: 1

P ( ∆S max / σ ) = 1/ N L = exp {– kβ k }

[6.5]

so that: 1

ln (N L ) = kβ k

[6.6]

Based on eqn [6.4], Table 6.2 shows the probability of exceedence for the Rayleigh distribution (k = 0.5) for a range of values of ∆S up to a value of σ

The influence of spectrum shape and block length

153

Table 6.2 Basic Rayleigh type distribution and derivation of corresponding stress spectrum Stress ∆S σ

Probable exceedence

0

1.0

0.2

0.9802

0.4

0.92312

0.6

0.83527

0.8

0.72615

1.0

0.60653

1.2

0.48675

1.4

0.37531

1.6

0.27804

1.8

0.19790

2.0

0.13534

2.25

0.07956

2.5

0.04394

2.75

0.02279

3.0

0.01111

3.25

0.00509

3.5

0.00219

3.75

0.00088

4.0

0.00034

4.25

0.00012

4.5

0.000040

4.75

0.000013

5.0

0.000004

5.25

0.000001

No. of peaks

At average stress

0.0198

0.1

0.05708

0.3

0.08785

0.5

0.10912

0.7

0.11962

0.9

0.11978

1.1

0.11144

1.3

0.09727

1.5

0.08014

1.7

0.06256

1.9

0.05578

2.125

0.03562

2.375

0.02114

2.625

0.01169

2.875

0.00602

3.125

0.00290

3.375

0.00131

3.625

0.00054

3.875

0.00022

4.125

0.000080

4.375

0.000027

4.625

0.000009

4.875

0.000003

5.125

2    exp  – 1  ∆S     2 σ  

From equation [6.5] the required values of peak stress for various block lengths are: Block length 9 19 100 103 104 105 524 284 ∆S max 2.10 2.34 3.03 3.72 4.29 4.80 5.13 σ

154

Cumulative damage of welded joints

5.25. It also shows how corresponding approximate block spectra can be derived from those values. This can be done by subtracting successive values of P ∆S from each other to find the number of peaks occurring at the midpoint σ stress level. Thus, at a stress level equal to 1/2  P ∆S + P ∆S  the σ σ i i+1  corresponding number of peaks is  P ∆S – P ∆S  × NL where NL is σ i+1 σ i  the number of cycles in the block. As shown in eqn [6.6] NL is related to the peak stress in the spectrum. Hence, knowing the block length, the peak stress is defined by eqn [6.6] and the number of cycles at each lower stress level can be calculated by proportionality so as to give one cycle at the peak stress level. For example, referring to Table 6.2, if the required block length is 100 cycles (for which eqn [6.6] shows that the corresponding peak stress is ∆S = 3.03) then the σ effective top limit stress is at 2.875 and the corresponding number of cycles is 0.01169 × 100 = 1.169 ≈ 1. At the next lowest stress level (2.625) the number of cycles is 0.2114 × 100 ≈ 2 etc. The experimental programme was, as noted previously, primarily intended to demonstrate the influence of block length, so tests were carried out with each of the block lengths shown in Table 6.3. However, it follows from eqn [6.6] that block length and clipping ratio are related to each other, so that by changing the block length the clipping ratio was also changed, as also shown in Table 6.3. Furthermore, as will be apparent from the explanation above, a change of block length also necessarily changed the number of stress levels in the spectra. With each block length, fatigue test results were obtained for three levels of peak stress range, namely 275, 225 and 175 N/mm2. In each instance the individual cycles in the block were applied in random order, all at R = 0. Before going on to consider the influence of block length it is interesting to note that it has been suggested by various investigators (Schilling et al. (1975); Tilly and Nunn (1980); White and Lewszuck (1971)) that either the r.m.s (of peaks) stress or the cube root mean cube (of peaks) stress, defined

( )

( ) ( )

( )

Table 6.3 Relationship between block length and clipping ratio for the spectra used in the investigation Block length, cycles

9

19

100

103

104

105

524 284

Clipping Rayleigh ratios

2.10

2.43

3.03 6.03

3.72 6.93

4.29 9.20

4.80

6.13

The influence of spectrum shape and block length

155

 Σ s 2i n i   Σ s 3i n i  respectively as  and    , are suitable parameters to  Σ ni   Σ ni  define the relevant ‘stress’ in variable amplitude test results. (In some instances the r.m.s of the ‘process’ rather than of the ‘peaks’ has been used, but they are related by the simple factor 2 .) In order to check this suggestion the results, expressed in terms of the r.m.s (of peaks) stress are shown in Figs 6.5 and 6.6. In these figures the plots of the test results are identical, but Fig. 6.5 shows their relationship in terms of the block length of the spectrum and Fig. 6.6 the corresponding relationship with the peak stress in the spectrum. Figures 6.7 and 6.8 show the corresponding relationships when the results are expressed in terms of the cube root mean cube stress. In each figure the relevant constant amplitude fatigue lives are also shown. It is obvious that fatigue life is a function both of the block length and of the peak stress and that neither the r.m.s nor the cube root mean cube stress represents a unique parameter for defining fatigue behaviour. As a further check on this finding, a few tests were carried out with a different form of loading spectrum, namely the Laplace distribution. The results are again plotted in Figs 6.5–6.8, but it is obvious that they are not consistent with the results obtained using Rayleigh distributions. This further confirms that life is clearly a function both of the stress level and of spectrum shape, and the r.m.s and r.m.c. stresses are not independent of those parameters; in other words there is little benefit to be gained by analysing results on the basis of r.m.s. or r.m.c. stresses. Returning now to the specific problem of the influence of block length and of the applicability of Miner’s rule and of the area rule, in situations involving Rayleigh type stress spectra, the results obtained are summarised in Figs 6.9 and 6.10. Since only one or two specimens were tested under each combination of block length and peak stress it is not surprising to note a considerable degree of scatter, but nevertheless the general trend of the results is quite clear. Figure 6.9 shows a distinct tendency for the value of n to increase with block length and also to increase with the magnitude ΣN of the peak stress. The replot of the results on the basis of the life predicted by the area rule (see Fig. 6.10) is interesting in that it shows a completely different pattern of behaviour from that shown in Fig. 6.9. For each of the three peak stresses the value of the ratio of actual life/predicted life rises to a maximum at a block length of about 102–103 cycles and then falls again at larger block lengths, with a value in the region of 1.0 at a block length of about 0.5 × 106. As was the case with Σ n , for a given block length, the higher the applied peak N stress the higher was the value of the ratio. Unfortunately, there seems to be remarkably little other similar experimental 1/2

1/3

r.m.s. (of peaks) stress, N/mm 2

280

70 104

80

90

100

120

140

160

180

200

220

240

260

2

3 4 5

105 1.5 Endurance, cycles

9 cycles

2 3 4

101 cycles

19 cycles

5

50 106

60

1.5

2

104 cycles

103 cycles

3

4

5

9 19 101 103 104 105 524284

107

6.5 Test results for Rayleigh type spectra expressed in terms of r.m.s. stress, showing the effect of block length.

1.5

Constant amplitude S-N curve (from Fig. 6.3)

Block length

r.m.s. (of peaks) stress, N/mm2

280

70 104

80

90

100

120

140

160

180

200

220

240

260

2

3 4

5 105 1.5 Endurance, cycles 2 3 4 5

50

60

106

1.5

2

Rayleigh peak = 275

3

4

5

Rayleigh peak = 225

Peak stress, N/mm2 Rayleigh Laplace 275 225 175

107

6.6 Test results for Rayleigh and Laplace type spectra, expressed in terms of r.m.s. stress, showing the influence of peak stress.

1.5

Rayleigh peak = 175

Constant amplitude loading (lives read from S-N curve (shown in Fig. 6.3)

Cube root mean cube stress, N/mm2

70 104

80

90

100

120

140

160

180

200

220

240

260

280

2

3 4 5

105 1.5 Endurance, cycles 2 3 4 5

50 106

60

103 cycles

101 cycles

1.5

2

104 cycles

3

4

5

107

6.7 Test results for Rayleigh type spectra, expressed in terms of cube root mean cube stress, showing effect of block length.

1.5

9 cycles

19 cycles

9 19 101 103 104 105 524284

Block length

Cube root mean cube stress, N/mm2

280

70 104

80

90

100

120

140

160

180

200

220

240

260

2

3 4

5 105 1.5 Endurance, cycles 2 3 4 5

50 106

60

1.5

2

3

4

5

Peak stress, N/mm2 Rayleigh Laplace 275 225 175

107

6.8 Test results for Rayleigh and Laplace type spectra expressed in terms of cube root mean cube stress, showing effect of peak stress.

1.5

Constant amplitude loading (lives read from S-N curve (shown in Fig. 6.3)

160

Cumulative damage of welded joints

Peak stress, N/mm2 275

3.0

225

2.8

175 2.6

2.4 2.2

Σ

n N

2.0 225 1.8 275 1.6

175

1.4

1.2

1.0

0.8 1

10

102 103 104 105 Block length, cycles

6.9 Influence of block length on the value of

106

Σ n obtained with

N Rayleigh type spectra with all stress cycles at R = 0.

evidence against which to compare these results. The only investigation in which tests with more than one block length have been carried out seems to be one which again involved specimens with longitudinal edge attachments subjected to Rayleigh distributions of stress range with block lengths of 1024 (=210) and 131 072 (=217) cycles. In both instances there was again a tendency for the value of Σ n to increase with increasing peak stress (as in N Fig. 6.9), and at Smax = 225 N/mm2 the values of Σ n were about 1.3 for a N block length of 1024 cycles and 3.1 for 131 072 cycles. While the former value is very similar to that indicated by Fig. 6.9, the latter result is very

The influence of spectrum shape and block length

161

3.2

3.0

Peak stress, N/mm2 275 225

2.8

175

2.6

Actual life Life predicted by eqn [6.2]

2.4

2.2 275 2.0 225 1.8 175

1.6

1.4

1.2

1.0

0.8 1

10

103 104 105 102 Block length, cycles

106

6.10 Influence of block length on the ratio of actual life to life predicted by the area rule for specimens subjected to Rayleigh type stress spectra.

much higher. The corresponding values of RG were about 2.0 and 1.65 respectively which is fairly similar to the results shown in Fig. 6.10. However, the main interest of this particular part of the investigation is that it also included comparative tests using the same spectra with all cycles applied (in the same order as in the tests at R = 0) at R = –1, and this appears to have been the only investigation in which that has been done for a Rayleigh distribution of stress ranges. Unfortunately, it proved necessary to use two different batches of specimens, and unusually for this type of specimen, the constant amplitude S-N curves for the two batches were significantly different. For both batches the S-N curves were well defined, with little scatter, and the

162

Cumulative damage of welded joints

Stress range, N/mm2

300 280 260 240 220 200 180 160 1st batch R = 0 R = – 1

140 120

2nd batch R = – 1 R = 0

100 90 80 70 60 1 0 4 1.5 2

3

4 5

1 0 5 1.5 2 3 4 5 1 0 6 1.5 2 Endurance, cycles

3

4 5

107

6.11 Comparison of constant amplitude S-N curves for the two batches of specimens.

tests at R = –1 gave steeper S-N curves than at R = 0. A summary of the mean S-N curves is shown in Fig. 6.11. At first sight, on the basis of these results, one might expect the variable amplitude tests at R = –1 also to give longer lives than at R = 0. In fact the opposite was the case (Fig. 6.12) particularly for the longer block length. It seems probable that the reason for this behaviour lies in the influence of residual stresses. For the same peak stress range the actual peak stress at R = 0 will be double that at R = –1 and will therefore create higher residual compressive stresses around the weld ends, thereby tending to blanket these more effectively against the lower stresses in the spectrum. The implication of this finding is that, under variable amplitude loading, one should expect alternating loading to give lower lives than tensile loading under identical spectra (expressed in terms of stress range). The same general trend is apparent when the results are expressed in terms of Σ n (Fig. 6.13), with the tests at R = –1 giving lower values of N n Σ N than tests under tensile loading. The results in Fig. 6.13 are shown in two forms, firstly with each value of Σ n based upon the relevant constant N amplitude S-N curve (specimen batch and stress ratio), and secondly with all values based upon the S-N curve for R = 0, regardless of the actual stress ratio. To a large extent this simulates the situation in most national design codes, where fatigue strengths are assessed in terms of stress range, regardless

The influence of spectrum shape and block length

163

Top limit range in spectrum, N/mm2

500 400 R=0 300 R=–1 200

150

100

R=0 R = –1 105

106

107

10

Cycles

500 R=0

400 300

N/mm2

R=–1 200

150

100

R=0 R = –1 105

106

107 Cycles

6.12 Variable amplitude test results for specimens subjected to a Rayleigh spectrum with stresses applied in random order. (a) Block length = 1024 cycles (b) 131 072 cycles.

10

164

500

Cumulative damage of welded joints

Block length = 1024 cycles

Block length = 131 072 cycles

Max stress range N/mm2

400

300

200

150

100 R=0 R = –1 1.0

1.0

10

6.13 Comparison of values of Σ n tests at R = 0 and R = –1. (a) Block N length = 1024 cycles (b) 131 072 cycles.

of stress ratio. Qualitatively, this change does not affect the general form of the results. It can be seen that there was a general tendency for the value of n to decrease as the stress decreased. For the shorter block length the ΣN value of Σ n ranged from about 1.1–1.6 at R = 0 and 0.6–1.2 at R = –1 N (based on the actual relevant S-N curve for the specimen in question). For the longer block length the corresponding values were about 2.5–4.6 at R = 0 and 0.8–1.4 at R = –1. Incidentally, some corresponding tests using these two spectra were also carried out with them arranged in narrow and wide band forms; those results are considered in more detail in Chapter 7. However, it is interesting to note in passing that the narrow band loading again gave significantly longer lives at R = –1 than at R = 0 but wide band loading, involving a wide range of stress ratios, gave longer lives at R = 0 than at R = –1. As discussed previously, although the tests in the main investigation were all carried out with the same peak stresses, the spectra had different numbers of stress levels depending upon the block length; the clipping ratio, r.m.s. and r.m.c. stresses also varied with block length. This can be seen in Table 6.4 where the main characteristics of the various spectra are summarised. In view of this situation some additional tests were carried out using modified Rayleigh type stress sequences. These consisted of the same sequences

Block length, cycles

0.199 0.144 0.109

* Assuming slope of S-N curve m = 3.00

0.281 0.204 0.154

Laplace distributions LAP 2 99 LAP 3 1000 LAP 4 104 5.03 6.93 9.20

0.241 0.205 0.196 0.196 0.195

r.m.s. of process

Rayleigh distributions with one additional peak per block RAY 19A 20 4.14 0.341 RAY 2A 102 4.87 0.29 RAY 3A 1001 5.11 0.277 RAY 4A 10001 5.10 0.277 RAY 5A 100 001 5.12 0.276

r.m.s. of peaks

0.328 0.268 0.233 0.209 0.196

3.05 3.73 4.28 4.79 5.11

Clipping ratio

0.463 0.377 0.330 0.295 0.277

Rayleigh distributions RAY 2 101 RAY 3 1000 RAY 4 104 RAY 5 105 RAY 524 524 284

Spectrum

Table 6.4 Characteristics of stress spectra

0.346 0.260 0.197

0.416 0.332 0.305 0.305 0.303

0.507 0.417 0.363 0.325 0.304

r.m.c. of peaks

24.23 56.95 130.46

13.86 27.32 35.07 35.15 35.78

7.66 13.81 20.92 29.22 35.55

By Miner*

Predicted life Nc

12.07 32.93 102.15

7.35 15.63 35.39 55.97 74.57

4.76 10.89 22.08 47.04 82.63

By area rule

166

Cumulative damage of welded joints

Table 6.5 Modified Rayleigh type stress distributions with an additional high stress cycle in each block p

0.020 0.059 0.098 0.137 0.176 0.215 0.254 0.293 0.332 0.371 0.415 0.463 0.512 0.561 0.610 0.659 0.707 0.756 0.805 0.854 0.902 0.951 1.0

Block length 102

103 + 1

104 + 1

2 6 9 11 12 12 11 10 8 6 6 4 2 1 1 – – – – – – – 1

20 57 88 109 120 120 111 97 80 63 56 36 21 12 6 3 1 – – – – – 1

198 571 879 1091 1196 1198 1115 973 801 626 558 356 211 117 60 29 13 5 2 1 – – 1

105 + 1 1980 5708 8785 10 912 11 962 11 978 11 144 9727 8014 6256 5578 3562 2114 1169 602 290 131 54 22 8 3 1 1

524 284 10 381 29 926 46 059 57 210 62 715 62 799 58 427 50 997 42 016 32 799 29 245 18 675 11 083 6129 3156 1520 687 283 115 42 14 5 1

as used earlier but with a single additional (higher) peak stress. In this way the lower stress levels in the various spectra were reduced to the same levels as those in the 524 284 cycle block and the single peak stress was introduced at the same peak stress level as in that block. Some of the levels immediately below the peak were, however, missing, as can be seen from the summary of the modified spectra shown in Table 6.5. (Effectively this Table also gives details of the basic spectra if one ignores the additional peak stress.) As can be seen from Table 6.4, for the modified spectra with the additional higher peak stress the clipping ratios, r.m.s. and r.m.c. stresses were each virtually constant, certainly for block lengths ranging from 1000 up to 524 284 cycles. For shorter block lengths the clipping ratios were reduced and the r.m.s. and r.m.c. stresses were increased. The test results obtained with these spectra are summarised in Figs 6.14 and 6.15. As can be seen from these figures the spectra with block lengths between 1000 and 524 284 cycles gave very similar lives and values of Σ n N while, for the same spectra, the area rule (Fig. 6.15) gave progressively reducing values of RG. It was noted previously that, for the basic spectra with

The influence of spectrum shape and block length

167

3.2 3.0 2.8 2.6 2.4

No additional peak stress

2.2

Σ

n N

2.0 1.8 1.6 1.4 1.2 1.0 0.8 1

10

6.14 Values of

102 103 104 105 Block length, cycles

106

Σ n for specimens tested at a peak stress of 275 N/

N mm2 using Rayleigh type loading with an additional peak stress per block.

no overload cycle, there was good correlation between the results expressed in terms of r.m.s. and r.m.c. stresses and either block length (Figs 6.5 and 6.7) or the peak stress in the spectra (Figs 6.6 and 6.8). The new results, however, do not correlate very well with those relationships. In other words, as with the Laplace spectra, a change in spectrum ‘shape’ destroys the relationship. Nevertheless, it is interesting to compare the lives obtained under Rayleigh distributions with and without the additional overload cycle, and such a comparison is shown in Table 6.6. The most obvious feature is that, for each block length except the shortest (19/20 cycles), there was a large increase in life and in the value of Σ n N resulting from the inclusion of the overload cycle. On the other hand the values of RG for the two types of spectra were fairly similar to each other. These results are interesting for the fact that they tend to mirror the behaviour

168

Cumulative damage of welded joints 3.2 3.0 2.8 2.6

Actual life Life predicted by eqn [6.2]

2.4 2.2 2.0 1.8 1.6 1.4 1.2 No additional peak stress

1.0 0.8 1

10

102 103 104 105 Block length, cycles

106

6.15 Influence of block length on the ratio of actual life to life predicted by the area rule for specimens subjected to Rayleigh type loading with an additional peak stress per block.

of joints under constant amplitude loading, where it is well known that intermittent overloading tends to lead to an increase in life; some typical results for two different levels of overload, are shown in Fig. 6.16. It appears that this may also be true under variable amplitude loading and that the application of intermittent overloads may be a relatively simple way of increasing the life of structures subjected to variable amplitude loading in service. This does, however, appear to be the only available experimental evidence relating to the technique and it would probably be wise to carry out some check tests before actually applying deliberate overloading in service. It was noted previously that, in this particular investigation, a few tests were also carried out under Laplace type spectra. It is interesting to note (see Fig. 6.17) that the values of Σ n obtained appear to behave in a very similar N manner to those for the Rayleigh type spectra; in other words they tend to increase with block length. The results for the two types of spectra are not,

The influence of spectrum shape and block length

169

Table 6.6 Comparison of results for Rayleigh spectra with and without an additional peak stress (Smax = 275 N/mm2) Block length

Σn

Endurance (×103) Normal spectrum

With additional peak stress

RG

N Normal With spectrum additional peak stress

Normal spectrum

With additional peak stress

19/20

138 168

386

1.09 1.33

1.01

1.79 2.18

1.95

100

267

1134 1296

1.29

1.51 1.72

2.08

2.69 3.07

103

605

1904 2799

1.62

1.96 2.88

2.06

1.99 2.93

104

1176

2715 2978

2.08

2.78 3.05

1.97

1.80 1.97

105

2132

2808 3198

2.70

2.83 3.22

1.68

1.39 1.59

524284

2496

2.53

1.12

300 280 260 240 220

Overload stress, N/mm2 275 175

200

Stress, N/mm2

180 160 140 120 Constant amplitude S-N curve for R = 0 (from Fig. 6.3)

100 90 80 70 60 104 1.5 2

3

4

5

105 1.5 2 3 4 5 Endurance, cycles

106 1.5 2

3

4

5

107

6.16 Results for specimens tested under constant amplitude loading at R = 0 after a single overload.

however, identical; in other words, the relationship between Σ n and block N length depends on the type of spectrum. In summary, therefore, the results outlined above, obtained at different times in two separate investigations appear to confirm, at least for Rayleigh and Laplace spectra that:

170

Cumulative damage of welded joints 3.2 Peak stress, N/mm2

3.0

275 225 175

2.8 2.6 2.4 2.2

Σ

n 2.0 N 1.8

1.6 1.4 1.2 1.0 0.8 1

10

102 103 Block length, cycles

104

6.17 Influence of block length on the value of subjected to Laplace type spectra at R = 0.

Σ n for specimens N

1. Neither the r.m.s. stress nor the r.m.c. stress is capable, by itself, of acting as a unique stress parameter by which to represent the results. Life was found also to be dependent on the peak stress and the form of the stress distribution. It is therefore dubious whether there is anything to be gained by the use of these parameters. 2. The values of Σ n showed a clear tendency to increase with block length N and also with the peak stress (see Fig. 6.9). For block lengths of less than about 100 cycles combined with peak stresses less than 225 N/mm2, Σ n N tended to be less than 1.0. 3. The area rule, represented by eqn [6.3], gave safe predictions of life at short and intermediate block lengths but tended to become unsafe at long

The influence of spectrum shape and block length

171

block lengths combined with low stresses. As with Miner’s rule, this rule also gave increasingly safe answers as the peak stress increased (see Fig. 6.10). 4. Tests using a Rayleigh stress distribution with an additional single higher peak stress per block gave values of Σ n which were even more dependent N on block length (Fig. 6.14). Similarly, the area rule gave safer predictions than for the unmodified Rayleigh spectra, particularly for block lengths in the region of 102–103 cycles (Fig. 6.15).

6.4

Tests under Weibull stress spectra

In the light of the results discussed above, for Rayleigh, modified Rayleigh and Laplace distributions of stress, it is clear that the value of the Miner summation is very dependent on spectrum shape. It is therefore a subject which merits further investigation. In order to do this an extensive series of tests was carried out on the same type of specimen as was used in the tests involving Rayleigh and Laplace distributions, namely a plate with welded edge attachments (see Fig. 5.10), but using loading defined by the twoparameter Weibull distribution. This is usually written in the form:

( ) 

 P (S) = exp  – S  θ

a

[6.7]

where P(S) is the probability that a peak exceeds stress, S, a is the Weibull slope or modulus (shape parameter), θ is the characteristic value or scale parameter. It has been suggested by Pook (1984) that this can more conveniently be written as:

( )

( ) 

 P S = exp  – b S σ  a θ

a

[6.8]

where σ is the r.m.s. of the process, assumed to be sinusoidal b is then a function of a. Pook also showed that, to a close approximation, over the range 0.71 ≤ a ≤ 2.36, b = 1.0 – 0.076(a2 –3a + 2)

[6.9]

It will be noted that, with a = 2.0 or 1.0, we get b = 1.0 so that, as special cases, eqn [6.8] gives the Rayleigh and Laplace distributions respectively. For other values of the parameter a, Pook has tabulated the corresponding

172

Cumulative damage of welded joints

values of b. Over the range 0.5 ≤ a ≤ 0.71 it appears that b can be adequately represented by b = 0.833 + 0.1925a. For consistency with eqn [6.4] it is convenient to write k = 1/a and to express the stress in terms of the stress range, ∆S. Then, from eqn [6.8], the probability of load exceedence becomes:

( )

( )

 P ∆S = exp  – kb ∆S σ σ 

1/k

  

[6.10]

so that eqns [6.5] and [6.6] now become: ∆S max  P = 1 = exp {– kbβ 1/k}  σ  NL and Ln (N L ) = kbβ 1/k

[6.11]

Clearly, therefore, the family of spectra represented by eqn [6.10] can easily be varied both in terms of shape (by changing the value of k) and clipping ratio. It therefore provides a convenient basis for a test programme to examine the influence of those two variables. For convenience, the influence of the value of k on spectrum shape is shown, for a range of values of k, in Fig. 6.18. In the earlier tests, considered previously, a start was made on investigating the influence of spectrum shape by carrying out tests with two different values of k (0.5 and 1.0) while varying the block length and clipping ratio. In the follow-up tests, however, an attempt was made to carry out tests at several pre-defined block lengths, and, predominantly at six different values of clipping ratio (approximately 2.3, 3.0, 3.5, 40, 4.8 and 6.0). The experimental plan is outlined in Fig. 6.19. As was the case in the tests using Rayleigh and Laplace spectra, the stresses within each block were applied at R = 0 in a pseudo-random sequence and the block was then repeated (in the same sequence) until specimen failure, characterised by complete rupture from one of the weld ends. For each combination of k/clipping ratio/block length tests were carried out at two values of maximum stress, namely 275 and 175 N/mm2, with the objective of defining the number of cycles to failure and hence the values of Σ n and N n were calculated assuming that all stress levels RG. All the values of Σ N were damaging and that there was no cut-off or bend in the constant amplitude S-N curve. An overall summary of the results showing the values of Σ n and of the N ratio (RG) of the actual life to that predicted by the area rule is shown in Figs 6.20 and 6.21. In each instance the results are given separately for the tests carried out at peak stresses of 275 and 175 N/mm2. From Fig. 6.20 it can be

The influence of spectrum shape and block length

173

1.0

Relative stress

0.8 k = 0.65

0.6

k = 1.19

0.4

0.2

k = 2.36

0 2

3 4 5 10 15 Number of exceedences (a)

20 25 32

1.0

Relative stress

0.8

0.6

k= 0.7 8

k = 0.46

0.4 k = 1.0 0.2 0 1

10 100 Number of exceedences (b)

1000

6.18 Some typical stress spectra used in the experimental programme, showing the influence of the value of k. (a) Block length = 32 cycles, (b) block length = 1000 cycles.

seen that there is a tendency for Σ n to increase as the loading changes N from a high value of k with low clipping ratio to a lower value of k with higher clipping ratio (i.e. as one proceeds towards the bottom right-hand corner of the diagram or, alternatively, as block length increases). On the other hand, as far as the value of RG is concerned, the influence of k and clipping ratio is far less pronounced (Fig. 6.21). Perhaps the most obvious feature of the results was the tendency to obtain a higher value of Σ n when, with the same spectrum, the peak stress was N 275 rather than 175 N/mm2. The value of the ratio between the two values of n is summarised in Fig. 6.22, from which no obvious influence of spectrum ΣN

Cumulative damage of welded joints

N

L

1.8

32

2.0

r

2.2

0.12 0.10 0.08 0.06 0.04 0.02

NL r 24

2.4

Minimum stress range in spectrum

NL r 15

174

1.6

Value of k

CR = 6.95 1.4

12

8

r

1.2

NL

1.0

NL

0.8

NL r

0.6 0.4

NL

r

32

1625

NL r

500

r

100

0

0

13050

0

0.2 0 2.0

3.0

4.0 Clipping ratio

5.0

6.0

6.19 Summary of planned tests, showing relationship between the value of K, clipping ratio at block length (N2).

shape is apparent. It therefore seems reasonable to regard the variation in the value of the ratio as being due to experimental scatter rather than of any consistent influence of the type of loading. A summary of the values obtained is shown in the form of a histogram in Fig. 6.23, the mean value being 1.18. Clearly, exactly the same ratio applies to the corresponding values of RG. For interest it may be noted that the extreme values of the ratio which were recorded could result from the two relevant endurances being above and below the mean by only about 20%; obviously this is not significant in the context of fatigue test results. In order to attempt to define the relative importance of each of the three variables (spectrum shape, clipping ratio and block length) let us consider the results in terms of each of the three pairs of those parameters.

Value of k

1.08 0.97

1.16 1.42 1.26

1.18

1.32

1.14

1.11

1.17

0 2.0

0.6

5.0

4.72 2.03

1.71

1.26

1.15

1.42 1.12

1.20

6.0

2.09 1.64 4.70 2.78

1.58 1.79 1.22

1.12

1.10

1.21

1.15

0.98

1.20

0.92

1.06 1.05

1.11

1.13

0 2.0

3.0

1.47 1.59 1.49

1.42

1.25

1.11

0.81

0.95

4.0 Clipping ratio (b)

1.26 1.26 0.4 1.42 1.11 1.69 1.12 1.36 1.29 1.17 0.2 1.26 1.06

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

N

5.0

1.68 3.07 1.54

1.63

1.02

1.13

1.09

6.0

1.54 1.65 1.26 1.85 2.40

1.06

0.98

1.36 1.00

0.87

Σ n obtained in tests with peak stress of (a) 275 N/mm2, (b) 175 N/mm2.

4.0 Clipping ratio (a)

6.20 Values of

3.0

1.61 1.52 1.84 1.62 1.22 1.43 1.33 2.21 0.4 2.65 1.22 2.51 1.28 1.21 1.74 1.27 1.51 0.2 1.25

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

Value of k

0.99

Value of k

1.89

0 2.0

2.59 2.31

2.11

2.35

2.15

2.08

2.08

5.0

4.21 1.81

2.53

2.36

2.84 2.23

2.08

6.0

2.91 2.47 1.90 1.92 1.51 2.78 1.64

2.18

2.10

2.20

2.24

2.35

2.07

1.61 2.02

2.03

2.21

2.09

1.52

1.69

0 2.0

3.0

4.0 Clipping ratio (b)

1.79 1.77 1.92 1.51 1.95 1.63 1.25 1.87 0.4 1.70 1.48 0.94 1.56 0.99 1.03 0.89 0.97 0.2 1.42 1.06 0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

6.21 Values of RG for tests with peak stress of (a) 275 N/mm2 and (b) 175 N/mm2.

4.0 Clipping ratio (a)

2.72 2.56 2.49 2.12 2.21 1.72 1.89 1.69 1.58 1.60 1.33

1.86 1.67

2.03

3.0

1.87 1.69 0.2 1.68 1.17

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

Value of k

2.06

1.87

5.0

1.50 1.81 0.91

2.41

1.92

2.59

1.60

6.0

2.40 2.68 1.96 1.68 1.42

2.26

1.89

2.03

The influence of spectrum shape and block length

1.23

177

1.39

2.4

1.10

1.12

2.2 2.0 0.85 1.8 1.6

Value of k

1.04

1.37

1.16

1.4 1.2

1.12

1.03 1.06

1.0

1.26 1.24 1.14 1.08

0.8

1.06 0.81 0.90

1.52 1.43 1.30 1.24 1.13 1.06 1.50 1.29 1.72 0.4 1.10 1.38 1.14 1.09 1.49 1.11 0.2 1.18 0.6

0 2.0

3.0

4.0 Clipping ratio

1.05

1.00 1.21

1.21

1.16

5.0

6.0

Σ n for tests with Smax = 275 N/mm 2 6.22 Values of the ratio

6.5

Σ

N . n for tests with S 2 max = 175 N/mm N

Influence of spectrum shape and clipping ratio combined

When the results relating to the various combinations of spectrum shape (as defined by the value of k) and clipping ratio, both for Σ n and for RG, were N plotted, the most obvious feature was that, for values of k greater than 1.0 (i.e. for spectra with p v. ln NE exceedence diagrams which are concave upwards) the values both of Σ n and of RG were approximately constant. In N the case of Σ n the value when the peak stress range was 275 N/mm2 was N slightly greater than 1.0 (typically 1.07) while for a peak stress range of

178

Cumulative damage of welded joints

Number of occurrences

Note. one result at ratio = 1.72 Arithmetic mean = 1.18 6

4

2

0 0.8

0.9

1.0

Value of the ratio

1.1

1.2

1.3

1.4

Σn

with peak stress = 275 N/mm 2

Σn

with peak stress = 275 N/mm 2

N N

1.5

6.23 Histogram showing the influence of peak stress on the value of Σn. N

175 N/mm2 it tended to be slightly less than 1.0. Meanwhile the value of RG was approximately 2.0 for a peak range of 275 N/mm2 and typically 1.75 (although with more scatter) for a peak range of 175 N/mm2. For values of k less than 1.0 (i.e. for spectra which are convex upwards) the situation is more confused. For those spectra the values of Σ n were N much more scattered and lay typically between 1.0 and 2.0 with a few results less than 1.0. On the other hand there seemed to be a tendency for RG to increase from 1.0 at k in the region 0.25–0.3 to a peak in the region k = 0.7– 0.8 and for the value then to decrease to the ‘plateau value’ at k = 1.0. No results are available for values of k between 0 and 0.23, but at k = 0 we must again have RG = 1.0 since that corresponds to constant amplitude loading. The general form of the relationship suggests that RG might be less than 1.0 at small values of k. Unfortunately, there are insufficient results to place too much reliance on these apparent trends but one way in which the database may be increased is to consider the tests at the two different stress levels (275 and 175 N/mm2) together. Since, as noted previously, there appears to be no influence of spectrum shape on the ratio between the values of Σ n or RG, for the two values of N peak stress, the results can be made approximately self-consistent by multiplying by 1.18 (the mean value of the ratio, see Fig. 6.23) the values of n and of R obtained with a peak stress of 175 N/mm2. The amalgamated ΣN G results, showing the influence of the value of k, are summarised in Figs 6.24 and 6.25.

The influence of spectrum shape and block length *CR = 9.20

2.2 β = 4.8

β = 5.4–7.0

2.0

∑n/N

1.8 β = 4.0

Note: Symbols marked / indicate results from Ref. 4 shown with approximate clipping ratio.

179

Clipping ratio 2.3 3.0 3.5 4.0 4.8 5.4 6.0 7.0

1.6 β = 3.0 1.4 1.2

β = 2.3

1.0 0.8 0

Constant amplitude loading 0.2

0.4

0.6

0.8

1.0

1.2 1.4 Value of k

6.24 Influence of spectrum shape on

1.6

1.8

2.0

2.2

2.4

2.6

Σ n . (Results for all specimens

N normalised to peak stress range of 275 N/mm2).

RG

3.0

2.0 Clipping ratio

1.0 Constant amplitude loading 0

0.2

0.4

0.6

0.8

1.0

2.3 3.0 3.5 4.0 4.8 5.4 6.0 7.0

Note: Symbols marked / indicate results from Ref. 4 shown with approximate clipping ratio. 1.2 1.4 Value of k

1.6

1.8

2.0

2.2

2.4

6.25 Influence of spectrum shape on RG. (Results for all specimens normalised to peak stress range of 275 N/mm2).

2.6

180

Cumulative damage of welded joints

The results in Fig. 6.24 tend to confirm the lack of influence of spectrum shape and clipping ratio for spectra with k > 1.0. However, with k less than 1.0 the values of Σ n are still scattered between approximately 1.0 and 2.0, N but it is possible to discern a tendency for the results to be related to the value of the clipping ratio. For a particular shape of spectrum (value of k) Σ n N clearly tends to increase as clipping ratio increases. However, at this stage it is pertinent to note that, for a given spectrum shape, block length also increased with clipping ratio (see Fig. 6.19), and reconsideration for the results in terms of block length suggests that the value of Σ n is primarily a function N of block length and spectrum shape rather than clipping ratio (see section below). With regard to the values of RG, Fig. 6.25 confirms that most of the results lie within a fairly broad scatterband with a peak in the region of k = 0.8. With more results, however, a different picture might emerge. For example, the results of the tests using Rayleigh and Laplace spectra, discussed previously, seemed to suggest that at higher values of clipping ratio, lower values of RG would be obtained at low values of k and higher values at higher values of k. Nevertheless, with the data shown in Fig. 6.26 it is difficult to discern any particular influence of clipping ratio on the results.

6.6

Influence of block length and clipping ratio combined

When the results for each individual magnitude of peak stress range were plotted separately they showed considerable scatter. In order to try to produce more consistent relationships with a larger database the results obtained with a peak range of 175 N/mm2 were multiplied by 1.18 (see Fig. 6.23) in order to obtain the approximate corresponding results for a peak range of 275 N/mm2 and were then combined with the actual results relating to 275 N/mm2. While this is necessarily an approximation it is noticeable that scatter for the overall set of results was virtually unchanged. Using the combined database, Figs 6.26 and 6.27, which refer respectively to the values of Σ n and of RG, show the combined influence of block N length and clipping ratio. It will be seen that there are no very obvious trends apart from a general tendency for Σ n to increase with block length and for N RG to increase to a peak in the region of 102–103 cycles and then to decrease with increasing block length. There is no marked influence of clipping ratio either on Σ n or on RG. N

∑ n/N

10 10 10 10 Block length, cycles (a) 10

7

6.26 Influence of block length and clipping ratio on (b) 175 N/mm2.

1

101

102

103 104 105 106 Block length, cycles (b)

107

Clipping ratio 2.3 3.0 3.5 4.0 4.8 5.4 6.0 7.0

N

Σ n for specimens tested with peak stress range of (a) 275 N/mm2,

0.8 6

0.8

10

5

1.0

1.0

10

4

1.2

1.2

1

3

1.4

1.4

2

1.6

1.6

2.0

2.2

2.4

2.6

2.8

3.0

1.8

1

Clipping ratio 2.3 3.0 3.5 4.0 4.8 5.4 6.0 7.0

1.8

2.0

2.2

2.4

2.6

2.8

3.0

∑ n/N

∑ n/N

102

103 104 105 106 Block length, cycles (a) 107

1

101

102

103 104 105 106 Block length, cycles (b)

Constant amplitude loading 107

2.3 3.0 3.5 4.0 4.8 5.4 6.0 7.0

Clipping ratio

6.27 Influence of block length and clipping ratio on the value of RG for specimens tested with peak stress range of (a) 275 N/mm2 and (b) 175 N/mm2.

101

1.0

1.0

1

1.2

1.2

0.8

1.4

1.4

0.8

1.6

1.6

2.0

2.2

2.4

2.6

2.8

3.0

1.8

Constant amplitude loading

Clipping ratio 2.3 3.0 3.5 4.0 4.8 5.4 6.0 7.0

1.8

2.0

2.2

2.4

2.6

2.8

3.0

RG

The influence of spectrum shape and block length

6.7

183

Influence of block length and spectrum shape combined

When the results for each of the individual values of peak stress range were plotted by themselves they were relatively scattered, although it was possible to identify some general trends. However, as before, the database can be increased (with some degree of approximation) by including with the results relating to a peak stress of 275 N/mm2 those obtained at 175 N/mm2, with the corresponding values of Σ n and of RG multiplied by 1.18. The results N are shown in Figs 6.28 and 6.29, which also include the results of the Rayleigh and Laplace spectra (k = 0.5 and 1.0) considered earlier. So far as the values of Σ n are concerned, the curves shown in Fig. 6.28 N are intended to show the general trends rather than to represent exact results. However it is clear that Σ n tends to increase with block length, at least for N k < 1.0. It can also be seen that the peak value of Σ n for a particular block N length appears to occur at a value of k which increases with block length. It should, however, also be noted that the peak shown as occurring at approximately k = 0.5 for a block length NL = 16 cycles may well be spurious, since it is not supported by the low results at k = 0.65 for NL = 32 cycles. 104

2.2

104

16250

2.0 ≥ 32500

∑ n/ N

1.8 1.6 1.4 102

Approximate Note: Symbols marked / indicate block length results shown with 16 approximate clipping ratio. 24 32 64 128 1000 1000 16250 32500 130500 128

1.2 102 1.0

16

0.8 0

0.2

0.4

0.6

0.8

1.0

1.2 1.4 Value of k

1.6

1.8

2.0

6.28 Influence of spectrum shape and block length on

2.2

2.4

Σ n for

N specimens tested with peak stress range of 275 N/mm2, including 2 corrected values for specimens tested at 175 N/mm .

2.6

184

Cumulative damage of welded joints

3.0

100

19

100

RG

100 104

2.0

19

105

105 524284 524284

1.0

0

0.2

0.4

0.6

0.8

Note: Symbols marked / indicate results shown with approximate clipping ratio. 1.0

1.2 1.4 Value of k

1.6

1.8

2.0

Approximate block length 16 24 32 64 128 1000 16250 32500 130500 2.2

2.4

2.6

6.29 Influence of spectrum shape and block length on the value of the ratio RG for specimens tested with peak stress range of 275 N/mm2, including corrected values for specimens tested at 175 N/mm2.

With regard to the values of RG (Fig. 6.29) the majority of the results fit into a relatively narrow scatterband with a peak at approximately k = 0.7. The earlier results for Rayleigh and Laplace spectra suggest that, at long block lengths, RG may be reduced at low values of k and increased at higher values of k, but in general it is difficult to discern any consistent influence of block length on the results when they are plotted in this form. The results can also be viewed in a different way such that the influence of block length is considered in the context of spectrum shape, as shown in Figs 6.30 and 6.31. In this instance the results for the two values of peak stress are shown separately, with the values of Σ n and of RG plotted to a N log scale. In order somewhat to simplify the diagrams it has been assumed that the values of k within the ranges 0.23–0.55, 0.6–0.8 and 0.85–1.5 can be considered as being similar. The most obvious feature of Fig. 6.30 is that the value of k seems to have remarkably little influence on the value of Σ n , with the majority of the N results lying in a relatively narrow scatterband involving Σ n increasing N with block length. It is notable that the results for Rayleigh distributions lie on an approximately straight line running diagonally across the scatterband. As might be expected in the light of the obvious influence of peak stress on n (see Fig. 6.23) the scatterbands relating to the two peak stresses are ΣN

The influence of spectrum shape and block length

185

3.0 2.8 2.6 2.4 2.2

2.0 Results for Rayleigh distribution spectra 1.8

Σn

1.6

N 1.4

1.2

1.0

0.8 1

10

102

103 104 105 Block length (cycles) (a)

106

107

6.30 (a) Influence of block length and spectrum shape (k) on

Σ n for

N specimens tested with peak stress range of 275 N/mm , (b) influence of block length and spectrum shape (k) on Σ n for specimens tested N with peak stress range of 175 N/mm2. 2

186

Cumulative damage of welded joints 3.0 2.8 2.6 2.4 2.2 Results for Rayleigh spectra 2.0

1.8

Σn

1.6

N 1.4

1.2

1.0 k = 0.23–0.55 k = 0.6–0.8 k = 0.8–1.5 0.8 1

10

102

103 104 105 Block length (cycles) (b)

106

107

6.30 Continued

displaced relative to each other, but they are virtually parallel. They each represent values of Σ n with a scatter (at any particular block length) of N approximately ±20% from the mean. The equations of the mean curves can if the peak stress is 275 N/mm2 and 0.78 be written as Σ n = 0.92 N 0.0787 L N 0.0787 N L if it is 175 N/mm2. It will be noted that these values differ by a factor of 1.18, as found in Fig. 6.23. Clearly these relationships must be spurious at very short block lengths since, for all values of peak stress, we must have

The influence of spectrum shape and block length

187

n = 1.0 at N = 1 (i.e. under constant amplitude loading). Nevertheless ΣN

they may have their uses at longer block lengths. In the case of RG (Fig. 6.31) the results, although fairly scattered, show a general trend which is very similar to that found in the tests using Rayleigh spectra (see Fig. 6.10). Apart from the fact that there is a general trend for the value of RG to increase with increasing value of k (i.e. as the exceedence diagram changes from convex upwards towards a Laplace or concave upwards spectrum), there is also a tendency for RG to increase to a peak at a block 3.0 2.8 2.6 k = 0.8–1.5 2.4 2.2

2.0

1.8

RG

1.6

1.4

1.2 Rayleigh spectra 1.0

Value of k 0.23–0.55 0.5 0.6–0.8 0.8–1.5

0.8 1

10

102

103

104

105

106

107

(a)

6.31 Influence of block length at spectrum shape (k) on value of RG: (a) for peak stress = 275 N/mm2 (b) 175 N/mm2.

188

Cumulative damage of welded joints 3.0

Value of k

2.8

0.23–0.55 0.5 0.6–0.8 0.8–1.5

2.6 2.4 2.2

k = 0.8–1.5

2.0

1.8

1.6 RG

1.4 Rayleigh spectra 1.2

1.0

0.8 1

10

102

103

104

105

106

107

(b)

6.31 Continued

length of about 103 cycles and then to decrease again at longer block lengths. For ease of comparison the results obtained with Rayleigh spectra are also included in Fig. 6.31 and it will be seen that the curve representing those results comes close to forming an upper bound to the results for spectra with k = 0.23–0.55, albeit with a few rogue results above the boundary. It is clear that virtually all the results for k = 0.8–1.5, which only relate to quite small block lengths, lie well above the Rayleigh curve, while those for k = 0.6–0.8 are scattered around it. In summary, however, with a peak stress range of 275 N/mm2 all the results (up to a block length of about 105 cycles) gave values of RG which were greater than 1.0, while at 1.75 N/mm2 RG only fell below 1.0 at block lengths exceeding 104 cycles.

The influence of spectrum shape and block length

189

Although these findings are based upon test results showing relatively little scatter it has to be acknowledged that there are other results which are somewhat different. In particular, attention needs to be drawn to the results obtained by Schilling et al. (1975) who tested plain welded beams, specimens with longitudinal non-load-carrying fillet welds and beams with cover plates welded in two different ways (i.e. with and without fillet welds across the ends of the cover plates). In that investigation it was notable that the scatter of test results was substantial; for plain welded beams that is not particularly surprising but for beams with partial length cover plates it is unusual. In spite of the scatter it is clear that there was a general tendency for the value of Σ n to increase as the stress level increased, as in the investigation N considered above (see for example Fig. 6.10). However, the actual values of n were somewhat different. For example, at peak stress ranges of 275 ΣN and 175 N/mm2, the values of Σ n for plain welded beams were approximately N 0.75 and 0.6 respectively, while for beams with welded cover plates they were approximately 1.4 and 1.2 when the cover plate ends were welded or 2.6 and 2.1 when they were unwelded. These values may be compared with approximately 1.5 and 1.1 derived from Fig. 6.9. Clearly, there are some major differences. However, one of the main points of interest of that particular work is that it is one of the few investigations in which tests were carried out on three different types of specimen, with very different basic fatigue strengths, using identical loading spectra. As can be seen from Fig. 6.32 the values of Σ n N did not seem to be affected by joint type, provided that they were based on the constant amplitude S-N curve relevant to each particular joint. The comparison between the results for the three different forms of Rayleigh distribution is also of interest. It seems clear that there is certainly a tendency for the value of Σ n to increase as the value of Srd /Srm increases, or as p N decreases. The constitutions of the three spectra are summarised in Table B4 of Appendix B, from which it is clear that the higher values of Σ n seem to N be associated with the spectra having the greater number of small stress ranges.

6.8

Summary

All the work described in this chapter involved tests on as-welded specimens using stress spectra in which all the individual stress cycles were applied in random order at the same stress ratio. The great majority were tested at R = 0 but a few comparative tests were at R = –1. For loading involving short

190

Cumulative damage of welded joints

4.0

3.0

Σn

N 2.0

1.0

0 Srd Srm

0.25 0.5 1.0

0.25 0.5 1.0

0.25 0.5 1.0

0.25 0.5 1.0

0.25 0.5 1.0

p

0.5 0.25 0

0.5 0.25 0

0.5 0.25 0

0.5 0.25 0

Plain LNLCFW Cover beam (Class F) plate (class C) (Class G)

0.5 0.25 0

Cover plate

0.25 0.5 1.0

LNLCFW

0.5 0.25 0

Plain beam

(a)

6.32 Summary of mean values of

(b)

Σ n obtained by Schilling et al. for

N three types of joint subjected to Rayleigh distributions with stress ranges = 0.25, 0.5 and 1.0 at block length = 500 cycles. (a) Values based on mean experimental S-N curves (b) based on mean British design S-N curves.

block lengths (up to 512 cycles/block) it was established that the greatest difference between the life predictions provided by Miner’s rule and the area rule occurred in spectra with concave upwards exceedence diagrams (i.e. involving a larger than usual number of cycles of small amplitude). For such spectra Miner’s rule almost always gave unsafe life predictions with Σ n N

The influence of spectrum shape and block length

191

ranging from 0.65–1.04; in contrast, the area rule was consistently safe, with RG lying in the range 1.13–2.11. Tests involving a Rayleigh distribution of stress ranges, with block lengths up to 524 284 cycles, showed that Σ n increases with block length and also N with the value of the peak stress range in the spectrum. Values of Σ n less N than 1.0 tended to occur at short block lengths (less than about 100 cycles) combined with relatively low peak stresses (less than about 225 N/mm2). In contrast, the area rule gave safe life predictions at short and intermediate block lengths, but the value of RG reached a peak at a block length of about 1000 cycles and then tended to fall. The area rule became unsafe at long block lengths (about 105 cycles) combined with lower peak stress ranges. Tests with Laplace load spectra, using block lengths up to 104 cycles, showed results with very similar trends. Σ n increased with block length, as did the N value of RG. However, it should be noted that the results are not consistent with those for the Rayleigh distributions (see Figs 6.5–6.8). Comparative tests at R = 0 and R = –1 were carried out using Rayleigh spectra with block lengths of 1024 and 131072 cycles. In these tests the results for R = –1 consistently showed a shorter life to failure than those for R = 0, probably due to the influence of residual stresses induced by the higher stresses in the spectra. Those induced at R = 0 would have been higher, for the same total stress range, than at R = –1 and would therefore have had a greater retardation effect on the lower stresses in the spectrum. Following the tests using Rayleigh and Laplace spectra, an extensive series of tests was carried out using Weibull spectra (which include the Rayleigh and Laplace distributions as special cases) in order to examine the relative influences of spectrum shape (value of k), block length and clipping ratio. The most obvious features of the results were: • that for spectra with concave upwards exceedence diagrams, which necessarily involved short block lengths, the values of Σ n and of RG N were virtually constant over the range 1.0 ≤ k ≤ 2.0 with approximate values of Σ n = 1.1 and RG = 2 (see Figs 6.28 and 6.29); N • that for convex upwards spectra the relationships between either Σ n or N RG and block length were very similar to those found for Rayleigh and Laplace spectra, with Σ n progressively increasing (Fig. 6.30) while RG N increased to a peak at a block length of about 1000 cycles and then decreased (Fig. 6.31). In this region Σ n clearly increases with clipping N ratio but that appears to be due primarily to the fact that clipping ratio

192

Cumulative damage of welded joints

increases with block length rather than because clipping ratio is the critical factor per se; indeed, clipping ratio by itself seems to have relatively little influence on Σ n or RG. N In the course of the work it has been proved beyond all reasonable doubt that the suggestion that all test results (for different spectra) can be ‘normalised’ by plotting the results on the basis of either r.m.s. stress or cube root mean cube stress is definitely incorrect.

7 The influence of narrow band, wide band and service loading

7.1

Introduction

It was noted in Chapter 3 that, so far as the fatigue problem is concerned, the most important feature of narrow band loading is that it has the special characteristic that each positive half-cycle is followed by an approximately equal and opposite half-cycle, which returns to the same mean level. Thus it involves a steady variation in the envelope of peaks and troughs, with a constant mean stress; very often all cycles are at R = –1. In other words narrow band loading has the appearance of a sinusoid of discrete frequency (f0) but with slowly varying random amplitude and phase. In loading of this type it is meaningful to talk in terms of individual cycles and an envelope. Although the instantaneous value of load or stress at any moment cannot be predicted, the probability that any given amplitude falls within a particular range is statistically predictable provided that the process is stationary (i.e. that it does not change in a statistical nature with time). It is particularly important to recognise that such loading does not include any sudden occurrences of high (overload) cycles or of small cycles. In the context of a Markov matrix of peaks and troughs narrow band loading has all the non-zero entries either on, or close to, the diagonal. A typical example is shown in Fig. B7 in Appendix B. This relates to the Gaussian spectrum with irregularity I = 0.99, where the ‘irregularity’ is the ratio of the number of upward going mean crossings to the number of peaks. In other words, for all practical purposes, there are as many mean crossings as peaks. With this type of loading the peak and range distributions are of the Rayleigh type and it results from random excitation of structures with low damping capacity that vibrate around their resonant frequency. In contrast, wide band loading has an irregularity significantly less than 1.0. In other words the mean stress of the various cycles varies widely so that the Markov matrix has non-zero entries scattered over a major part of the diagram. For example, Fig. B8 in Appendix B relates to a Gaussian spectrum with I = 0.7. A typical example of this type of loading is from random 193

194

Cumulative damage of welded joints

excitation of structures with medium damping capacity (e.g. half critical viscous damping) such as the frames of transport vehicles with shock absorbers. It is interesting to note that, in terms of the numbers of cycles of each stress range these two spectra (i.e. those with I = 0.7 and I = 0.99) are virtually identical (see Fig. 7.1); the difference lies primarily in the degree of variation in mean stress of the individual cycles. This is clearly apparent in Fig. 7.2, which shows typical sections of the loading traces for each of the two spectra. Returning to the problem of narrow band loading, the process was analysed by Rice (1945), assuming it to be Gaussian, and he showed that the expected number (fs) of crossings of stress level S with positive slope per unit time is:  – S2  fs = fo exp    2σ 2 

[7.1]

where σ is the r.m.s. value of the process. The expected number of peaks occurring in the interval between S and (S + dS) will be  – S2  df s = f o S2 exp   dS σ  2σ 2 

[7.2]

The probability density function of a peak of amplitude S occurring is therefore the Rayleigh probability density function, namely  – S2  p (S) = S2 exp   σ  2σ 2 

[7.3]

20

I = 0.99 I = 0.7 10

1

10

100

103

104

105

106

7.1 Comparison of exceedence diagrams for Gauss spectra with I = 0.99 (narrow band) and I = 0.7 (broad band).

I = 0.314

7.2 Typical loading sequences for Gauss spectra with different values of I (after Overbeeke (1990).

I = 0.706 (wide band)

I = 0.994 (narrow band)

196

7.2

Cumulative damage of welded joints

Comparing loading types

In many respects it is obviously logical to consider the effects of narrow and wide band loading separately. However, before doing that, it is useful to summarise the results of one particular investigation in which a direct comparison was made between the two loading types using spectra with identical rainflow counts. In that work the particular type of specimen which was used consisted of a plate with fillet welded edge attachments (Fig. 7.3). The reason for that choice was that it was desired to use a joint with a low inherent fatigue strength, so as to enable the loading spectra to contain a wide range of stresses while keeping the upper limit stress well below yield strength and yet obtain relatively short fatigue lives. At the same time it was useful to employ a joint giving low scatter in fatigue lives in order to simplify the drawing of valid conclusions. The investigation (Gurney, 1993) had several objectives.

150

8 mm leg length fillet welds on each side of main plate

90

12

12

125

1. To obtain comparative fatigue lives under four different forms of loading: (a) random order loading with all cycles applied at the same stress ratio; the results of this part of the work were considered in Chapter 6 (see Figs 6.12 and 6.13) (b) narrow band loading (c) wide band loading (d) block programme loading. For each of these four types of loading the same two spectra (in terms of rainflow count) were used in each case. So far as is known this is the only investigation in which such a direct comparison has been made. 2. To compare test results relating to two different spectra (both Rayleigh distributions but with different block lengths, 1024 and 131 072 cycles).

Dimensions in mm

7.3 Details of test specimens with fillet welded longitudinal edge attachments.

The influence of narrow band, wide band and service loading

197

7.4 Typical example of part of simulated ‘narrow band’ stress history (R = –1).

3. To investigate the effect of applying the stresses at different stress ratios, notably with the peak stress range, and in some instances some of the other stress ranges as well, at R = 0 and at R = –1. The first objective, therefore, was to investigate the influence of the form of loading on fatigue life and it will be recalled that the first part of this work, relating to tests under random order loading at a constant stress ratio, was considered in Chapter 6. The results were shown in Figs 6.12 and 6.13 and it was found that the life at R = –1 was significantly shorter than at R = 0, due to the action of residual stresses induced by the high loads in the spectrum. The next step was to carry out tests with the same spectra with the cycles arranged in the form of simulated narrow band loading (Fig. 7.4). In those tests either all the cycles were applied at R = –1 or a superimposed tensile mean stress was applied to the whole spectrum so that the peak stress range was at R = 0 and all the other stresses were fully tensile. For this type of loading the results are shown in Figs 7.5 and 7.6. From Fig. 7.5 it is clear that, as with random order loading, the tests at R = 0 gave significantly longer lives than those at R = –1. In terms of Σ n (Fig. 7.6) it N is apparent that, at R = –1, all specimens gave values of Σ n which were N close to 1.0, regardless of block length. In the tests at R = 0, however, the values of Σ n were significantly greater than 1.0, particularly with the N longer block length, but there was a distinct tendency for the value of Σ n N to decrease as the applied stress decreased. Qualitatively, it seems probable that the reason for this behaviour is that, at R = 0, the high ranges in the spectrum generate larger compressive residual stresses at the crack tip, which are more effective in ‘blanketing’ the effect of the lower stress ranges. In this respect it has to be remembered that, in a test at R = –1 with the same peak range, the actual peak tensile stress will only be half that at R = 0; hence the residual stress generated will be much smaller in extent. The larger effect that is evident in the case of the longer block length presumably stems from the fact that, in that spectrum, there is a much

Peak stress range N/mm2

60 105

70

80

90

100

120

140

160

180

200

220

240

260

280

300

2

3 4 5

106 1.5 2 3 4 5

Endurance, cycles

107

1.5

2

3

4

5

108

7.5 Test results for specimens with longitudinal edge attachments under narrow band loading.

1.5

R=–1

NL = 1024 R=0

R=–1

NL = 131027 R=0

N/mm2

100

150

200

300

400

500

1.0 2.0

7.6 Narrow band test results expressed in terms of

0.5

Block length = 1024

N

Σn.

0.5

1.0

2.0

Block length = 131 072

5.0

R = –1 R =0

200

Cumulative damage of welded joints

larger proportion of small stress cycles, which are capable of being completely blanketed. So far as the tests under wide band loading are concerned, the order of application of the various cycles was identical to that used in the earlier tests under random order loading, but in the wide band versions the stress ratios of the various cycles were changed, keeping the rainflow counts the same. By way of example Fig. 7.7 shows part of the longer spectrum with all cycles at the same stress ratio (R = 0 and R = –1) while Fig. 7.8 shows the wide band versions for each block length. For comparison the block programme form of the spectrum, which in some respects is akin to narrow band loading, is shown in Fig. 7.9. So far as lives are concerned, the results obtained under wide band loading are summarised in Fig. 7.10. The most interesting feature of these results is that, in contrast both to random order loading and to narrow band loading, the specimens subjected to spectra in which the peak range was applied at R = –1 gave longer lives than when the peak range was at R = 0. Presumably this stems from the fact that, under wide band loading, a significant number of the small stress ranges were applied at high mean stress, such that the upper limits of those ranges were close to the upper limit stress of the peak range. In those circumstances the blanketing effect exercised by the peak range would be expected to be less beneficial. As a result the generally high mean stress of the cycles in the spectra at R = 0, compared with those in the R = –1 spectrum, would be expected to lead to an enhanced rate of crack propagation and consequently a shorter life. The values of Σ n obtained in these tests did not vary greatly between N the four test series, the overall range of values being 0.45–1.09. The ranges of results for each individual series are shown in Table 7.1. In each case there was a clear tendency for the value of Σ n to have a relatively high value N when the peak stress range was high and for the value then to decrease as the applied stress decreased. Nevertheless, the important fact remains that wide band loading gives low values of Σ n with most values being well below N 1.0. Given that this is the only investigation on which tests were carried out using the same spectra in different forms, it is interesting to note the influence of spectrum form on the results. Expressed in terms of the values of Σ n , N the results are summarised in Figs 7.11 to 7.14. Figures 7.11 and 7.12 give the results for the shorter block length while Figs 7.13 and 7.14 relate to the longer block. The obvious feature of these results is that, in the tests at R = –1, all forms of loading, except block programme loading with the larger block length, gave values of Σ n which were quite close to 1.0. The overall N

7.7 Typical examples of stress histories with the cycles in random order at constant stress ratio. (Note: the two traces relate to identical parts of the history, NL = 131 072 cycles) (a) R = 0 (b) R = –1.

(b)

(a)

7.8 Typical examples of wide band stress histories (a) NL = 1024 cycles (b) NL = 131 072 cycles.

(b)

(a)

7.9 Typical example of a simulated ‘block programme’ stress history (NL = 1024 cycles, R = 0).

Peak stress range N/mm2

300

60 105

70

80

90

100

120

140

160

180

200

220

240

260

280

2

3 4

131,072

5

R=0 R=–1

Block length

106 1.5 2 3 4

5

Endurance, cycles

7.10 Test results obtained under wide band loading.

1.5

1024

R=0 R=–1

107

R=0 R=–1

1.5

2

3

4

5

108

The influence of narrow band, wide band and service loading

205

Table 7.1 Comparing fatigue life under four different forms of loading Block length

1024

131 072

R=0 R = –1

0.6–0.85 0.45–1.09

0.5–0.92 0.74–1.05

400 Form of loading Block programme Random order Simulated narrow band Wide band

Peak stress range, N/mm2

350 300 280 260 240 220 200 180 160 140 120 100 90 80 0.1 1.5 2

3

4 5

1.0

1.5 2

3

4 5

10 1.5 2

3

4 5

100

Σn N 7.11 Comparison of values of (NL = 1024 cycles, R = 0).

Σ n for four types of loading N

spread of values was from 0.45 to 1.35 with a block length of 1024 cycles and from 0.74 to 1.34 for a block length of 131 072 cycles. On the other hand at R = 0 the values of Σ n for the various types of loading were well N separated with wide band loading giving the lowest values and narrow band, random order and block programme loading giving progressively better results. So far as block programme loading is concerned the tests at R = 0 gave values of Σ n ranging from about 2.66 to 1.5 with the shorter block, and N values in the region of 10.0 with the longer block. This clearly confirms what has been known for many years, namely that block programme loading can lead to wildly optimistic values of Σ n . A summary of Fischer’s comparison N of the effects of block programme and random order loading is shown in Fig. 7.15. It was as a result of this work that it was decided to ignore all block

206

Cumulative damage of welded joints

400

Peak stress range, N/mm2

350

Form of loading Block programme Random order Simulated narrow band Wide band

300 280 260 240 220

Black programme

200 180

Wide band

160 Narrow band 140 120 100 90 80 0.1 1.5 2

3

4 5

1.0 1.5 2

3 4

5

10 1.5 2

3

4 5

100

Σn N 7.12 Comparison of values of (NL = 1024 cycles, R = –1).

Σ n for four types of loading N

400

Peak stress range, N/mm2

350 300 280 260 240 220 200 180 160 140 Form of loading Block programme Random order Simulated narrow band Wide band

120 100 90 80 0.1 1.5 2

3

4 5

1.0 1.5 2

3 4 5

10 1.5 2

3

4

5

Σn

N

7.13 Comparison of values of (NL = 131 072 cycles, R = 0).

Σ n for four types of loading N

100

The influence of narrow band, wide band and service loading

207

400 Form of loading Block programme Random order Simulated narrow band Wide band

Peak stress range, N/mm2

350 300 280 260 240 220 200 180 160 Wide band 140 120 Narrow band 100 90 80 0.1 1.5 2

3

4 5

1.0 1.5 2

3 4

5

10 1.5 2

3

4 5

100

Σn

N

7.14 Comparison of values of (NL = 131 072 cycles, R = –1).

Σ n for four types of loading N

programme fatigue test results when compiling the British fatigue design rules, even though the evidence suggests that at R = –1 the results of such tests are more acceptable. Even in that situation it will be seen from Fig. 7.14 that, with the longer block length, the values of Σ n were still larger than N those for the other types of loading by a factor of 2 or more. Having considered the results that were obtained when a direct comparison was made between the various types of loading spectra, an attempt will now be made to summarise, separately, the results that have been obtained in other investigations involving narrow and wide band loading.

7.3

Tests under narrow band loading

So far as welded joints are concerned, the first major investigation involving narrow band loading appears to have been that carried out by White and Lewszuk in the late 1960s and published in 1971. This involved relatively small specimens, typically 35 mm wide and 9.5 mm thick, containing transverse load carrying fillet welds (Fig. 7.16). In one set of tests the specimens were tested in the as-welded condition and were subjected to axial loading (White and Lewszuk, 1971) while in the other the weld toes were ground to improve the fatigue strength and the specimens were tested in bending (Lewszuk and White, 1970). In both instances it was verified that the frequency of occurrence

208

Cumulative damage of welded joints 30

Absolute frequency of occurrence

25

20

15

10

5

0 0.2

0.3 0.4 0.5 1 2 3 4 Life under programme loading

5

10

Life under random loading

7.15 Comparison of results obtained under block programme and random order loading (adapted from Fischer et al., 1977).

of the stress peaks followed a Rayleigh distribution at least up to four times the r.m.s. value; this implies that the block length was 2980 cycles. This investigation stemmed from fatigue failures which had occurred in the gas circulation system at Hinkley Point Nuclear Power Station and which were reported by Rizk and Seymour (1965). They were associated with acoustic excitation from the gas circulators and site measurements proved that various structural parts, such as ducts, bellows and cascade corners, were responding in a predominantly narrow band manner with the frequency restricted to the neighbourhood of one characteristic frequency. As a result it became clear that information on the fatigue strength of welded joints under this type of loading was urgently required. Both parts of the investigation date from a time when there was a tendency to plot fatigue test results with axes of stress, (rather than log stress), and log life. Inevitably this led to the test results lying on a curve, which was relatively difficult to define with any accuracy. In view of this situation all the results have been re-analysed in the course of writing this book, so that it should not come as a surprise to note that the positions of the various S-N curves,

The influence of narrow band, wide band and service loading

209

203 mm (8 in.)

7 mm (9/32 in.) 44 mm (13/4 in.) 9.5 mm (3/8 in.) 7 mm (9/32 in.) 9.5 mm (3/8 in.)

35 mm (13/8 in.)

(a)

121 mm (43/4 in.) 7 mm (9/32 in.)

67 mm (25/8 in.)

9.5 mm (3/8 in.)

25.4 mm (1 in.)

50.8 mm (2 in.) 16 mm (5/8 in.) dia.

44.5 mm (13/4 in.)

(b)

7.16 Details of specimens tested by Lewszuk and White (1970).

together with the resulting values of Σ n quoted here, are somewhat different N from those in the original papers. To confuse the issue still further the original results are reported in terms of ‘r.m.s. stress’. As has been noted previously (Gurney, 1988b) this is a very

210

Cumulative damage of welded joints

ambiguous quantity, and in any event it has no physical meaning in the fatigue context. By way of example it may be taken to mean: 1/2

1. 2.

3. 4.

 N  the r.m.s. of the peak stress ranges i.e. S rms =  1 Σ S 2i  1 N   the r.m.s. of the ‘process’, again expressed in terms of stress ranges; for S a sinusoidal signal this gives S rms = max 2 the r.m.s. of the peak stress amplitudes, where ‘amplitude’ is defined as half of the range the r.m.s. of the process expressed in terms of amplitudes, as often used in electricity.

In the author’s opinion there is no justification for the use of ‘r.m.s. stress’ (of any sort) in reporting fatigue test results, and it would clearly avoid a lot of confusion if the practice were abandoned. Given that, in the last resort, all fatigue tests are carried out to provide information for designers, either for a particular structure or for use in the improvement of fatigue design Standards, and that structures are not normally designed on the basis of r.m.s. stress but in terms of the actual stresses (or stress ranges) which are likely to occur, it seems pointless to introduce r.m.s. stresses at all. In spite of this problem it is hoped and believed that it has proved possible, in the course of the reanalysis, to deduce what the published results actually were. Considering first the tests on as-welded joints under axial loading, the investigation involved tests both under constant amplitude loading and under narrow band loading at three different mean stresses, namely 0, 77 and 154 N/mm2. Since the results at a mean stress of 77 N/mm2 lay between the other two it is convenient to look simply at the results obtained at R = –1 and at a mean stress of 154 N/mm2. The two sets of results obtained under constant amplitude loading are summarised in Figs 7.17 and 7.18. This shows that the application of a tensile mean stress produced a significant reduction in fatigue strength of about 25%. The mean fatigue strengths (expressed as stress ranges) are shown in Table 7.2. The corresponding results obtained under narrow band loading are shown in Figs 7.19 and 7.20, and superimposed upon them are calculated lines indicating Σ n = 1.0. These can easily be deduced from eqn [6.2], which N showed that, for Σ n = 1, the predicted life to failure was: N N ⋅N [7.4] N pred = c mL Σ vp where NL is the number of cycles per block and Nc is the life under constant amplitude loading with the peak stress range in the spectrum.

The influence of narrow band, wide band and service loading

500

211

TLCFW axial R = – 1 CA As welded m = 3.509 C = 6.5193 × 1013

400

SVSM range, N/mm2

300

200

150

100

105

106

10

Cycles

7.17 Constant amplitude test results for transverse load-carrying fillet welds under axial loading at R = –1 (White and Lewszuk, 1971).

CA Mean = 10 Ksi

300 280 260

TLCFW mean σ = 10 tsi Arial m = 3.342 c = 0.1 × 1014

240 200

Range of stress, N/mm2

220

S-N curve for R=–1 (from Fig. 7.4)

180 160 140 120 100 90

3.3

42

80 70 60 104 1.5

2

3

4 5

105

1.5

2

3

4

5

106

1.5

2

3

4 5

107

Endurance, cycles

7.18 Constant amplitude test results for transverse load-carrying fillet welds under axial loading with mean stress = 154 N/mm2 (White and Lewszuk, 1971).

212

Cumulative damage of welded joints Table 7.2 Fatigue strengths under constant amplitude loading for aswelded joints Mean stress

0 154 N/mm2

Fatigue strength 105

2 × 106

325 248

138 101

Now, for a given stress spectrum, Nc =

N pred  NL     Σ vp m 

NL is constant so that Σ vp m [7.5]

Hence, if we define two values of Npred the corresponding values of Nc can be calculated, and entering the constant amplitude S-N curve with those values the relevant peak stresses in the spectrum can be calculated. By way of example consider the results for R = –1. In this case NL = 2980 = 23.57 [7.6] Σ vp m 126.43 Thus for a life under variable amplitude loading (Npred) of 105 cycles we get, 5 from eqn [7.5], Nc = 10 = 4242 cycles and for Npred = 107 cycles. Nc = 23.57 424 268 cycles. Given that the equation of the relevant constant amplitude S-N curve (Fig. 7.4) is S mN = C

[7.7]

with m = 3.509 and C = 6.5193 × 1013, it is easy to deduce that N (= Nc) = 4242 cycles gives S = 799.8 N/mm2 and that N = 424 268 cycles gives S = 215 N/mm2, and these are the peak stresses in the spectrum corresponding to variable amplitude lives of 105 and 107 respectively. Hence the straight line defining Σ n = 1 is known. The same method can be used to define the N corresponding line for a mean stress of 154 N/mm2, the values of maximum stress range being 617 N/mm2 at 105 cycles and 156 N/mm2 at 107 cycles. From Figs 7.19 and 7.20 it is easy to deduce that, at R = –1, the value of n ranged from approximately 0.34 to 2.6, while at a mean stress of 154 ΣN N/mm2 it ranged from about 0.49 to 3.5. In other words there was no great difference between the values of Σ n for the two different mean stresses. N Furthermore, at R = –1, there was no obvious influence of applied stress

The influence of narrow band, wide band and service loading

Maximum stress range in spectrum N/mm2

1000

213

• R = –1

800

600

400

Σ

n = 1.0 N

300

200

120 104

105

106

107

Cycles

7.19 Results for axially loaded transverse load-carrying fillet welds under narrow band random loading at R = –1.

Maximum stress range in spectrum N/mm2

1000 800 600

Σ

n = 1.0 N

400

300

200

120 104

105

106

107

Cycles

7.20 Results for axially loaded transverse load-carrying fillet welds under narrow band random loading with mean stress = 154 N/mm2.

214

Cumulative damage of welded joints

range on the value of Σ n although when the mean stress was increased to N 154 N/mm2 there seemed to be a tendency for the higher values of Σ n to N be associated with high applied stresses while lower applied stresses tended to give lower values of Σ n . N Turning now to the tests on toe ground joints in bending, the fatigue strengths were, as might be expected, significantly higher. Toe grinding produces an increase in fatigue strength for two reasons; firstly it reduces the stress concentration at the weld toe and secondly it has the effect of eliminating weld toe defects. This latter effect means that instead of the fatigue life being occupied solely in crack propagation, with the crack initiating from one or more of the toe defects, it results in the crack first having to initiate and then to propagate; in other words an initiation period is introduced into the total life. In addition it is well known that tests in bending give significantly higher fatigue strengths than tests under axial loading; typically the increase is equivalent to approximately two joint classes in the British design rules, although that increase is not yet recognised in the design Standard. In fact, the tests under constant amplitude loading (see Fig. 7.21) gave the mean fatigue strengths (stress ranges) shown in Table 7.3. Comparison with the corresponding results for the as-welded joints under axial loading shows that these results represent an increase in strength of about 100%. A similar

Stress, N/mm2 (semi range)

300 280 260 240 220

Σ

n = 1.0 N

200 180 160 140 120 100 90 80 70

R=–1 σmean = 232

60 4 10 1.5

2

3

4 5

105 1.5 2 3 4 5 Endurance, cycles

106 1.5

2

3

4 5

7.21 Results for toe ground transverse load-carrying fillet welds tested in bending under constant amplitude loading.

107

The influence of narrow band, wide band and service loading

215

Table 7.3 Fatigue strengths under constant amplitude loading for toe ground joints Mean stress

Fatigue strength

0 232 N/mm2

105

2 × 106

610 496

292 240

1200 R=–1 Mean stress = 232 n/mm2

Peak stress range in spectrum (N/mm2)

1000

800

600

Σ

n = 1.0 N

R

=

–1

400

M 300 240 105 1.5 2

3

4 5

106 1.5 2 3 4 5 Endurance, cycles

ea

n

=

23

2

107 1.5 2

N/

m

m2

3

4 5

108

7.22 Results for toe ground transverse load-carrying fillet welds tested in bending under narrow band random loading at R = –1 and with mean stress = 232 N/mm2.

increase in strength was also obtained under narrow band loading, for which the results are shown in Fig. 7.22. This also shows the predicted lives for n = 1.0 and it will be seen that the actual values ranged from about 0.32– ΣN 5.7 at R = –1 and from about 0.48–1.8 at a mean stress of 232 N/mm2. At R = –1 there was a general tendency for the results at high stresses to give relatively low values of Σ n and at lower stresses for the values to be N greater than 1.0. At the higher mean stress there was no distinct trend. A substantial set of test results was also reported by Haibach et al. (1980). For the most part the specimens were transverse load-carrying fillet welds, 50 mm wide and 10 mm thick, and the results were reported in relation to the

216

Cumulative damage of welded joints

Upper limit stress in spectrum, N/mm2

600

Σ

n = 1.0 N

400

300

Σ

200

n = 0.2 N

150

104 1.5 2

3

4 5

105

1.5 2

3

4

5

106 1.5 2

3

4 5

107

Endurance, cycles

7.23 Results of tests on transverse fillet welded specimens under narrow band loading at R = –1 as reported by Haibach et al.

stress range expressed as a multiple of the constant amplitude fatigue strength at 2 × 106 cycles. The tests also included some specimens with transverse non-load-carrying fillet welds with the results expressed in the same manner, so the two sets of results were then amalgamated. The slope of the constant amplitude S-N curve was m = 3.75 and the results obtained under narrow band loading at R = –1 (Gaussian spectrum with I = 0.99, see Appendix B) are summarised in Fig. 7.23. This also shows the theoretical ‘S-N curve’ corresponding to Σ n = 1, assuming a block length of 106 cycles and a N clipping ratio of 5.25. Examination of Fig. 7.23 shows that all the results gave values of Σ n N which were less than 1.0, the minimum value being typically 0.2 with a few outlying results being even lower than that. Figure 7.24 shows a summary of the results expressed in terms of probability of occurrence, the mean value of n for 54 results being 0.39. On the basis of these results Haibach suggested ΣN that the design value of Σ n should be reduced to about 0.3. It will be seen N that there was no obvious tendency for the value of Σ n to be related to the N magnitude of the peak stress range in the spectrum. Virtually the same loading spectrum was also used by Overbeeke et al., (1990) in tests on three different types of specimen, namely transverse butt

The influence of narrow band, wide band and service loading

217

20 18

Mean = 0.39 (54 results)

No. of occurrences

16 14 12 10

I = 0.99

8 6 4 2

No. of occurrences

0

0

6

1.0 n N

2.0

Σ

Mean = 0.39 (14 results)

4

I = 0.7

2 0 0

1.0 n N

2.0

Σ

7.24 Results reported by Haibech et al. for transverse non-loadcarrying and load-carrying fillet welds subjected to Gauss loading.

welds, longitudinal non-load-carrying fillet welds and single lap load-carrying fillet welds, but in each case the test programme was restricted to (typically) 7–9 specimens. The results are summarised in Fig. 7.25(a)–(c) in the form of relationships between the values of the top limit stress in the spectrum and the corresponding values of Σ n . In this investigation some comparisons N were made for two of the joints, between the results for narrow band loading and those for wide band loading (Gauss spectra with irregularity factors of 0.99 and 0.71). Considering first the results for narrow band loading, both the transverse butt welds (Fig. 7.25(a)) and the fillet welded lap joints (Fig. 7.25(c)) showed a tendency for the value of Σ n to increase as the applied stress decreased. N This trend was not apparent with the longitudinal non-load-carrying fillet welds (Fig. 7.25(b)) which, for all practical purposes, showed no effect of applied stress level. The difference between the two test series of longitudinal non-load-carrying fillet welds lay only in the method of welding. In the

218

Cumulative damage of welded joints

300 280 260 240 220

I = 0.99

Top limit stress, N/mm2

200 I = 0.99 I = 0.7

180 160 140

I = 0.7

120 100 90 80 70 60 0.1 1.5 2

3

4 5

1.0

1.5 2

3

Σ

4

n N

5

106 1.5 2

3

4 5

107

(a)

500

Top limit stress, N/mm2

400 300 Type C Type A Both I = 0.99

200

150

100

0.1

1.0

Σ

n N

10

(b)

7.25 Tests on (a) transverse butt welds, (b) longitudinal non-loadcarrying fillet welds, and (c) fillet welded lap joints. (Overbeeke 1990).

10

The influence of narrow band, wide band and service loading

219

300 280 260 240 220

Top limit stress, N/mm2

200 180 I = 0.99 I = 0.7

160 140 I = 0.7

120

I = 0.99

100 90 80 70 60 0.1 1.5 2

3

4 5

1.0 1.5

2

3

Σ

(c)

4 5

106 1.5 2

3

4 5

107

n N

7.25 Continued

joints designated Type C, the welds around the ends of the longitudinal stiffeners were made using pulsed MIG welding while joints of Type A were made by manual metal arc welding. Under constant amplitude loading they gave very different strengths; at 2 × 106 cycles they were approximately 120 N/mm2 (stress range at R = –1) for Type C (pulsed MIG) but only about 74 N/mm2 for Type A (MMA welds). At first sight these results seem slightly suspect, but the higher values of Σ n for Type A almost certainly stem from N the fact that the base strength was significantly lower. In view of this situation, at least as far as the values of Σ n are concerned, it would probably be more N realistic to accept that, for this type of joint, Σ n was approximately 1.0 for N all applied stress levels. The results obtained under wide band loading for the transverse butt welds and lap joints will be considered later in the chapter. It is also of interest to consider these results relating to narrow band loading in the context of the area rule. It will be recalled that, in Chapter 6, it was noted that the predicted life defined by Miner’s rule (i.e. with Σ n = N 1.0) could be written as (from eqn [6.2]).

Predicted life =

Nc ⋅ NL Σ (vp m )

Equally, using the area rule (eqn [6.3]), predicted life = Nc · NLe-Area, hence

220

Cumulative damage of welded joints

Table 7.4 Characteristics of narrow band spectra Joint

Mean stress

Slope (m) of S-N curve

TLCFW

R = –1 154

3.509 3.342

R = –1

3.75

Trans butt

R = –1

5.47

LNLCFW (A)

R = –1

2.71

LNLCFW (C)

R = –1

4.01

LC Lap Jt

R = –1

3.16

R R R R

2.656 3.0 2.656 3.0

Block length (NL) (cycles) 2980 106

∑vpm

126.43 140.58 13 019

Fillet welded edge attachment

–1 0 –1 0

5348

e Area

NL

9.125 109.7

3.513 9.125 109.7

10 402

1.141

22 253

131 072

1.428 0.363

32 023

1024

Area life Miner life

14.176 0.601 0.669

3308

106

= = = =

Area

104.1 7084 4396

2.441 4.614

10.15

7.729

57.68

1.032 0.735 3.117 1.935

Predicted life by area rule = Σ (vp m ) ⋅ e –Area Predicted life by Miner’s rule

It will be seen that the value of this ratio depends solely on the slope of the S-N curve and the characteristics of the loading spectrum under consideration; in particular it does not depend upon the magnitude of the stress. In other words, if the results obtained using a particular loading spectrum, but with variable levels of peak stress, are plotted in the form of an ‘S-N curve’ (see for example Fig. 7.5) the curve defining RG = 1.0 will be parallel to the curve for Σ n = 1.0. The characteristics of the various spectra referred to in this N summary, and the resulting values of the ratio of predicted lives, are summarised in Table 7.4. Using these values of the ratio and the values of Σ n referred to earlier N in this chapter, the corresponding values of RG have been deduced and are summarised in Table 7.5. For consistency with the values discussed in earlier chapters the values of RG shown here relate to the situation where the peak stress range in the spectrum was 275 N/mm2 although, where available, the values relating to a peak range of 175 N/mm2 are also shown. No results are given for the transverse butt welds tested by Overbeeke for which all the applied stresses were considerably higher. There was considerable difficulty about deducing realistic ranges of Σ n (and hence of RG) for the results N

The influence of narrow band, wide band and service loading Table 7.5 Values of Joint

Mean stress

Σ n and RG for narrow band loading N

Results in Fig.

Peak range = 275 N/mm2

Peak range = 175 N/mm2

Σn

RG

Σn

RG

0.53–1.75

0.79–2.6

N

TLCFW

221

N

R = –1 1.54

.6 .7

0.52–2.5 0.7–2.0

0.86–4.2 1.05–3.0

R = –1

.10

0.25–0.75

0.18–0.52

LNLCFW (A)

R = –1

.11b

1.6

0.45

LNLCFW (C)

R = –1

.11b

1.0

0.87

LC Lap Jt

R = –1

.11c

0.35

0.14

0.9

0.37

Fillet welded edge attachment

R R R R

.12

1.0 1.8 1.05 4.5

0.97 2.45 0.34 2.33

0.95 1.5 1.0 3.7

0.87 2.04 0.32 1.91

= = = =

–1 0 –1 0

reported by White and Lewszuk (1971) (see Figs 7.19, 7.20 and 7.22) because the scatter was uncharacteristically large. The values shown in Table 7.5 were therefore derived after ignoring the outlying test results and to that extent they should be regarded as approximations. In addition to those results it may be noted that Agerskov and Ibso (1993), Marsh et al. (1975), Tilly and Nunn (1980) and Tubby and Razmjoo (1991) also carried out tests under narrow band loading on various types of fillet welded joints. Insufficient information was given to enable the various values of RG to be calculated but, taking all four investigations together, the mean value of Σ n ranged N from 0.65 to 1.08 and the overall range of individual values was from 0.19 to 2.38. Apart from the low value of 0.19 the remainder of the values are in good agreement with those shown in Table 7.5. Ignoring the scattered results reported by White and Lewszuk, the most obvious feature of the values shown in Table 7.5 is the large difference in the magnitudes of RG between the tests carried out at R = 0 and at R = –1. Overall, the range of values at R = –1 was 0.14–0.97 while at R = 0 it was 1.91–2.45. For Σ n they were 0.25–1.6 and at R = –1 and 1.5–4.5 at R = 0. N For higher values of R, Σ n ranged from 0.19 to 2.0. N

7.4

Tests under wide band loading

Unfortunately, useful tests relating to wide band loading appear to be remarkably sparse. Many of the results which have been obtained have not been associated

222

Cumulative damage of welded joints

with comparable tests under any other type of loading, even constant amplitude loading, so they are comparatively useless as it is obviously impossible to draw any conclusions from the results. However, returning to the work discussed at the beginning of the chapter, in which tests were carried out using two sizes of loading block and four different types of loading, including wide band, there was some doubt as to whether the wide band results were wholly realistic. Examination of Fig. 7.8 shows that, in many instances, the small stress ranges in the spectrum were applied in such an order that the mean stress of successive cycles increased progressively over most of the groups of small cycles. This was done deliberately, since it was anticipated that such an order of loading was likely to be the most damaging, but on the other hand it was certainly possible that, since the groups of small cycles were essentially attached to the trough of the preceding large cycle, the large cycle might tend to ‘blanket’ the small cycles. In order to investigate this problem some further tests were carried out using two other wide band spectra for which the exceedence diagrams are shown in Fig. 7.26. In both instances the block length was 128 cycles and spectrum WR7440 was convex upwards while spectrum WB 1151 was slightly concave upwards; in order words WB1151 had a somewhat greater proportion of small stress ranges. In both of these spectra the initial basic tests (referred to as Type A1) were carried out with the peak of each cycle applied before the corresponding trough. However several variants were also used (see Fig. 7.27). • Type A2: the whole spectrum was applied in reverse order (i.e. the last cycle of the basic spectrum was applied first) so that troughs were now applied before peaks. • Type A5: the whole spectrum was applied ‘upside down’ (i.e. with peaks and troughs reversed). With this type of loading troughs again occur before peaks, but the peak stress range is still applied at R = 0. • Type A6: the same as Type A2 but again upside down. In this case peaks precede troughs. Four more variants (A3, A4, A7 and A8) were identical to A1, A2, A5 and A6 respectively, but with the mean stress of the whole spectrum shifted so that the peak range was at R = –1 instead of at R = 0. It will be noted that, in terms of a rainflow or reservoir count of stress ranges, none of these variations have any effect; all the spectra give the same cycle counts. For comparison purposes tests were carried out using each of the eight spectrum variants at two levels of peak stress range, 275 and 175 N/mm2, and a summary of the results is shown in Table 7.6. The first point that is immediately obvious is that for both stress ratios and both levels of peak stress the concave upwards spectrum gave, as anticipated because of its higher proportion of small stress ranges, a much larger life to failure, typically by a factor ranging from about 1.6–1.9. This was true not only for wide band

The influence of narrow band, wide band and service loading

223

150 12

125

8 mm leg length fillet welds on each side of main plate

12

90 Dimensions in mm

1.0

0.8

Relative stress

Spectrum 7440 0.6

0.4

Spectrum 1151 0.2

0 1

10 Number of exceedences

100 128

7.26 Tests under wide band loading.

loading but also when all stress cycles were applied at the same stress ratio. On the other hand the corresponding values of Σ n are remarkably similar N to each other. Overall the values ranged from 0.50–1.03 under wide band loading, from 1.24–1.39 when all cycles were applied at R = 0 and from 0.71 to 0.82 when all cycles were at R = –1. It will therefore be seen that for these two relatively short block length spectra the values of Σ n were consistently N

224

Cumulative damage of welded joints B

B

F

F

H

H

A

A D

D

G

G E

E

C 1. Basic spectrum (Type A1)

C 2. As 1., but in reverse order (Type A2)

c

c

e

e g

g d

d

a

a h

f

b 3. As 1., but upside down (Type A5)

h

f

b 4. As 3., but in reverse order (Type A6)

7.27 Variants of the basic ‘wide band’ spectra. (Note: all the above variants ran from left to right.)

*Mean of three results

Tests at R = –11.22 All cycles at R = –1 A3 basic wide band A4 in reverse order A7 as A3, but upside down A8 as A4, but upside down Mean of wide band

Tests at R = 0 All cycles at R = 0 A1 basic wide band A2 in reverse order A5 as A1, but upside down A6 as A2, but upside down Mean of wide band

0.82 0.84 0.77

490

492

450

0.77

309

0.58

0.70

284

343*

0.77

309

0.73 0.83

0.78

313

428 487*

1.31 0.82

N

Σn

528 330

Life ×103

WR7440

1.07

1.17

1.14

0.81

1.02 1.16

1.36

1.25

1.36

1.38

2.33 1.45

RG

Peak range = 275 N/mm2

738

609

725

765

818 853

583

520

548

614*

956 649*

Life ×103

WB1151

0.74

0.61

0.73

0.77

0.82 0.86

0.85

0.76

0.80

0.90

1.39 0.94

N

Σn

Table 7.6 Comparison between results for spectra WB1151 and WR7440

1.13

0.93

1.11

1.17

1.25 1.30

1.65

1.47

1.55

1.74

2.71 1.84

RG

1.64

1.24

1.51

2.23

1.91 1.75

1.89

1.88

1.77

1.96

1.81 1.97

Life ×103 N

Σn

1.00

0.73

0.89

1.33

1.12 1.04

1.10

1.08

1.04

1.15

1.06 1.15

Ratio 1151 7440

1.08

0.79

0.97

1.44

1.22 1.12

1.21

1.18

1.14

1.26

1.16 1.27

RG

1340

1546*

1396

1021*

1394 1397*

1164

1203

1251

1108

1937 1094

Life ×103

WR7440

0.72

0.83

0.75

0.55

0.75 0.74

0.74

0.77

0.80

0.71

1.24 0.70

N

Σn

1.00

1.15

1.04

0.76

1.04 1.04

1.32

1.36

1.42

1.25

2.19 1.24

RG

Peak range = 175 N/mm

226

Cumulative damage of welded joints

less than 1.0. In terms of the area rule, however, only 9 of the 27 individual test results gave values of RG less than 1.0, the overall ranges of results being 1.24–1.84 when the peak range was applied at R = 0 and 0.69–1.30 where it was at R = –1. There was certainly some tendency for the value of RG to be somewhat higher for the concave upwards spectrum (WB1151) than for spectrum WR7440, although the difference was not great. (Note that the reason for the apparent differences between the ranges of values quoted here and in Table 7.6 is that some of the values in the table are the means of three individual results). In order to elucidate the effect of block length under wide band loading, it is interesting to compare the above results with those obtained, for the same peak stresses, using block lengths of 1024 and 131 072 cycles. The relevant results are summarised in Table 7.7 and in Fig. 7.28. From these it seems clear that the value of Σ n appears to remain virtually constant over N the range of block lengths from 128 to 131 072, but it is at least possible that there may be a rise or fall in value between 1024 and 131 072 cycles; in that region there is at the moment no data. In contrast, so far as the area rule is concerned, there appears to be no major change up to 1024 cycles, but there is a very large decrease in the region 1024 up to 131 072 cycles. Another important conclusion which can be drawn from the results shown in Table 7.6 is that the order of application of the peaks and troughs seems to have little effect on life. However, when all the cycles were applied at R = 0 (i.e. not in wide band form) Σ n ranged from 1.32 to 1.39, implying a N reduction of (typically) about 45% due to wide band loading. At R = –1, however, there was no significant difference between the values of Σ n N under wide band loading and when all cycles were applied at R = –1, where n was typically 0.75; in that instance both types of loading gave low ΣN values of Σ n . It will be noted that, even through these results relate to a N different block length, they are remarkably similar to the results shown in Figs 7.11–7.14. As noted previously, the investigations by Haibach and by Overbeeke both involved tests using the Gauss spectrum in both narrow and wide band forms. The results obtained under narrow band loading were considered above. Under wide band loading Haibach obtained almost identical results (see Fig. 7.24) to those obtained under narrow band, all the individual results giving values of Σ n less than 1.0 and the mean value again being 0.39. In N the case of Overbeeke’s tests however, the wide band results tended to give rather greater values of Σ n than were obtained under narrow band loading N (see Fig. 7.25), although the scatter was somewhat increased.

N

1.31–1.39 1.57 3.24

2.33–2.71 2.13 1.6

Values of RG 128 1024 131 072

Σn

2.19–2.56 1.95 1.66

1.24–1.32 1.43 3.22 1.25–1.99 1.01 0.46

0.70–1.02 0.74 0.90 1.24–144 1.06 0.39

0.66–0.8 0.78 0.76 1.02–1.25 1.03–1.14 0.38

0.73–0.82 1.07–1.18 1.17

275 N/mm2

1.04–1.07 0.99–1.30 0.39

0.71–0.75 1.02–1.35 1.22

175 N/mm2

275 N/mm2

275 N/mm2 175 N/mm2

All cycles at same stress ratio

Wide band

All cycles at same stress ratio 175 N/mm2

R = –1

N

0.75–1.34 0.80–0.83 0.36

0.54–0.96 0.83–0.85 0.95

275 N/mm2

Wide band

0.69–1.43 0.94–1.05 0.34

0.50–1.03 0.97–1.09 0.91

175 N/mm2

Σ n and RG under wide band loading and with all cycles at same stress ratio

R=0

128 1024 131 072

Values of

Block length (cycles)

Table 7.7 Influence of block length on values of

228

Cumulative damage of welded joints

1.0

Value of

Σ

n N

1.5

0.5

Value of RG

0 1

10

100 103 Block length (cycles)

104

105

2.0

Legend

1.5

R = 0 Filled points R = – 1 open points Peak range = 175 = 275

1.0

0.5

0 1

10

100 103 Block length (cycles)

104

105

7.28 Results for specimens with fillet welded edge attachments under wide band loading showing effect of block length.

7.5

Tests under service loading spectra

From the fatigue testing point of view the types of loading in which there has been most interest seem to have been those associated with transport and with offshore structures. So far as transport is concerned the loading of interest has been both that suffered by the vehicles themselves and the loading applied by the vehicles to such structures as bridges. In the case of offshore structures interest has primarily centred on the effect of wave loading and

The influence of narrow band, wide band and service loading

229

was triggered, in particular, by the requirement to produce structures suitable for service in the North Sea, which has a much more aggressive wave environment than had been encountered by earlier structures in, for example, the Gulf of Mexico. Dealing firstly with ‘transport’ loading, it is well known that vehicle manufacturers have spent, and do still spend, a considerable amount of time and energy defining loading spectra that their vehicles are expected to encounter and carrying out associated fatigue tests, but in general such work is regarded as commercially confidential and relatively few details of it have been published. However, a substantial amount of information has been published relating to fatigue of bridges. As was noted in Chapter 6, much of the work in the USA under Rayleigh spectra was aimed primarily at bridges, but American interest was centred mainly on the behaviour of the main longitudinal beams and, in particular, on the strength of beams with welded cover plates, in which some service fatigue failures occurred. In passing, it has to be added, however, that the tests appeared to involve loading block lengths that were much too short to be realistic so some account needs to be taken of that in assessing the significance of the results. In Britain, and to a lesser extent in Europe, interest centred more on the fatigue behaviour of orthotropic decks and details of the British work, much of which involved constant amplitude loading, has been summarised elsewhere. The main stimulus for the development of orthotropic decks for steel bridges in Europe stemmed from the desire to reduce to a minimum the dead weight of the superstructure, particularly in the case of long span and lifting bridges. One way to assist in the achievement of that objective is, of course, to avoid the use of heavy bridge decks. Although there is more than one approach to that problem, a fairly obvious one is to make use of a welded steel deck plate, since that eliminates the relatively heavy concrete deck which is an inevitable feature of composite construction. Typically a concrete deck might be expected to weigh about four times as much as an orthotropic steel deck. The orthotropic deck consists of a deck plate supported in two mutually perpendicular directions by a system of transverse ‘crossbeams’ and longitudinal stiffeners. It may therefore be likened to a plate with dissimilar elastic properties in the two directions; in other words it is effectively an ORTHOgonal anisoTROPIC plate or, for brevity, an ‘orthotropic plate’. In this form of construction the deck itself can form the top flange of the main load-carrying member making up the bridge; in some instances it has, for example, formed the top flange of a box girder. The basic layout of an orthotropic bridge deck is shown in Fig. 7.29. The most notable variant is the form of the longitudinal stiffeners which may be either open ribs, such as flat bars, angles or bulb sections, or closed sections (usually of trapezoidal, V or rounded sections). Clearly the latter are much stiffer in torsion.

230

Cumulative damage of welded joints Wearing surface

Types of stiffeners Types of stiffeners (a) Deck with open stiffeners

(b) Deck with closed stiffeners

7.29 Two basic types of orthotropic steel bridge decks.

So far as is known, the first use of orthotropic decks occurred in Germany after the Second World War, when there was a need to rebuild various longspan bridges against a background of steel being in short supply. The first such bridge is believed to have been the Kurpfalz bridge over the River Neckar in Mannheim (1950), while the first suspension bridge to have an orthotropic deck was the Cologne–Muelheim Bridge over the Rhine, completed in 1951. Both these bridges had open-rib longitudinal stiffeners; the use of torsionally stiff closed ribs came a few years later. Further impetus to the use of orthotropic decks was provided by their relatively low weight, which made them attractive where a high degree of pre-fabrication or rapid erection was required. In addition, the development of weldable high-strength steels enabled further desirable reductions in deck weight to be obtained. As far as Britain is concerned the first two major bridges to have orthotropic decks were the Forth road bridge, opened in 1964, and the Severn crossing, opened in 1966. The latter consists essentially of three adjacent structures, the Severn bridge, Beachley Viaduct and the Wye bridge, but all have the same design of orthotropic steel deck. More recently there have been several others, such as Avonmouth (1974), Erskine (1971), Humber (1981) and Kessock (1982), to name a random selection. A summary of some of the joints which can and do exist in orthotropic decks is shown in Fig. 7.30. It is obvious that most of these are potentially liable to cause fatigue problems, and fatigue cracking has indeed been reported in several bridge decks in many different countries. Among others it certainly has occurred, for example, in France, Germany, Holland, Turkey and the UK and has been found in most of the details outlined in Fig. 7.30. In terms of loading there is a major difference between the main longitudinal girders of a bridge and an orthotropic deck. In general, details on the main

The influence of narrow band, wide band and service loading

231

Transverse deck plate butt weld Longitudinal trough to crossbeam weld

Longitudinal deck plate butt weld

Longitudinal trough to deck plate weld

Web of box to deck plate weld

Trough splice welds Crossbeam to deck plate weld

Alternative connections

Crossbeam Diaphragm

7.30 Main welded connections in a typical orthotropic bridge deck.

girder, such as the end of a partial length cover plate, will have relatively long influence lines so that, in the fatigue context, each passage of a truck is likely to produce a single loading cycle. However, for the details in an orthotropic deck the influence lines are usually short, so that each wheel or axle will produce a cycle. In consequence, details in the deck will suffer far more cycles than details on the main girder flange. Indeed, it is often said that the closer one gets to the wheels the greater is the potential fatigue problem. Clearly, knowledge of the expected traffic loading is an obvious essential prerequisite for the adequate fatigue design of an orthotropic deck, or indeed any other highway structure. However, with an orthotropic deck the problem is more complicated than simply requiring a knowledge of the total traffic flow. Since fatigue cracking, if it occurs, is essentially caused by local loading conditions, there is certainly a need to know the distribution of loading between traffic lanes and, indeed, the lateral distribution of wheel loads within a lane. In addition, since the stiffening effect of the surfacing is dependent upon temperature, it is necessary to know the likely distribution of traffic flow over the year and also through the day. Nevertheless, the starting point obviously has to be the Standard Load Spectrum and information on the number of commercial vehicles per traffic lane per year. This will necessarily vary between countries, and in Britain it is defined in the Bridge Design Standard as a spectrum of 106 vehicles ranging from five-axle articulated trucks to two-axle rigid trucks, with a variety of axle loads in each case; smaller vehicles, such as cars, are assumed

232

Cumulative damage of welded joints

to make an insignificant contribution to fatigue damage. The Standard also defines the number of traffic lanes and the annual traffic flow per lane to be assumed for each type of road. Thus the British assumption is that a dual carriageway motorway with three lanes per carriageway should be designed for an annual flow of 2 × 106 HGVs in the slow lane and 1.5 × 106 in each adjacent lane. However, it is notable that, since that Standard was written, the allowable gross weight of articulated vehicles has increased significantly and there has also been the advent of a large number of six-axle lorries. Designers obviously have to bear in mind the potential impact of likely changes in traffic on the expected life of the structure. Given the Standard Load Spectrum and the corresponding distribution of axle loads for each type of vehicle it is obviously possible to deduce a realistic axle load spectrum. In fact in the British bridge design Standard (BS5400) that is also specified and is presented in terms of the number of axles corresponding to each axle load, the whole spectrum containing about 2.8 × 106 axles. A comparison between that spectrum and spectra measured on three British bridges is shown in Fig. 7.31. The most significant difference is that the BS5400 spectrum contains no axle loads exceeding 100 kN (the legal axle load limit in the UK) whereas all three bridges showed some axle loads exceeding that limit; in the case of Forth Bridge 2.9% of axle loads were above the limit. Obviously, if the bridge designer expects a significant number of axles to exceed the legal limit he should consider using a nonstandard spectrum for his calculations. Incidentally, it is often assumed that single-tyred front wheels produce far less damage than twin-tyred rear wheels. However, in an orthotropic deck a front (single) wheel may well produce higher stresses than a more heavily loaded twin rear wheel. Computer modelling of this problem, allowing for the potential transverse distribution of vehicle positions on the bridge, showed that, at least in some positions (such as adjacent to the longitudinal stiffeners), the front wheels can in fact produce far greater fatigue damage than the rear wheels. Clearly this is a variable which needs to be taken into account. The lateral distribution of wheel loads is obviously an important factor. In the UK it has been found that about 88% of HGVs travel in the slow lane and 12% in the adjacent lane. The number in the fast lane of a three-lane carriageway (illegally!) is extremely small. However, since stresses are very sensitive to the wheel position it is also necessary to consider the lateral distribution within a traffic lane. A summary of the measured positions of the offside rear wheels of commercial vehicles as found on three bridges is shown in Fig. 7.32, where normal curves have been fitted to the data. The mean distance from the outer edge of the lane and the standard deviation (sd) of the calculated curves are as shown in Table 7.8. The differences are probably due to drivers’ reactions to the layout of the carriageway. For example, in the Wye Bridge the footpath or cycle path

The influence of narrow band, wide band and service loading

233

160

140

120

Axle load (KN)

100

80

60

40

20

0 0.01

Forth bridge Manchester road bridge Wye bridge M6 BS 5400 0.1 1 10 Percentage of axles (log scale)

100

7.31 Comparison between British axle load design spectrum and corresponding spectra measured on four British bridges.

restricts the view of the river, while in the Forth Bridge the Firth can easily be seen 50 m below, so vehicles tend in that case to travel further from the edge. Finally, in view of the immense influence of the stiffness of the surfacing, which is very dependent on temperature, it is necessary to know how traffic is distributed in relation to temperature. In effect this means knowing the distribution with respect to time of day, since traffic flow is not uniform through the day. Clearly this distribution will vary from site to site. Often there will be a morning peak in the region of 07.00–09.00 h and an afternoon peak at 16.00–18.00 h, but that will not always be the case. In general, however, it has been found that surfacing (38 mm of mastic asphalt) tends to increase the life of an unsurfaced deck by a factor of about three.

234

Cumulative damage of welded joints

60

Stress level (1 level = 3 N/mm2)

50

40

30

20

10

0

1

10

100 103 Number of exceedences

104

105

7.32 Exceedence diagram for theoretical deck to box web spectrum.

Table 7.8 Mean distance of lorry offside rear wheels from outer edge of lane Bridge

Mean distance from outer edge of lane (mm)

sd (mm)

Wye Forth Manchester Road

894 592 716

238 219 250

Turning now to the fatigue tests which have been carried out on orthotropic deck details, results have been obtained for: (a) the deck to longitudinal box-web joint using both the theoretically derived and the measured spectrum relating to Avonmouth Bridge (UK). The exceedence diagram for the measured spectrum is shown in Fig. 7.33. This was based on 21 weeks of records, seven in summer (so as to include the effect of high temperatures on the stiffening effect of the deck surfacing), seven weeks simulating spring and seven weeks in the autumn. (b) the trough splice joint in the same bridge, using the measured spectrum adjacent to the splice on the soffit of the trough; the exceedence diagram is shown in Fig. 7.34. (c) the trough to cross-girder joint at the soffit of the trough using the theoretically derived spectra both for the Avonmouth Bridge and for the Rheden Bridge (Holland). For the latter tests the magnitude of the stresses

The influence of narrow band, wide band and service loading

235

60

Stress level

50 40 30 20 10 0 1

10

100

103 104 Number of exceedences

105

106

7.33 Exceedence diagram for Avonmouth Bridge measured at deck to box web spectrum. 60

Stress level

50 40 30 20 10 0 1

10

100

104 103 Number of exceedences

105

106

7.34 Exceedence diagram for Avonmouth Bridge measured at trough splice spectrum.

was increased and small stress ranges (less than 50 N/mm2) were ignored so as to produce fatigue cracking in a shorter time. Exceedence diagrams for the two Rheden Bridge spectra are shown in Fig. 7.35; the Avonmouth bridge spectrum was of the same general type (i.e. slightly convex upwards). For both the deck to longitudinal box-web joint and the trough splice the tests gave values of Σ n which were greater than 1.0, but the trough to N cross-girder joints, tested under the theoretically derived spectra, gave values which were significantly less than 1.0 (down to 0.39). It is probable that the reason for this is that the trough to cross-girder spectrum is ‘wide band’, with a large number of small stress cycles at a wide variety of high mean

236

Cumulative damage of welded joints

Proportional stress range

1.0

0.8

Test spectrum

0.6

0.4

Calculated from simulated traffic flow

0.2

0 1

10

100 103 Number of exceedences

104

105

7.35 Exceedence diagram for theoretical spectrum for trough to cross girder joint in Rheden bridge (Apex of trough).

stresses. In contrast, the deck to box-web spectrum consisted almost entirely of cycles with a constant lower limit stress. Interestingly, for both spectra, the area rule gave values of RG which were, except for one specimen, greater than 1.0. A surprising feature of the results was that the trough splice spectrum was also of the wide band type and yet gave values of Σ n which were greater N than 1.0. A possible reason for that behaviour is that in the very hot summer weather there was an unexpectedly large number of higher than expected stresses. It may well be that the effect of those high stresses was to introduce compressive residual stresses around the ‘notch’ and thereby give increased lives. If that is the correct explanation it would probably be wise to guess that, in the absence of very hot summer weather, the life might be considerably lower. So far as the work relating to offshore structures is concerned, virtually all the variable amplitude testing has been carried out using loading spectra which were intended to simulate North Sea wave loading, and the tests have mainly been performed either in Britain or the Nordic countries. The spectra are described in some detail in Appendix B. The test results which have been obtained in the British test programme are summarised in Fig. 7.36(a) and in the Nordic programme in Fig. 7.36(b). The main objective of the British tests, primarily carried out using the C/12/20 spectrum, was to investigate the influence of the environment and, to that end, results were obtained for three different environmental conditions; they were free corrosion, cathodic protection at –0.85 V and intermittent immersion (simulating conditions which might exist in the tidal or splash zones near the water surface). In each case the values of Σ n were based upon constant amplitude fatigue tests N

Free corrosion

Free corrosion Cathodic protection (–0.85 V) Intt immersion

Air

R=–1 CP = – 0.86 V

0.1 Value of

Σ

n N

1.0

5.0

7.36 Summary of test results obtained in the (a) British offshore test programme and (b) Nordic offshore test programme.

k butt, c/12/20 toe ground bending

Air CP = – 0.85 V

Wide band 54 R = – 1

CP = – 0.86 V

Wide band 54 R = – 1

R=–1

c/12/20

k Butt

(–0.85 V) Intt immersion

440 N/mm2 Cathodic protection

Smax =

R=–1

R=–1

Intt immersion

Bending

Bending

c/12/20

Free collosion

Smax =

c/12/20

LNLCFW bending

k Butt

Free corrosion Cathodic protection Intt immersion

R=–1

440 N/mm2 Cathodic protection

Air

R=–1

c/12/20

LNLCFW Axial

Environment

Stress ratio

Joint and Spectrum loading

PmmoD 64

BRoAD 64

PmmoD 64 BRoAD 64

TNLCFW

(T = 10)

TNLCFW (T = 5)

7.36 Continued

PmmoD 64 Pm 32 BRoAD 64

TNLCFW (T = 5)

Narrow 64

PmmoD 64 Pm 32 BRoAD 64

Narrow 64

BRoAD 64 BRoAD 64

LNLCFW

TNLCFW (T = 10)

SP2 SP2 SP3

VAmP SP1 SP2 SP3 SP4 (SP5)

Spectrum

LNLCFW

LNLCFW

Joint and loading Stress ratio

Air

Air

Environment

Value of

0.1

Σ

n N

300

1.0

5.0

The influence of narrow band, wide band and service loading

239

carried out under the same conditions as the corresponding variable amplitude tests; the relevant constant amplitude S-N curves for K butt welds are summarised in Fig. 7.37. It will be noted that ‘free corrosion’ results in a life reduction, compared to air, by a factor of about two. The tests by Holmes and Booth (1981) involved two different types of joint, a longitudinal non-load-carrying fillet weld (LNLCFW) and a K butt weld, both tested in bending at approximately 0.16 Hz. For each joint results were obtained both at R= –1 and also at high mean stress. The constant amplitude tests were, however, only carried out at R= –1, so the values of n obtained at high mean stress were calculated using the ‘wrong’ S-N ΣN curve. As can be seen from Fig. 7.36(a) there was a distinct tendency for the longitudinal fillet welded joints, in particular, to give low values of Σ n at N high mean stress, but that can probably be attributed largely to the use of the ‘wrong’ S-N curve. Thus, allowing for that, it will be seen that for most environmental conditions the values of Σ n were typically in the range 1.0– N 2.0; this suggests that environment, of itself, is not a critical factor in determining the value of Σ n , provided that the S-N curve used in the calculation is N realistic for the conditions under consideration. The Nordic test programme (Fig. 7.36(b)), on the other hand, only involved tests in air and was concerned primarily with defining the influence of spectrum ‘shape’. One part of it, using both longitudinal and transverse non-loadcarrying fillet welded specimens, involved comparative tests using both narrow and broad band loading and also loading based on the Pierson-Moscowitz wave elevation spectrum. At first sight there appears to be remarkably little difference between the values of Σ n obtained with any of these three N n spectra, most results giving Σ less than 1.0; in particular there appears to N be little evidence to suggest that broad band loading gave significantly shorter lives than narrow band loading. On the other hand closer inspection shows that both the lower limit values of Σ n and the mean values do tend to N decrease as the loading changes from narrow to wide band, with the PiersonMoscowitz spectrum giving results in between. Indeed Agerskov and Ibso (1993) suggest that Σ n appears to follow the relationship N n Σ N = 2I – 1 where I is the irregularity factor (number of rising zero crossings divided by the number of peaks). For the three spectra involved in these tests the values of I were

Range of stress, N/mm2

300

60 104

70

80

90

100

120

140

160

180

200

220

240

260

280

2

3 4 5

105 1.5 3 4

5

Endurance, cycles

2

106

Intermittent immersion

1.5

2

3

4

CP = – 0.85 V

5

107

7.37 Comparison of constant amplitude S-N curves for K butt welds in various environments (R = –1).

1.5

Free corrosion

Air

The influence of narrow band, wide band and service loading

241

1.2

1.0

0.8

T = 5 mm T = 10 mm

0.6

0.2

0 0.5

0.6

0.7 0.8 Irregularity factor (I)

Narrow 64

BRoAD 64

0.4

n = 2I – 1 N

PmmoD 64

Σ

Pm 32

Value of

Σ

n N

Transverse non-loadcarrying fillet welds

0.9

1.0

7.38 Relationship between Σ n and irregularity factor obtained in the Nordic test programme. N

NARROW64 = 0.987 PMMOD64 = 0.817 BROAD64 = 0.745. Figure 7.38 shows the relationship between the experimental values of n and I for the transverse non-load-carrying fillet welds (the bands indicate ΣN the range of values and the ‘dot’ the mean value for the series). Clearly the scatter is large but the relationship does show promise, although it also has to be noted that some other results for which the values of I are known, such as those obtained by Gurney (1988b), do not fit the proposed relationship. It is also clear that the formula cannot be correct for low values of I, since (according to the formula) Σ n becomes zero at I = 0.5. Some of the spectra N used by Gurney had a large number of small stress ranges at very high and very low stress ratios (i.e. with rising ranges that did not cross the mean stress) so that in those cases the value of I was very low (less than 0.5), but the values of Σ n did not tend to zero. The other notable feature of these N tests was that in most instances there was a definite tendency for Σ n to N decrease as the stress level decreased, as also found by Gurney. With small specimens, however, the opposite was the case.

242

Cumulative damage of welded joints

The second part of the Nordic tests programme involved tests on longitudinal non-load-carrying fillet welds using spectrum VAMP and variants of it. All the spectra had the same range pair count and the same value of minimum stress range (p = 1/6) but there were variations both in mean stress and in the value of I. However, these seem to have had remarkably little effect on the values of Σ n which were obtained, most results being very similar to each N other. The only obvious major difference is provided by spectrum SP4, which gave very high values of Σ n , in the region of 4–5. That was the only N spectrum in which all cycles were at R = 0, the others being at higher or lower stress ratios (see Appendix B), but there is no obvious explanation for this behaviour.

7.6

Summary

A comparison between test results using two spectra, with block lengths of 1024 and 131 072 cycles, with each spectrum arranged both in narrow and wide band forms, showed three significant features: 1. Under tensile loading wide band loading consistently gave values of Σ n N n which were less than 1.0, while narrow band loading gave Σ greater N than 1.0; 2. At R = –1, however, the difference between the two types of loading was much smaller, although wide band loading still gave Σ n less than 1.0 N and was more damaging than narrow band; 3. In terms of life, narrow band loading gave a longer life at R = 0 than at R = –1, for both block lengths. Under wide band loading the opposite was true. In the same investigation block programme loading with the same two spectra gave very optimistic values of Σ n in tests at R = 0, particularly with the N longer block length (131 072 cycles). At R = –1, however, the difference compared with the other types of spectra was very much smaller. Nevertheless, this optimism of block programme loading has been found by other investigators, which is why results obtained with this form of loading were ignored when preparing the British fatigue design rules. In other investigations narrow band loading gave values of Σ n ranging N from about 0.25 to 1.6 at R = –1 and from 1.5 to 4.5 at R = 0. In principle this is in general agreement with the results outlined above. For comparison the area rule gave RG ranging from about 0.14 to 0.97 at R = –1 and from 1.9 to 2.45 at R = 0. It should, however, be noted that the great majority of the

The influence of narrow band, wide band and service loading

243

tests on as-welded joints under narrow band loading were carried out at R = –1 and virtually all of those gave values of Σ n either substantially less N than 1.0 or approximately equal to 1.0. Relatively few results have been under an applied tensile mean stress, although those that do exist have generally given values of Σ n which were somewhat greater than 1.0. N Meanwhile, wide band loading seems consistently to have given low values of Σ n , with nearly all results less than 1.0. Overall, values obtained N in the British tests have ranged down to about 0.55 (although with a few even lower results) and there was no great difference between the results obtained at R = 0 and at R= –1. On the other hand Haibach obtained considerably lower values, with a mean of 0.39, which is more in line with the low British results. In contrast, the area rule has tended to give somewhat lower values of RG at R = –1 than at R = 0, although that was not always the situation. Overall the values of RG ranged from about 0.28 to 1.99, but the great majority lay in the range 1.10 to 1.60; at R = –1 the overall range was about 0.28 to 1.56, with the majority of results between 0.5 and 1.35. In other words, the values of RG were rather lower at R = –1 than at R = 0. It was interesting to note that, compared with tests in which all cycles were applied in random order, wide band loading gave a reduction in life of about 50% at R = 0, but there was no great difference at R= –1. Under wide band loading the order of application of the peaks and troughs appeared to have very little influence on life. The tests carried out under service loading spectra gave similar results but do, however, highlight the need to view the laboratory test results with a little caution. For example, there certainly seems to be some evidence that unexpected peak stresses may modify the local residual stresses sufficiently to reduce, or possibly to increase, fatigue damage, even under wide band loading. However, it would probably be unwise to assume that such changes will necessarily occur.

8 The influence of cycles of small stress range

8.1

Introduction

It is obvious that one of the crucial decisions that has to be made before Miner’s rule can be used is what constant amplitude S-N curve should be chosen as a basis for the calculation. Most usually the values of N in the summation of Σ n tend to be derived from the mean –2 standard deviations N design curve given in the relevant design standard. However, so far as the treatment of the low stresses in the spectrum is concerned, there are considerable variations between standards. In most of the British rules (e.g. BS 5400, Part 10:1980 and BS 7608:1993) it is assumed that, for joints in air, the S-N curve is bent at 107 cycles from a slope m to a shallower slope (m + 2), so that the values of N for the lower stresses are increased (see Fig. 8.1). Below the bend it is assumed that the S-N curve continues linearly downwards (on a log-log basis) ad infinitum. On the other hand if all stress ranges are less than the constant amplitude ‘fatigue limit’, which is assumed to be the stress range corresponding to 107 cycles, then the possibility of fatigue failure can be ignored. In the existing European design standard, however, the bend is assumed to occur at 5 × 106 (rather than 107) cycles and the curve is then assumed to become horizontal at 108 cycles. In other words, cycles at stress ranges smaller than that corresponding to 108 cycles on the bent S-N curve are assumed to do no damage. For the lowest class of joint (Class G) and the mean –2SD design curve this stress is 13.5 N/mm2. At the other extreme, the American AASHTO bridge rules (1994 edition) require that the S-N curve is extrapolated ad infinitum with no slope change. Relaxation is permitted if the maximum stress range due to traffic loading does not exceed half the constant amplitude fatigue limit, which corresponds to endurances from 2 × 106 to 2 × 107 cycles depending on the weld detail. In such circumstances, the fatigue life can be assumed to be infinite. It has to be noted, however, that the justification for the use of a bent S-N curve is rather limited, since extremely few variable amplitude (or even constant 244

The influence of cycles of small stress range

245

Slope = m

Log S

Slope = m + 2 N is increased in this ragion

Log N

107

8.1 General form of S-N curve assumed for cumulation damage calculations in the British design rules (BS 5400 and BS 7608).

amplitude) fatigue test results have been obtained under long life conditions. Until quite recently hardly any data had been obtained for lives exceeding about 2 × 107 cycles. In consequence, it is largely unknown how much, if any, damage is done by cycles below the constant amplitude fatigue limit and how small such cycles have to be before they can definitely be ignored. One of the earlier investigations to produce any long life data (up to about 4 × 108 cycles) was that reported by Tilly and Nunn (1980) who tested longitudinal non-loading-carrying fillet welded joints under axial loading with a narrow band spectrum. In that instance the use of the bent S-N curve was found to give a good prediction of life and several investigators have subsequently claimed that those particular test data justify the method. However, it should be borne in mind that the investigation involved only a very small number of specimens and one type of loading, so in fact the degree of validation is not very convincing by itself. Shortly afterwards some work by Fisher et al. (1983) cast some doubt on that conclusion, and led to the more stringent approach now embodied in the AASHTO bridge rules referred to earlier. They tested eight large beams with welded cover plates and longitudinal attachments on the web under a Rayleigh load spectrum consisting of 1024 cycles per block all at R = 0, applied in random order. In this work the number of cycles exceeding the assumed constant amplitude fatigue limit varied between about 0.1% and 12%. However,

246

Cumulative damage of welded joints

it was found that, if any of the stress cycles exceeded the constant amplitude fatigue limit, crack propagation was likely to occur and design should be based on the assumption that all cycles would then contribute to fatigue damage and that an extrapolated linear S-N curve should be used. In that investigation failures were mainly in the region 107–2 × 108 cycles. (It should be noted that there are small differences between the assumed values of the constant amplitude fatigue limit in the British and American fatigue design rules, although they are probably not sufficient or invalidate the conclusion being generally applicable). Some ten years later this conclusion was reinforced in another similar investigation (Fisher et al., 1993) on large beam specimens, although in that case the block length of the loading varied from 103 to 105 cycles and in some instances only 0.01% of the cycles exceeded the assumed constant amplitude fatigue limit. Once again some of the lives exceeded 108 cycles. This has been confirmed by Marquis (1995b) both in tests on box beam components under in-plane bending and in work on specimens with longitudinal non-load-carrying fillet welds. In this work all the specimens were tested at R = –1 using a concave upwards spectrum with a relatively large peak range (368 N/mm2), but the spectrum was truncated at three different levels, namely at 33%, 50% and 100% of the measured constant amplitude fatigue limit. (This was equivalent to approximately 10%, 16% and 31% of the peak range). The respective block lengths were 408 000, 99 459 and 3532 cycles with irregularity factors of 0.86. 0.88 and 0.99. In both cases it was found that the use of Miner’s rule in conjunction with a bent S-N curve led to nonconservative damage estimates while the use of a straight line extrapolation of the S-N curve gave a conservative upper bound to the test data. Further confirmation of the fact that the use of a bent S-N curve, as defined in BS 5400 or BS 7608, is unsafe, was obtained by Dahle and Larsson (1992) and Dahle (1993, 1994) who tested both longitudinally welded half scale box beams of the type used in railway bogie frames and also specimens with longitudinal non-load-carrying fillet welds. This finding seems to have been particularly the case in tests using a concave upwards spectrum. In all the above investigations the main objective, as in most other work on cumulative damage, was, in effect, to define the value of the Miner summation, Σ n , as the stresses applied to successive specimens were N progressively reduced. Alternatively, which amounts to much the same thing, one can plot an ‘S-N curve’ for the results relating to the chosen spectrum, based upon the top limit stress in the spectrum, and try to define where it diverges from linearity. Unfortunately, the problem which necessarily arises is that one is committed to very long, and therefore expensive, tests. As a result, that may not be the most appropriate method of defining the influence of the low stresses in a spectrum.

The influence of cycles of small stress range

8.2

247

Block testing of low stresses

An alternative approach, used by Gurney (1983a) was to examine the results in terms of the number of blocks to failure rather than (effectively) the number of cycles to failure. In this context a ‘block’ contained one cycle of the peak stress range together with the appropriate numbers of cycles of each of the lower ranges in the spectrum. The whole spectrum was then formed by repeating the block until failure occurred. As a simple explanation of this approach, consider a very simple loading spectrum consisting of only two levels of stress, σ1 and σ2, with each block consisting of one cycle of each. With only one stress (σ1) present the life will be Nc. But with σ2 also present the number of blocks to failure (i.e. the number of cycles of each of σ1 and σ2) will, if σ2 does damage, be less than Nc. However, if σ2 does no damage the number of blocks to failure will remain Nc, although the life will now be 2Nc (= Nc cycles of σ1 + Nc cycles of σ2). Obviously this explanation assumes no scatter. Hence a method of defining the minimum damaging stress range is to start by testing with a spectrum σ1 to σi (σ1 > σ2 > σ3, etc.) in which all stresses are above the constant amplitude fatigue limit, and are therefore definitely damaging. Then a lower stress range (σj) is added and a check is made on the effect on the life in blocks. This can then be followed by adding a lower stress, σk, etc., until the cutoff point is reached. Alternatively, to reduce testing time, σj can be deleted and replaced by σk, albeit with a somewhat larger number of cycles in order to compensate for the fact that σk will be less damaging than σj. Both approaches were used in the course of the investigation. For this sort of investigation, where one is particularly interested in the influence of the low stresses in the spectrum, it is appropriate that the spectrum should contain a relatively large number of cycles of low stress range and should therefore be of the concave upwards type. It is also useful to employ a spectrum where the theoretical damage caused by each stress level, as measured by the ratio n , is approximately the same. Luckily these two N requirements are consistent with one another, at least under tensile loading, and Table 8.1 shows the spectra which were actually used, specified nondimensionally in terms of p; Fig. 8.2 shows the corresponding exceedence diagrams. It will be noted that, for the top eleven stress levels, the four spectra were identical; they only differed in respect of the lower stress ranges. The majority of the specimens were tested under spectra in which all the stress ranges were fully tensile, with the maximum range applied at R = 0. However some comparative tests were carried out with the peak range applied at R = –1. The spectra were all of wide band form in that the small ranges were at a wide variety of mean stresses, although the mean stress for the whole spectrum was approximately equal to the mean stress of the peak

248

Cumulative damage of welded joints

Table 8.1 Details of test spectra used by Gurney Spectrum

A

B

C

Relative stress

Number of cycles per block

D Relative damage σ3 N

1.0 0.8 0.68 0.62 0.56 0.48 0.4 0.32 0.24 0.2 0.16 0.12 0.1 0.08 0.06

1 2 3 4 6 9 16 30 72 125 244 512 – – –

1 2 3 4 6 9 16 30 72 125 244 – 1000 – –

1 2 3 4 6 9 16 30 72 125 244 578 – 1953 –

1 2 3 4 6 9 16 30 72 125 244 – 1000 – 4632

Total

1024

1512

3043

6144

∑σ N 3

11.854

Predicted number of blocks to failure

11.968

2167

Predicted life Nc

2147

86.37

126.3

12.968 1981

1.0 1.024 0.943 0.953 1.054 0.995 1.024 0.983 0.995 1.0 0.999 0.884 (512) 1.0 1.0 1.0

12.968 1981

234.6

Assuming Σ n = 1.0 N

473.5

Relative stress =

Stress range Peak stress range

Notes: (1) In this table the ‘predicted number of blocks to failure is derived from Miner’s rule assuming σmax = 280 N/mm2 and the mean (British). Class F S-N curve with no bend or cut-off, giving Nc = 25 692 cycles (2) Relative stress = stress range/ peak stress range. 1.0

0.8

0.6

0.4

0.2

A

0.0 1

10

B

C D

102 103 104 Number of exceedences

8.2 Exceedence diagram of test spectra.

105

The influence of cycles of small stress range

249

Load

8.3 Trace showing typical section of the various spectra.

stress range. In fact the ranges between approximately p = 0.25 and p = 0.8 all had mean stresses which were exactly equal to the mean stress of the largest range (i.e. at p = 0.5 when the peak range was at R = 0 or at zero mean stress when the peak range was at R = –1); it was only the smaller ranges which varied widely in mean stress. In all instances the various ranges in a block were applied in random order. The block was then continuously repeated, with the same order of individual cycles, until failure. A trace showing a typical section of one of the spectra is reproduced in Fig. 8.3. It will be noted from Fig. 8.3 that, in the initial tests under tensile loading and also in the tests with the peak range at R = –1, the cycles with small ranges tended to occur in groups with progressively increasing mean stress in each group. In order to try to ensure that this particular arrangement of the small stress cycles did not have a significant influence on the results, some check tests were carried out using the same spectra but with the stress cycles applied ‘upside down’ (i.e. with the peaks becoming troughs and the troughs becoming peaks. In this situation the small ranges occurred in groups with progressively decreasing mean stress in each group although the order of stress ranges in the spectrum was identical to that in the earlier tests. Hence a comparison between the two sets of results should show the relative influence of increasing and decreasing mean stresses in the groups of small cycles. At this stage it is useful to consider what results one might expect if there were no scatter and if all the stress ranges were fully damaging. Let us start by considering two tests, with the same peak range, under Spectra A and B (Table 8.1). The main difference between them is that the lowest stress range, namely p = 0.12, in Spectrum A was replaced by stress range p = 0.10 in Spectrum B, the number of cycles being increased so that the ‘damage’

250

Cumulative damage of welded joints

(measured as p3n) due to the two stresses was nominally the same. (In reality there was a small difference between them, to which we will return later, but for the time being let us assume that they were actually the same, in which case the overall values of ∑(p3n) would also have been equal.) Now, from eqn [6.2] we know that the expected number of blocks to failure, assuming Σ n = 1.0, is given by: N Nc NB = [8.1] Σ (vp m ) where v and n are synonymous, m = 3 and Nc is the expected constant amplitude life under the top limit stress in the spectrum. Hence, for Spectra A and B, both Nc and ∑(vpm) are the same for each spectrum, so that the expected number of blocks to failure is the same for each. The same is also true for Spectra C and D (Table 8.1) but both have an additional stress level as compared with Spectra A and B, so that the corresponding values of ∑p3n are larger and the expected number of blocks to failure is decreased (although, of course, the expected number of cycles to failure would be increased). Returning to the fact that ∑p3n for Spectra A and B were not, in fact identical, the actual expected number of blocks to failure can be deduced by entering eqn [8.1] with the two relevant values of ∑p3n. This results in the life (in blocks) for Spectrum B being 1% less than that for Spectrum A. Meanwhile Spectra C and D give expected ‘lives’ (in blocks) which are 7.7% lower than for Spectrum B. By way of illustration, the expected lives for spectra with a top limit range of 280 N/mm2 (Nc = 25 692 cycles) are summarised in Table 8.1; clearly for different peak stress ranges the only change is in the value of Nc, so that the expected lives will scale pro rata. By carrying out tests with each of the four spectra at three different levels of peak stress range, and by careful choice of those values, it is possible to arrive at a test programme in which the stresses come close to repeating those relating to the next higher stress, but one step removed, and thereby providing a check on the results. The peak stress ranges actually selected were 280, 210 and 168 N/mm2, leading to the bottom limit stresses being as shown in Table 8.2. Thus it will be seen that the bottom limit ranges for Spectrum D with a peak range of 280 N/mm2, Spectrum C with a peak range of 210 N/mm2 and Spectrum B with a peak range of 168 N/mm2 were, in fact, identical. From Table 8.2 it can be seen that, overall, the bottom limit stress in the various spectra ranged from 33.6 to 10.1 N/mm2. For the particular type of specimen that was used (a plate with longitudinal fillet welded edge attachments) the hypothetical bend in the mean-2SD design curve is at 29 N/mm2, so the first specimen, tested under Spectrum A with a peak range of 280 N/mm2, should certainly have been subjected to stresses which were fully damaging. The objective was to try to determine at what point that

The influence of cycles of small stress range

251

Table 8.2 Bottom limit stress ranges for the spectra used in the test programme Top limit stress range, N/mm2 280 210 168

Corresponding bottom limit ranges Spectrum A

Spectrum B

Spectrum C

Spectrum D

33.6 25.2 20.2

28 21 16.8

22.4 16.8 13.4

16.8 12.6 10.1

ceased to be the case. When one reaches a test in which the bottom limit stress does no damage, or even little damage, there should be an increase in the number of blocks to failure. For example, if the lowest stress in Spectrum D (p = 0.06) does no damage it effectively becomes the same as Spectrum B, so the number of blocks to failure should be much the same as for Spectrum B and certainly more than for Spectrum C. The results obtained for as-welded specimens tested with a peak range at R = 0, and with all other cycles fully tensile, are summarised in Table 8.3. From that it will be seen that the numbers of blocks to failure for each of the two pairs of spectra (A and B) and (C and D), were in all instances virtually the same, apart from one specimen tested under Spectrum C with a peak range of 210 N/mm2. However, since this anomaly did not carry through to Spectrum D that can safely be ascribed to experimental scatter. From this set of results it seems safe to conclude that, for those loading conditions at least, all stresses down to 10 N/mm2 (and maybe lower) are fully damaging. That is far lower that the currently assumed ‘knee’ in the relevant mean –2SD design curve which is at 29 N/mm2. Since no obvious cut-off was detected in these tests, and since the expected number of blocks to failure should be the same for Spectra A and B and also for Spectra C and D, it seems reasonable to combine the various pairs and calculate the mean. They are also shown in Table 8.3, after making the 1% correction in the results for Spectrum A to allow for the slightly smaller than required number of cycles at the lowest stress level (p = 0.12) as discussed previously. It is then very clear that the lives under Spectra C and D are consistently smaller, as expected, than the lives under Spectra A and B.

8.3

Comparative tests on stress relieved joints

In these tests all the stress ranges would necessarily have been highly tensile, due to the presence of residual stresses. It was therefore decided to carry out some comparative tests on stress relieved joints and the results are summarised in Table 8.4. They are somewhat more confusing to interpret. Considering first the tests with a peak range of 280 N/mm2, all the stresses in Spectrum A, and probably also in Spectrum B, should have been damaging so both

1024 1512 3043 6144

1024

1512

3043

6144

1024 1512 3043 6144

A B C D

A

B

C

D

A B C D 168 168 168 168

210

210

210

210

280 280 280 280

Predicted

33.6 28 22.4 16.8

20.2 16.8 13.44 10.08

12.6

16.8

21

25.2

33.6 28 22.4 16.8

1600 1391 1161 1299

5379 5377 4626 4660

2766 2729 2701 2787 2315 2809 2597 2336

1412 1450 1228 1186

Actual

Blocks to failure

Maximum Minimum

Stress range (N/mm2)

(b) Tests using inverted spectrum A 1024 280 B 1512 280 C 3043 280 D 6144 280

Block length

Spectrum

Table 8.3 Summary of test results for as-welded joints tested with peak range at R = 0

1230

1488

4643

5353

2514

2733

1207

1425

Mean of (A and B) and (C and D)

0.827

0.867

0.920

0.847

Ratio of Means (C and D) (A and B)

0.74 0.65 0.59 0.65

0.53 0.54 0.50 0.51

0.53 0.52 0.52 0.54 0.49 0.59 0.55 0.49

0.65 0.67 0.62 0.59

N

Σn

Block length

1024 1512 3043 6144

1024

1512

3043

6144

1024 1512 3043 6144

Spectrum

A B C D

A

B

C

D

A B C D 168 168 168 168

210

210

210

210

280 280 280 280

Maximum

20.2 16.8 13.44 10.08

12.6

16.8

21

25.2

33.6 2.8 22.4 16.8

Minimum

Stress range (N/mm2) Predicted

4517 5428 5333 4461

3011 2984 2981 2490 2797 2234 3157 2144 2642

1171 1061 1006 989

Actual

Blocks to failure

Table 8.4 Summary of test results for as-welded joints tested with peak range at R = 0

2393

2695

2643

2962

4897

4972

2544

2803

998

1110

Mean of (A and B) and (C and D)

0.867

0.907

0.899

(C and D) (A and B)

Ratio of Means N

Σn

254

Cumulative damage of welded joints

should have given approximately the same number of blocks to failure. In fact they differed by about 9%, with Spectrum B giving the smaller number. However, if the smallest stress in Spectrum B (p = 0.1) had been nondamaging the life (in blocks) under Spectrum B should have been greater than under Spectrum A, so the 28 N/mm2 stress level must have been damaging. Based upon the result for Spectrum B (1061 blocks) the predicted life for Spectra C and D, allowing for their additional stress level, becomes 979 blocks. The actual results (1006 and 989 blocks) were quite close to that prediction, so it seem reasonable to deduce that the stress range of 16.8 N/mm2 was fully damaging. Turning now to the tests with a peak range of 210 N/mm2, we again find that Spectrum B gave smaller lives than Spectrum A, when they should have been the same. However, as before, if the lowest stress in Spectrum B had been non-damaging that spectrum should have given a longer life, so the level of 21.2 N/mm2 was certainly damaging. For Spectra C and D the results are rather scattered and one might deduce, from a comparison of the results for Spectra B and C that, in this instance, the level of 16.8 N/mm2 is nondamaging, but it is clear from a comparison of the results for Spectrum D with either B or C that 12.6 N/mm2 is certainly fully damaging. The ‘odd’ results for Spectrum C must therefore be ascribed to the single unduly high result of 3157 blocks resulting from scatter. Finally, the results obtained when the peak stress was reduced to 168 N/mm2 are distinctly confusing and they certainly suggest that 10.1 N/mm2 may be non-damaging. Further work is required to prove that point. As was noted earlier a few tests were carried out on as-welded joints at R = 0 with the whole spectrum inverted. The results of these tests, all of which involved a peak stress range of 280 N/mm2, are summarised in Part (b) of Table 8.3. As we saw previously, the comparable tests using the basic (uninverted) spectra gave remarkably similar lives for each of the two specimens subjected to Spectra A and B and also for Spectra C and D. This was not repeated in these tests, although it is worth noting that the average number of blocks to failure for the two pairs of specimens were quite similar. For the basic spectra they were 1431 for Spectra A and B and 1207 for Spectra C and D. For the inverted spectra the comparable results were 1495 and 1230, giving differences of about 4% and 2% on life. It is also notable that, whereas in the tests with the basic spectra specimens B and C gave the longer lives (measured in terms of blocks of failure) in the two pairs of tests (A and B) and (C and D), the opposite was the case in the tests with the inverted spectra. This suggests that although the inverted spectra gave rather greater scatter than the basic spectra, there was no major difference between the two sets of results, which implies that the use of rising or of falling mean stresses in the groups of small cycles had no significant effect on the result. This is in agreement with the results of similar earlier tests using different basic spectra (Gurney, 1988a).

The influence of cycles of small stress range

255

Table 8.5 Summary of results for as-welded joints tested at R = –1 Spectrum

Stress range, N/mm2

A B C D

Minm stress range, N/mm2

Number of blocks to failure, mean

±16.8 ±14 ±11.2 ±8.4

1870 1638 1562 1573

300 280 260 240 220 200 180

Predicted blocks to failure Class G mean S-N curve, (m = 3.0)

Mean expl S-N curve for R = –1, (m = 2.66)

2167 2147 1981 1981

2818 2758 2445 2399

Rutile Low hydrogen

Source Other investigation This investigation Other previous investigations

160 140 120

Class G mean curve

100 90 80 70 60 50 104

105

106

107

Endurance, cycles

8.4 Results obtained for as-welded joints under constant amplitude loading at R = 0.

A few exploratory tests were also carried out using the same loading spectra at R = –1 with a peak stress range of ± 140 N/mm2. Unfortunately the results were very scattered which makes interpretation fairly difficult but Table 8.5 shows a summary of the results for the as-welded specimens. In each case the quoted experimental lives, in blocks, represent the mean of two or three results. To confuse matters still further it was found that, in constant amplitude tests, the S-N curve for R = –1 was steeper than that for R = 0, the slope being m = 2.66 (cf. Figs 8.4 and 8.5). As a result Table 8.5 contains two predicted lives for the two S-N curves. Whichever S-N curve is used, it is clear that Spectra C and D, with their additional stress level, would be expected to give a significant reduction in the number of blocks to failure compared with Spectra A and B. Comparing

Cumulative damage of welded joints

Stress range, N/mm2

256

300 280 260 240 220 200 180 S-N curve for R = 0 160 (from Fig. 8.4) = Class G mean curve 140

Separate investigations

120 100 90 80 70 60 50 104

105

106

107

Endurance, cycles

8.5 Results obtained for as-welded joints under constant amplitude loading at R = –1.

the results for Spectra A and C, which are effectively identical except for the addition of cycles at p = 0.08 in Spectrum C, it is clear that the lower stress must have been damaging since the experimental results did indeed show a large reduction (1562 rather than1870 blocks). It follows that the cycles at p = 0.10 (Spectrum B) must have been damaging, so that the apparent large difference between the experimental results for Spectra A and B can be attributed to scatter. Meanwhile the experimental results for Spectra C and D were virtually identical, as expected, which implies that stress level p = 0.06 (Spectrum D) was fully damaging. Hence, under alternating loading, it appears that stresses down at least to ±8.4 N/mm2 can be regarded as fully damaging. So far as tests at R = –1 on stress-relieved specimens are concerned only one test was carried out with each spectrum, again with a peak range of ±140 N/mm2. The results are shown in Table 8.6. In the corresponding constant amplitude tests it had been found (see Figs 8.6 and 8.7) that, although the S-N curve had the same slope (m = 2.66) as for as-welded specimens the life was increased by a factor of about 1.75. In consequence the lives predicted using that S-N curve were greatly increased compared with those for aswelded specimens. From Table 8.6 it is notable that the numbers of blocks to failure were significantly greater at R = –1 than at R = 0, for the same stress ranges, by factors of between 2.5 and 3.5. For as-welded specimens the difference was only about 1.25. As before, Spectra C and D, with their additional stress level, would be expected to give a significant reduction in the number of blocks to failure

The influence of cycles of small stress range

257

Table 8.6 Summary of results for stress-relieved specimens tested at R = –1 Spectrum

A B C D

Minm stress range, N/mm2

No. of blocks to failure

Corresponding life at R=0

Predicted blocks to failure Class G mean S-N curve, m = 3.0

Expl S-N curve for SR specns, R = –1

±16.8 ±14 ±11.2 ± 8.4

4122 2818 2606 2428

1171 1061 1006 989

2167 2147 1981 1981

5197 5087 4510 4425

Stress range, N/mm2

300 280 260 240 220 200 180 160 140 120

Separate investigations This investigation

S-N curve for as welded specimens (from Fig. 8.4) = Class G mean curve

100 90 80 70 60 104

105

106

107

Endurance, cycles

8.6 Results obtained for stress-relieved specimens under constant amplitude loading at R = 0.

compared with Spectra A and B, regardless of which S-N curve is used. Comparing the results for Spectra A and C, which are effectively identical except for the addition of cycles at p = 0.08 (Spectrum B), suggests that the apparent large difference between the experimental results for Spectra A and B must be attributed to scatter. Meanwhile the results for Spectra C and D were, as expected, not greatly different from each other, which implies that stress level p = 0.06 in Spectrum D was fully damaging. Hence, in stress relieved joints, just as in as-welded joints, it appears that, under alternating loading, stresses down at least to ±8.4 N/mm2 (and maybe lower) can be regarded as fully damaging.

258

Cumulative damage of welded joints

Stress range, N/mm2

300 280 260 240 220 200

Rutile

Low hydrogen

S-N curve for as180 welded speclmens at 160 R = –1 (from Fig. 8.5) 140 120 100

S-N curve for as-welded specimens at R = 0 (from Fig. 8.4) = Class G mean curve

90 80 70 60 104

105

106

107

Endurance, cycles

8.7 Results obtained for stress-relieved specimens under constant amplitude loading at R = –1.

8.4

Predicting fatigue life

As a result of this work, and taking into account the substantial supporting evidence provided by other investigations (outlined at the beginning of this chapter) it seems reasonable to conclude that: 1. In as-welded joints stress ranges down to 10.1 N/mm2 under tensile loading, or ±8.4 N/mm2 under alternating loading (and maybe lower in either or both situations) are equally damaging, ‘damage’ in this case being assumed to be proportional to σ3 · N. This is consistent, in principle, with the findings of Fisher et al., (1983, 1993); Marquis (1995b, 1996); Dahle and Larsson (1992) and Dahle (1993, 1994). 2. In stress-relieved joints a similar situation exists although under tensile loading further work is required to confirm the value of 10.1 N/mm2. It does, however, seem certain the 12.6 N/mm2 is damaging, as is ±8.4 N/mm2 under alternating loading. These stress ranges are far lower than the hypothetical bend in the S-N curve, which is assumed to exist for design purposes. In the British design code (BS 7608) the bend is assumed to occur at 107 cycles, giving a corresponding stress range on the Class G mean –2 standard deviations design curve of 29 N/mm2. For comparison, a stress range of 10.1 N/mm2 implies a life on the extrapolated straight line (no bend) Class G design curve of approximately 5.5 × 108 cycles, while the life corresponding to

The influence of cycles of small stress range

259

±8.4 N/mm2 is approximately 1.2 × 108 cycles. It must also be remembered that these are not necessarily lower limit stresses; they could well be lower still. In the light of these findings it seems necessary to consider whether any bend in the design S-N curve can be justified. It is therefore of interest to compare the calculated fatigue lives using three possible S-N curves, namely: 1. The straight line extrapolated curve. 2. The S-N curve as in BS 7608 with a bend from m = 3 to m = 5 at 107 cycles. 3. The S-N curve in Eurocode, with a bend from m = 3 to m = 5 at 5 × 106 cycles and a horizontal cut-off at 108 cycles. For the purposes of this comparison Spectrum D (see Table 8.1) with a peak stress range of 210 N/mm2 has been used, together with the Class G mean SN curve and assuming Σ n =1.0. As can be seen from the test results N (Tables 8.3 and 8.4) this is obviously an optimistic assumption but it is, nevertheless, the relevant design requirement and in any case the objective is merely to obtain comparative results. It will be noted that, for this particular spectrum and peak stress, the relevant minimum range is 12.6 N/mm2, for which the corresponding value of N using the extrapolated straight line S-N curve is 2.82 × 108. This stress has been proved to be damaging, but is below the cut-off level defined by the European Standard. The calculated lives, both in terms of numbers of cycles and number of blocks to failure, are shown in Table 8.7. Thus, for this particular spectrum, which was slightly concave upwards (see Fig. 8.1), the straight line S-N curve gives a predicted life equal to 0.86 times the life predicted by the S-N curve defined in BS 7608 and 0.82 times the life predicted by the Eurocode S-N curve. These represent considerable life reductions and, in the light of the experimental results, indicate that the use of bent S-N curves with this type of load spectrum is not safe. These results should not, however, necessarily be regarded as typical of all concave upwards spectra; the lives do depend on the actual form of the spectrum. By way of example consider Spectrum E (for which no experimental test results exist), which is the same as Spectrum D down to (and including) stress level p = 0.16 but then has a pronounced ‘tail’ of low stress ranges as Table 8.7 Comparing fatigue life using three possible S-N curves: Spectrum D S-N curve

Cycles to failure

Blocks to failure

Straight line BS 7608 Eurocode

28.8 × 106 33.6 × 106 36.6 × 106

4695 5464 5694

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Cumulative damage of welded joints

Table 8.8 Datails of (hypothetical) Spectrum E Level

Rlative stress, p

No. of cycles

∑n

p 3n

∑(p3n)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

1.0 0.8 0.68 0.62 0.56 0.48 0.4 0.32 0.24 0.2 0.16 0.12 0.1 0.08 0.06

1 2 3 4 6 9 16 30 72 125 244 2000 8000 105 9 × 105

1 3 6 10 16 25 41 71 143 268 512 2512 10 512 110 512 1 010 512

1.0 1.024 0.943 0.953 1.054 0.995 1.024 0.983 0.995 1.0 0.999 3.456 8.0 51.2 194.4

1.0 2.024 2.967 3.920 4.974 5.969 6.993 7.976 8.971 9.971 10.970 14.426 22.426 73.626 268.026

Table 8.9 Comparing fatigue life using three possible S-N curves: Spectrum E S-N curve

Cycles to failure

Blocks to failure

Straight line BS 7608 (bent) Eurocode

229.6 × 106 1305.8 × 106 5884 × 106

227.2 1292.3 5823

set out in Table 8.8. With this spectrum the predicted lives become as shown in Table 8.9, again for a peak stress range of 210 N/mm2. In this situation there is obviously a very large difference in predicted lives depending on which particular S-N curve is used. (Note, incidentally, that this is merely an example for demonstration purposes, the predicted lives would, in many instances, probably be far beyond the required design life.) For comparison purposes this example is also considered on the basis of fracture mechanics in Chapter 9, where it will be found that the predicted life turns out to be 1685 × 106 cycles or 1667 blocks, although that result is obviously dependent on the particular assumptions made in the analysis. The results set out above suggest that there may be a huge benefit, with this type of spectrum, in using one of the bent S-N curves. On the other hand, if the work outlined earlier in this chapter is still valid, their use could be extremely unsafe. Unfortunately, there is no direct proof one way or the other as to which is correct in these circumstances. Clearly there is a need for some experimental testing, using a realistic concave upwards spectrum with a longer ‘tail’ than was present in Spectrum D, to elucidate this problem.

The influence of cycles of small stress range

261

Table 8.10 Comparing fatigue life using three possible S-N curves: Rayleigh distribution S-N curve

Cycles to failure

Blocks to failure

Straight line BS 7608 Eurocode

1.779 × 106 1.790 × 106 1.810 × 106

17.795 17.901 18.096

Table 8.11 Comparing fatigue life of various joint types Test regime

Table

As-welded joints, R = 0 As-welded joints, inverted spectrum Stress relieved joints, R = 0 As-welded joints, R = –1 Stress relieved joints, R = –1

Σ n based on

Σ n based on

N Class G (mean curve)

N expl S-N curve

8.3 8.3

0.49–0.67 (0.55) 0.59–0.74 (0.66)

ditto ditto

8.4 8.5 8.6

0.45–0.67 (0.53) 0.56–0.96 1.22–1.90

ditto 0.44–0.78 (0.63) 0.55–0.79 (0.62)

A similar calculation based on a Rayleigh distribution of stress ranges (which has a convex upwards exceedence curve) gives quite a different result. Thus, assuming a block length of 105 cycles, for which the clipping ratio is 4.80, the calculated fatigue lives were as shown in Table 8.10. In this instance, therefore, the reduction in life using the straight line S-N curve as opposed even to the Eurocode form of the curve is only 1.7%, which in terms of the required change in design stress would be completely trivial. It therefore also follows that, for this type of spectrum, there is no significant benefit in using a bent, as opposed to a straight, S-N curve.

8.5

Summary

The situation can therefore be summarised as follows: 1. For a concave upwards spectrum it is unsafe to use a bent S-N curve. Obviously this assumes that the results set out above are equally valid for other types of joint, which still needs to be proved, but which nevertheless seems probable. 2. For a convex upwards spectrum there is no significant benefit in using a bent S-N curve rather than a straight one. One can therefore conclude that, subject to the proviso that the results can be shown to be equally valid for other types of joint, a straight line S-N curve with no bends should be used in all cases. This would also simplify design calculations. So far as the traditional analysis of variable amplitude test

262

Cumulative damage of welded joints

results is concerned, it is notable that all the specimens tested using Spectra A, B, C and D gave values of Σ n less than 1.0 based on the relevant N experimental S-N curve. The results can be summarised as shown in Table 8.11, the mean values of the various sets of results being shown in parentheses. Given that both the as-welded and the stress-relieved joints tested under constant amplitude loading at R = 0 gave results that were in close agreement with the Class G mean curve (see Figs 8.4 and 8.6) the corresponding values of Σ n were identical regardless of whether the Class G mean or the N experimental S-N curves were used. For the joints tested at R = –1, however, there were significant differences between the two S-N curves. With aswelded joints the curve was significantly steeper (m = 2.66) than at R = 0 and while the lives were virtually identical at a stress range of about 55 N/mm2 the tests at R = –1 gave longer lives (for the same stress range) at higher stresses (see Fig. 8.5). With stress-relieved joints the slope of the curve was similar to that for as-welded joints, but the whole curve was shifted to longer lives by a factor of about 1.75 or to a larger stress range by a factor of about 1.22. As a result of the shift to longer lives the values of N in the Miner summation were larger, resulting in a significant decrease in the values of n . Overall therefore, based upon the corresponding experimental S-N ΣN curve, the values of Σ n range from 0.44 to 0.79 with a mean value of 0.58. N It is worth noting, however, that when calculated on the basis of the Class G mean –2 standard deviations design curve rather than the mean curve, the values of Σ n are automatically increased by a factor of 2.29. In this situation N all the values are ‘safe’ (i.e. greater than 1.0) ranging from 1.03 to 2.19 (not including the results for stress-relieved joints at R = –1).

9 Design for variable amplitude loading

9.1

Introduction

Near the beginning of this book it was noted that design calculations for variable amplitude loading conditions are almost universally based upon Miner’s Rule, namely, Σ n ≤ 1.0 and in almost every subsequent chapter it N has been shown that Miner’s Rule based upon the best fit mean constant amplitude S-N curve for the joint in question is not ‘correct’. The question therefore arises as to whether, in the light of the experimental evidence now available, it is possible to derive a better rule. However, before considering that particular problem, it is worth considering how Miner’s Rule is currently employed in the design context. The main variant is the choice of constant amplitude S-N curve to use as a base. In most countries the design S-N curve is the assumed ‘lower limit’ curve to the experimental results, often referred to as the mean –2 standard deviations curve. The simplest and safest approach is to assume that that curve, plotted on a log stress v. log endurance basis, continues downwards as a straight line ad infinitum, and that is the form of curve that is used, for example, in the USA. However, in some design codes, for purposes of design for variable amplitude loading, the curve may be modified by introducing a hypothetical ‘knee’ in the curve, or a fatigue limit (so that the curve becomes horizontal) or both. The reason for introducing these bends stemmed from the thought that the lower stresses in a loading spectrum could not possibly be as damaging as the higher stresses, so that it would be sensible to allow for that by somehow increasing the values of N for the lower stresses in the summation of Σ n . At the time that the introduction of these bends was first N being considered (around 1970) there was no theoretical justification for them, but in modern fracture mechanics parlance they can be viewed as being associated with the threshold stress intensity for propagation. Clearly, as the crack propagates, the stress intensity factor associated with a particular 263

264

Cumulative damage of welded joints

stress increases so that, although that stress may be non-damaging at the start of life, it will become damaging as life progresses. Similarly, the introduction of a ‘fatigue limit’ (in some codes) stemmed from the thought that, just as in constant amplitude tests, there would be some stress which was so low that it would form a lower limit below which cracks would not propagate; in effect, for the purposes of carrying out this summation, that implies that for stresses below this ‘lower limit’, where the design curve was assumed to become horizontal, the values of N would be infinite and the corresponding values of n would be zero. N Typical examples of these two approaches can be seen in the British and European design codes. In the British code the S-N curve for design under variable amplitude loading is assumed to have a bend at 107 cycles from slope m = 3 to the shallower slope m = 5 (line ABC in Fig. 9.1), it being assumed that under constant amplitude loading 107 cycles corresponds to the fatigue limit or initial non-propagation stress. In the European code there is a similar bend from m = 3 to m = 5, but it is situated at 5 × 106 cycles rather than 107 cycles. Then at 108 cycles the curve becomes horizontal and forms a fatigue limit. Thus the European S-N curve has the form ADEF in Fig. 9.1. In passing it is worth noting that the first person to propose the introduction of a bend is believed to have been Haibach in Germany. His proposal was that, at the knee, the slope should change from m to (2m – 1). At that time Haibach concluded that in most instances m = 3.75 giving a slope of 6.5 for A

Log stress

m=5

m=5

D B F

E

C

104

105

106 107 Endurance (cycles)

108

109

9.1 Comparison of British and European design S-N curves for use with variable amplitude loading.

Design for variable amplitude loading

265

the lower part of the curve. It is notable, however, that for m = 3 the lower slope becomes 5. In the British rules the slope is assumed to change from m to (m + 2) which, for m = 3, happens to give the same answer. It should, however, be remembered that there is nothing ‘magic’ about either formula. At the time that they each originated, both were little more than wild guesses and to a large extent they still are.

9.2

Testing for different types of stress

It is of some interest to examine the implications of introducing these hypothetical bends and fatigue limits in relation to the calculated design stresses. The easiest way to do that is to consider the effect on design stresses for various types of spectra, namely convex upwards, Laplace and concave upwards. The important difference between these three types is the relative frequency of high and low stress ranges which they contain. It will be recalled that in Chapter 5, it was shown that for a loading spectrum containing NL cycles per block and with a life to failure of NB blocks

N

n = B Σ (v p m ) ΣN i i NC

[9.1] (from eqn [5.1])

where the spectrum is expressed in terms of proportional stress (pi) (= stress range/maximum stress range in the spectrum) with corresponding number of cycles (vi). NC is the constant amplitude life relating to the maximum stress range, as defined by the S-N curve under consideration. From that it follows that, for Σ n = 1.0, the life to failure (in blocks) is N given by NB =

NC Σ (v i p im )

[9.2]

or in cycles by N = N BN L =

NCNL Σ (v i p im )

[9.3]

In carrying out cumulative damage calculations it will usually be found to be simpler to work in terms of proportional stresses (pi) rather than in terms of absolute stresses. This is because, if it is found necessary to change the absolute stresses in order to obtain the required value of Σ n , the proportional N stresses do not change, so that the major part of the calculation is unaltered. The only change that occurs is in the value of Nc. Initially, let us consider what this means if the basic S-N curve is linear, with no bends and no fatigue limit (see Fig. 9.2). Normally the design process

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Cumulative damage of welded joints

Log S

KSmax

Smax

Nc′

Nc

Log N

9.2 S-N curve with no bends and no fatigue limit.

would involve guessing what the required top limit stress might be, which would automatically define Nc (the corresponding constant amplitude life read from the S-N curve), so that it would be possible (from eqn [9.2]) to calculate the expected life to failure in blocks or, multiplying by the block length (NL = ∑v), the expected life in cycles (from eqn [9.3]). Suppose that N this prediction is N1 but that the required life is N2 cycles  or 2 blocks .  No  It will therefore obviously be necessary to modify all the stresses by some factor, k, in order to obtain N2, and with a linear logS-logN curve it is easy to calculate the required value of k. Thus, since the equation of the S-N curve is Sm · N = C (where C is a constant)

[9.4]

we have

Smaxm.NC = (kSmax)m · N 1C

[9.5]

Hence

N 1C =

or

N  k =  1C   NC 

NC km

[9.6] 1/m

[9.7]

But, as can be seen from eqn [9.3], the predicted life to failure is directly proportional to NC, so it follows that

Design for variable amplitude loading

267

A Log S

SmN = C 1

K Smax Smax

B S nN = C 2

C

NC 2

NC 1

Nbend

Log N

9.3 S-N curve with hypothetical change of slope, as in the British design rules.

N 1/m k=  1  N2 

[9.8]

Clearly this will give the required value of the stresses regardless of how bad the initial guess of the stresses might have been. We will now consider the same problem when the S-N curve is bent and effectively consists of two straight lines (AB and BC in Fig. 9.3) with slopes m and n respectively. In the British design rules the two lines would intersect at N = 107 cycles and we would, for most types of joint, have m = 3 and n = 5, although those values may not be universal. As before, the first step is to make an inspired guess as to the required value of the top limit stress (Smax), which immediately enables the value of p corresponding to point B on the S-N curve to be determined as well as the values of NC1 and NC2 (see Fig. 9.3). Now, from eqn [9.1] , the value of Σ n relating to the stresses above the N transition point (B) can be written.

( Σ Nn )

A

=

NB ( Σ v i p im ) A N C1

where the subscript A refers to the stresses relevant to the part AB of the curve. Similarly,

( Σ Nn )

B

=

NB ( Σ v i p in ) B N C2

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Cumulative damage of welded joints

where subscript B refers to the stresses relevant to the part BC. In total, therefore,

( ) + ( Σ Nn )

n = Σn ΣN N

A

= B

NB N ( Σ v i p im ) A + B ( Σ v i p in ) B N C1 N C2

[9.9] However, the relationship between NC2 and NC1 is known for the particular value of Smax under consideration. Since

Sm max ⋅ N C1 = C 1

and

S nmax ⋅ N C2 = C 2

we have

N C2 Sm C = 2 ⋅ max N C1 C 1 S nmax

Hence eqn [9.9] can be rewritten as N



n = B (Σ v pm ) + (Σ v pn ) ⋅ ΣN A i i i i B N C1 

C 1S n–m max  C 2 

[9.10]

This is the equivalent equation to [9.1] when the S-N curve is bent instead of straight. In the same way it follows that, for Σ n = 1.0, N N C1 [9.11] NB = C 1S n–m  max  m n ( Σ v i p i ) A + ( Σ v i p i ) B C  2   and the number of cycles to failure is given by N1 = N B ⋅N o =

N C1 ⋅ N 0 C 1S n–m  max  m n ( Σ v i p i ) A + ( Σ v i p i ) B C  2  

[9.12]

Table 9.1 Method of calculating design stresses if using a linear curve of Log S v Log N 1. 2. 3. 4. 5.

Choose a trial value of Smax (effectively this also defines all the other stress levels in the spectrum). Since the S-N curve is known, the value Smax defines the value of Nc. Using the known loading history for one block, and assuming Σ n =1.0, use N eqn [9.2] to calculate the expected number of blocks to failure (NB). Multiply NB by the block length to obtain the expected life to failure in cycles (N1). Using eqn [9.8] calculate the factor k by which the stresses have to be modified in order to obtain the required life (N2).

Design for variable amplitude loading

269

It will be noted that, in this equation, all the values are defined so the predicted life can be calculated. In contrast to the situation that exists with a straight S-N curve with no bends, there is no way in which the correct value of the stresses can be deduced directly. This is because a change in the value of Smax will necessarily change the position of the bend in the S-N curve in relation to Smax, so that it is probable that some stress levels which were initially assumed to be below the knee will become above the knee, or vice versa. By reference to eqn [9.8] it is clear that the value of the factor k by which 1

1

N m N n the stresses need to be multiplied will lie between  1  and  1  . If  N2   N2  the initial guessed value of Smax was reasonably accurate it will usually be found that these values of k are not wildly different from each other. However, N in defining the value of k it is sometimes helpful to recognise that if 1 > N2 1.0 it is obviously possible to increase the design stresses. This implies that more stress levels than originally anticipated will tend to be above the knee and that the S-N curve of slope m will therefore become relevant for them. 1

1

N m N n This suggests that k is likely to be closer to  1  than to  1  . On the  N2   N2  N other hand the opposite is likely to be the case if  1  < 1.0 and the design  N2  stresses therefore need to be decreased. The design process for a bent S-N curve is summarised in Table 9.2. Table 9.2 Method of calculating design stresses if using a bent S-N curve 1. 2. 3. 4.

Choose a trial value of Smax (effectively this defines all the other stress levels in the spectrum and the value of p at the ‘knee’ in the curve). Since the S-N curve is known, the value of Smax defines the values of N c 1 and N c 2 relating to the two branches of the curve. ) A for the stresses above the knee and ( Σ v i ⋅ p in ) B for the Calculate ( Σ v i ⋅ p m i stresses below the knee. From eqn [9.12] calculate the predicted life to failure (N1) corresponding to the trial value Smax. 1

5.

 N m If N1 is significantly different from the design life (ND) calculate  1  and  ND  1

6.

 N1  n   These are the limiting values of the factor (k) by which all the stresses  ND  need to be multiplied to obtain the correct stresses. As the next trial value it would be usual to choose the mean of the two values. If there was a large difference between the two values of k it would be sensible to return to step 1 and choose a new value of Smax and repeat the process.

270

Cumulative damage of welded joints

Table 9.3 Cumulative damage calculation for Laplace spectrum with block length of 105 cycles 1

2

Stress level p

No. of exceedences

1.0 0.94 0.91 0.85 0.8 0.74 0.71 0.65 0.6 0.54 0.51 0.45 0.4 0.34 0.31 0.25 0.20 0.14 0.11 0.05 0

1 2 3 6 10 20 30 60 100 200 300 600 1000 2000 3000 6000 10 000 20 000 30 000 60 000 100 000

3

4 Corresponding spectrum

Cycles

Average applied stress

1 1 1 3 4 10 10 30 40 100 100 300 400 1000 1000 3000 4000 10 000 10 000 30 000 40 000

1.0 0.97 0.925 0.88 0.825 0.77 0.725 0.68 0.625 0.57 0.525 0.48 0.425 0.37 0.325 0.28 0.225 0.17 0.125 0.08 0.025

5

6

7

8

p3 N

∑p3N

p 5N

∑p5N

1.0 0.9127 0.7915 2.0444 2.2461 4.565 3.811 9.433 9.766 18.519 14.470 33.178 30.706 50.653 34.328 65.856 45.562 49.13 19.531 15.36 0.625

1.0 1.913 2.704 4.748 6.994 11.559 15.370 24.803 34.569 53.088 67.558 100.736 131.442 182.095 216.423 282.279 327.841 376.971 396.502 411.862 412.487

5.546 6.922 3.626 5.163 2.306 1.420 0.305 0.098 0

25.386 19.840 12.918 9.292 4.129 1.823 0.403 0.098

By way of example let us consider the design of a joint subjected to a Laplace distribution of stresses with a block length of 105 cycles. For the purposes of this exercise we will assume that the relevant design S-N curve is the Class F mean curve in the British rules, with the hypothetical knee in the curve being at 107 cycles at 56 N/mm2. The equation of the two branches of the curve arc S3N = 1.715 × 1012 and S5N = 5.5073 × 1015, (i.e. with slopes m = 3 and n = 5 above and below the knee respectively). For explanatory purposes the calculation is set out in some detail in Table 9.3. In this Table, columns 1 and 2 show the spectrum expressed in terms of ‘exceedences’. For example, it shows that 1 cycle equals or exceeds the relative stress p = 1.0, 100 000 cycles exceed zero stress and 100 cycles exceed p = 0.6. In columns 3 and 4 that information is converted into a statement of the number of cycles at each of the mean stresses between two known exceedence values. Thus we know from columns 1 and 2 that 600 cycles exceeded p = 0.45 and that 1000 cycles exceeded p = 0.4. From that it follows that 400 cycles (1000–600)

Design for variable amplitude loading

271

were at stresses between p = 0.4 and p = 0.45, so for convenience it is assumed that all were at p = 0.425. Obviously, that is an approximation but it is not an unreasonable one. At this stage the spectrum is defined by columns 3 and 4 and column 5 shows, for each stress level, the corresponding value of p3N while column 6 shows the progressive summation ∑p 3 N starting from the top. Column 7 shows the individual values of p5N for each stress level while column 8 shows the progressive summation of ∑p5N starting from the bottom. The reason why the summation ∑p3N and ∑p5N start from the top and bottom respectively (rather than both from the top) is that the figures that one ultimately needs to use are all the values relating to the stresses above and below the bend in the S-N curve, and the position of that has yet to be defined. Having tabulated the various values of ∑p3N and ∑p5N we will first calculate the anticipated life for Σ n = 1.0 of a joint with a peak applied N stress range of 200 N/mm2, assuming a straight S-N curve with no bend. In this situation we know, from eqn [9.1], that for NB = 1 block

(

)

3 n = Σ vp ΣN NC

[9.13]

Given that the equation of the S-N curve with slope m = 3 is S3N = 1.715 × 1012 it follows that, for S = 200, NC = 214 375 cycles, and since ∑vp3 for the whole spectrum is 412.487 we get, from eqn [9.13] n = 412.487 = 0.001924 per block. ΣN 214 375

Hence, for Σ n = 1.0, the anticipated number of blocks to failure is N NB =

1.0 0.001924

so that the number of cycles to failure (since NL = 105 cycles) is

N=

10 5 = 5.1975 × 107 cycles 0.001924

As a second example, using the same spectrum but a bent S-N curve, what stress is required to obtain the same life as when using the straight S-N curve (i.e. 5.1975 × 107 cycles)? If we start by assuming, obviously wrongly, that the required peak stress will be 200 N/mm2 then the bend in the S-N curve (at 56 N/mm2) will be at p = 56 = 0.28. Hence, referring to Table 9.3, we 200

272

Cumulative damage of welded joints

see that ∑vp3 (for the stresses above the knee) = 216.423 and that ∑vp5 (for the stresses below the knee) = 9.292. In eqn [9.10] the values of the other variables are C1 = 1.715 × 1012 and C2 = 5.5073 × 1015 (the constants defining the two branches of the bent S-N curve), Smax = 200 and NC1 = 214 375. Hence for one block (NB = 1). n = ΣN

12   1 216.423 +  9292 × 1.715 × 10 15 × 200 2      214 375  5.5073 × 10

= 216.423 + 115.743 214 375 = 332.166 = 0.0015495 per block. 214 375 Hence, for Σ n = 1.0, the predicted life is N 1.0 = 645.37 blocks 0.0015495

or 6.4537 × 107 cycles (since the block length is 105 cycles). Comparison with the result for the previous example shows that the use of a bent S-N curve rather than a stright one leads to an increase in predicted life by a factor of 1.2418. Let us now see what that means in terms of design stress if we aim for the same life rather than an increased one. At this stage we now have N1 (the predicted life) = 6.4537 × 107 and N2 (the design life) = 5.1975 × 107. Hence it is obviously possible to increase N the design stress  i.e. 1 > 1.0  and the relevant factor by which it can be   N2 N 1/3 N 1/5 increased will lie between  1  = 1.0748 and  1  = 1.044. Given  N2   N2  that in this case the initial guess of the value of Smax was quite accurate there is very little difference between these two factors and it would be tempting to use the average value (say k ≈ 1.06) making Smax = 212 N/mm2. In reality 213 N/mm2 is a slightly closer estimate, but the difference is hardly significant. It is interesting to note that a difference in life of about 24% equates to a difference in design stress of only about 6%. As a final example let us consider the situation when there is a much bigger difference between the initial calculated life (using the guessed value of Smax) and the design life. Suppose for example that the design life is 2 × 108 cycles but that, as before, the initial guess of Smax was again 200 N/mm2. In this situation we have N1 = 6.4537 × 107 (as in the last example) and N2 N = 2 × 108. Clearly a reduction in design stress will be required  1 < 1.0   N2 

Design for variable amplitude loading

273

1/3

7 and the reduction factor will lie between  6.4537 × 810  = 0.32271/3 =  2 × 10  0.686 and (0.3227)1/5 = 0.797. In this situation the correction factor k will lie between 0.686 and 0.797, but is likely (but not necessarily) to be closer to N 0.797, because 1 < 1.0. The mean of the two values is approximately 0.74 N2 so a reasonable first guess of the value of k might be about 0.75, giving a second estimate of Smax of 150 N/mm2. This change of Smax means that the knee of the S-N curve has changed to p = 56 = 0.373 . This implies (see Table 9.3) that ∑vp5 = 19.840, and that 150 NC1 = 508 148 cycles. Inserting these values, it turns out that the new predicted life is 18.79 × 107. This is slightly below the design life of 2 × 108 cycles, 7 although it is not a bad estimate, with 18.79 × 10 = 0.9395 . This leads to 8 2 × 10 1/3 second correction factors of (0.9395) = 0.979 and (0.9395)1/5 = 0.987 suggesting a revised value of about 0.983. Clearly this is trivial, and the required value of Smax will now be 150 × 0.983 ≈ 147.5 N/mm2.

9.3

The area rule

In Chapters 5 and 6 it was shown that one of the circumstances which tended to result in Miner’s rule giving unsafe predictions of life was when the loading spectrum involved short block lengths. On the other hand that was a situation where the Area rule consistently gave safe life predictions. It therefore seems reasonable to suggest that, in that situation, it is at least worth considering the use of the area rule, although it has to be remembered that it is not a rule that is even mentioned in any national design codes. It would therefore be necessary for the use of the method to be agreed by all interested parties before proceeding. In general, design using this method is considerably easier than by Miner’s rule. As we saw in Chapter 5 (eqn [5.20]) the life predicted by the area rule is Npred = Nc e–Area × Block length

[9.14]

Hence, if the spectrum can be defined for a single block (i.e. including a single application of the peak stress) the area under the p versus ln E (number of exceedences) graph can be calculated and the block length is also known. Then, putting Npred equal to the required service life, the only unknown in eqn [5.20] is NC, so that

NC =

e

–Area

N pred × Block length

[9.15]

274

Cumulative damage of welded joints

and knowing NC, which is the constant amplitude life under the peak stress range in the spectrum, the value of that stress can easily be derived from the relevant S-N curve.

9.4

Possible modifications to Miner’s rule

As we have seen in earlier chapters, the value of Σ n is clearly, in many N situations, influenced by the block length of the loading under consideration, with Σ n tending to increase as block length increases. Thus, although N Miner’s rule can be a useful design approach as it stands, it would obviously be more useful if the value of Σ n remained more or less constant, regardless N of block length. On the basis of the results obtained under tensile loading with the stresses applied in random order, it was suggested by Cottrell (private comm.) that the values of the various Miner summations might be brought to a relatively constant level if they were expressed in terms of the function n × (ln (N ))–0.5 ΣN L

[9.16]

and that this might be more useful for design purposes. In order to investigate this possibility the values of that function have been calculated for as many test results as possible, but in particular for those where the maximum applied stress range was either 275 or 175 N/mm2. It was thought that it might be misleading to take account of all test results, regardless of the peak range, since it has been established that the value of Σ n also depends on the peak N range (see, for example, Fig. 6.10). The results for tests carried out at the Welding Institute (UK) under fully tensile loading are shown in Figs 9.4(a) and (b), which refer to spectra with peak stress ranges of 275 and 175 N/mm2 respectively. In general, apart from a few very high results, the great majority of tests gave a reasonably consistent set of results with values of Σ n (ln (NL))–0.5 generally lying in the range N 0.4–0.8 when the peak range was 275 N/mm2 and 0.4–0.7 for a peak range of 175 N/mm2. This included tests under random order and narrow band loading and initially appeared to be a very hopeful result, giving a design value of about 0.4. However, some tests under wide band loading although few in number, gave very much lower values, typically in the range 0.22–0.4; they are also shown in Fig. 9.4. Clearly, they indicate that this particular function is not satisfactory for use as an overall design criterion. As an alternative approach it was decided to consider the possibility of using a combination of Σ n and RG. As can be seen from Figs 6.31 and N

Design for variable amplitude loading

275

6.32, Σ n tends to increase with block length while RG initially increases N and then decreases, while Σ n continues to increase. On the assumption N that this decrease in RG would be balanced by the continuing increase in n , consideration was given to the use of the function ΣN

(

0.5 Σ n + R G N

)

[9.17]

The relevant results, for the same set of data as used in Fig. 9.4, are shown in Fig. 9.5. It will be seen that, under fully tensile loading, the values of the function lay mainly in the range 1.3–2.2 when the peak stress range was 275 N/mm2 and in the range 1.1–2.0 for a peak range of 175 N/mm2. However, once again, this approach fails for tests at R = –1 or under wide band loading, where values down to about 0.6 were obtained (Gurney, 1988a; Gurney, to be published). Clearly the crux of the problem is to define a rule which works for loading with partly compressive or variable stress ratios, but that has not yet been achieved.

9.5

The fracture mechanics approach

It is easiest to describe this method by reference to a specific example. This deliberately incorporates a number of simplifying assumptions while at the same time demonstrating the method of approach. The problem is to calculate the predicted life of a plate 125 mm wide with two longitudinal edge attachments 150 mm long, as shown in Fig. 7.3, subjected to a loading spectrum with a peak stress range of 210 N/mm2. For convenience, this is the same as Spectrum D referred to in Chapter 8. This type of joint will normally fail from one or more of the weld ends. Initially the crack will consist of one or more quarter elliptical cracks centred on the corner of the main plate, but ultimately they will become a single through-thickness crack. Obviously, in reality these two forms of crack will have different K calibrations, but for demonstration purposes it will be assumed that, throughout the life, K = MK σ πa

[9.16]

and it will be assumed that the value of MK can be written as MK =

p a B

( )

[9.17]

q

where p and q are constants. Assuming that the specimen has a total width (2B) = 125 mm we have B = 62.5 mm so that if the initial defect size is a = 0.15 mm the corresponding value of a = 0.0024. Furthermore, if it is B

( )

0.2

0.4

0.6

0.8

Σ n (Ln NL ) –1/2 (log scale) N

1.0

1.5

1.75

1

102 103 104 105 106 Block length (cycles) (a) 107

0.2

0.4

0.6

0.8

1.0

1.5

1.75

1 10

102 103 104 105 106 Block length (cycles) (b)

13 results wide band, R = –1

8 results wide band, R=0

107

Filled points R = 0 Open points R = – 1 Rayleigh Wide band Weibull

9.4 (a) Relationship between Σ n (Ln (NL )) –1/2 and block length for tests with peak stress range = 275 N/mm2. (b) N Relationship between Σ n (Ln (NL )) –1/2 and block length for tests with peak stress range = 175 N/mm2. N

10

16 results wide band, R = –1

16 results wide band, R=0

Filled points R = 0 Open points R = – 1 Rayleigh Wide band Weibull

Σ n (Ln NL ) –1/2 (log scale) N

 n Σ + RG  (log scale)  N

1/2

0.5 1

1.0

2.0

3.0

4.0

leig

h

102 103 104 105 106 Block length (cycles) (a)

16 tests at R=–1 Mean = 0.91

Ray

107

0.5

1.0

2.0

3.0

4.0

1 10

h

8 wide band results Mean = 1.02

leig

102 103 104 105 106 Block length (cycles) (b)

13 results wide band R=–1 Mean = 0.89

Ray

Filled points R = 0 Open points R = – 1 Rayleigh Wide band Weibull

107

Relationship between 1/2  Σ n + R G  and block length for tests with peak stress range = 175 N/mm2.  N 

9.5 (a) Relationship between 1/2  Σ n + R G  and block length for tests with peak stress range = 275 N/mm2. (b)  N 

10

16 wide band results Mean = 1.18

Filled points R = 0 Open points R = – 1 Rayleigh Wide band Weibull  n Σ + RG  (log scale)  N 1/2

278

Cumulative damage of welded joints

assumed that failure corresponds to a = 37.5 mm that gives a = 0.6. The B values of p and q will therefore be chosen to give Mk = 3 at a/B = 0.0024 and Mk = 1 at a/B = 0.6. Inserting these two end points into eqn [9.17] we get p = 0.9587 q = 0.1891. The rate of crack propagation will be assumed to be represented by:

da = C ⋅ ( ∆K) m dN

[9.18]

with m = 3. For structural steel the corresponding mean value of C = 1.832 × 10–13; the upper limit value is C = 5.21 × 10–13. Finally, it will be assumed that the threshold value of ∆K for crack propagation is ∆Kth = 63 N mm–3/2. Given that the value of MK is expressed in terms of a (= α), and that the B initial and final crack sizes can also be written in that way, it is convenient to work non-dimensionally using the form of the crack propagation equation given in eqn [1.10], with Y = MK. Thus

∫

α2

α1

m –1 dα = C ( ∆σ ) m B 2 N m (M K πα )

[9.19]

so that, inserting the value of MK from eqn [9.17] the value of the integral becomes

I=

∫

α2

α1

=

dα p m α –qm π m/2 α m/2

1 p m π m/2

∫

α2

α1

dα α m(0.5–q) α2

=

 α 1–m(0.5–q)  1 m m/2  1 – m (0.5 – q)  p π  α1 

which gives, inserting the values of m, p and q, I=

1 [ α 0.0673 ]α 2 α1 0.3302

[9.20]

Putting this value into eqn [9.19] and inserting the values B = 62.5 and C = 1.832 × 10–13 (the mean value) we get

[ α 0.0673 ]αα12 = 0.3302 × 1.832 × 10–13 × 62.50.5 ∆σ3N = 4.782 × 10–13 ∆σ3N

[9.21(a)]

Design for variable amplitude loading

279

Alternatively, using the upper limit value of C = 5.21 × 10–13 we get [ α 0.0673 ]αα12 = 13.6 × 10–13 ∆σ3N

Note that in Table 9.5 the factor

[9.21(b)]

10 13 is written as F, thereby giving 13.6 ( ∆α ) 3

N = F × [0.0673]. By way of a check, it is interesting to compare the predictions derived from eqns [9.21(a)] and [9.21(b)] with the known fatigue strengths under constant amplitude loading. Thus, for α1 = 0.0024 and α2 = 0.6 eqn [9.21a] gives (∆σ)3N = 6.2708 × 1013, so that the corresponding S-N curve (with m = 3.0) gives strengths of 67.9 N/mm2 at 2 × 106 cycles and 39.7 N/mm2 at 107 cycles. This may be compared with 66 N/mm2 at 2 × 106 cycles for the British Class G mean curve, which is the curve relevant to the joint under consideration. Alternatively, if we use the upper limit value of C from BS7910, namely C = 5.21 × 1013, eqn [9.21(b)] with α start = 0.0024 and α end = 0.6 gives (∆σ)3N = 2.205 × 1011. For N = 2 × 106 cycles this gives ∆σ = 47.9 N/ mm2 and for N = 107 ∆σ = 28 N/mm2. The 2 × 106 cycles fatigue strength defined by the Class G mean – 2SD design curve is 50 N/mm2. The derived strengths at 107 cycles (39.7 and 28 N/mm2) also correlate quite closely with the mean and mean – 2SD values of fatigue strength for Class G at 107 cycles, which are 38 and 29 N/mm2 respectively. These comparisons suggest that the various assumptions outlined above are reasonable. Returning now to the example with which we started, the problem at this stage is to decide at what crack lengths the various stresses in the spectrum become active in propagating the crack, ignoring any prior overloading effects due to the higher stresses in the spectrum. Initially, with a defect of size a = 0.15 mm, where MK = 3, we have ∆K = MK σ πα = 3 σ 0.15 π = 2.059σ

[9.22]

so that, for ∆Kth = 63 N mm–3/2, σth = 63 = 30.6 N/mm2 which means that 2.059 all stresses above and including level 11 must be damaging. The point at which level 12 (21 N/mm2) becomes damaging can also be derived from eqn [9.22]. Eliminating Mk using eqn [9.17] and writing in terms of α rather than a we get p ∆σ πα αq

B = ∆K th

where ∆σ is the stress range under consideration. This equation can be rewritten as α

(0.5-q)

=

∆K th p∆σ πB

[9.23]

280

Cumulative damage of welded joints

and putting in the relevant values that gives

α

or For

0.3109

=

63 = 4.6897 ∆σ 0.9587 ∆σ 62.5 π

(

α = 4.6897 ∆σ

)

3.2165

[9.24]

∆σ = 21 N/mm2 this gives α = 0.00805

so that a = 0.00805 × 62.5 = 0.503 mm. By the same means it is easy to show that level 13 of the spectrum (12.6 N/ mm2) becomes active at α = 0.0416 or a = 2.60 mm. Clearly, each stage of propagation now has to be calculated separately and it is easiest to work with the equivalent stress for each stage. For the first stage, from α1 = 0.0024 to α2 = 0.0080, where only the first 11 stress levels are active, the equivalent relative stress is  Σ p 3i n i   N  i  

1/3

(

= 10.97 512

)

1/3

= 0.2778

[9.25]

giving an actual equivalent stress range of 0.2778 × 210 = 58.34 N/mm2. Hence, from eqns [9.19] and [9.20] 3 0.5 [ α 0.0673 ] 0.0080 N 0.0024 = 0.3302. C. (58.34) 62.5

giving N = 592 291 for C = 1.832 × 10–13 or

N = 208 260 for C = 5.21 × 10–13

However, these are only the damaging cycles. Associated with each block of 512 damaging cycles there are 5632 (in this instance) non-damaging cycles so that the total life expected in the initial stage of propagation becomes 6144 × damaging cycles. (It is very easy to forget to include these non512 damaging cycles in the calculations of the total life.) Hence the total lives become N = 592 291 × 6144 = 7 107 513 for C = 1.832 × 10–13 512 or

208 260 × 6144 = 2 499 127 for C = 5.21 × 10–13 512

The same method can now be used to calculate the corresponding lives for propagation from α = 0.0080 to α = 0.042 under 12 loads of the spectrum and from α = 0.042 to final failure at α = 0.6 where all 13 levels of the spectrum are damaging. The calculations set out above may appear long-

Design for variable amplitude loading

281

Table 9.4 Fracture mechanics calculation for spectrum D (see Table 9.4) with peak stress = 210 N/mm2 1

Lowest damaging stress range (N/mm2)

33.6

21

12.6

2

Damaging cycle per block

512

1512

6144

0.2778

0.1993

0.1283

41.85

26.94

3

Equiv σ σ max for damaging stresses

4

Equiv. σ of damaging stresses (N/mm2)

58.34

5

αstart

0.0024

0.0080

0.0416

6

αend

0.0080

0.0416

0.60

7

α 0.0673 start

0.66633

0.72257

0.80736

8

α 0.0673 end

0.72257

0.80736

0.96620

0.05624

0.08479

0.15884

3 703 069

10 031 703

37 606 967

α end α start

9

I

10

F=

11

Damaging cycles = I × F

208 260

850 588

5 973 490

12

Total cycles

× 6144 512 = 2 499 127

× 6144 1512 = 3 456 357

× 1.0

10 13 13.6 σ e3

= 5 973 490

Total = 11 928 974 (= 1941 blocks)

Assumptions 1

K = Mk σ πa

2

Mk =

3 4 5

p  a  B

q

with p = 0.9587, q = 0.1891 and B = 62.5 mm

ae = 0.15 mm and af = 37.5 mm m = 3.0 and c = 5.21 × 10–13 (typical upper limit value) ∆kth = 63 N mm–3/2

winded, but many of them repeat at each stage and the whole calculation can be set out in tabular form, as shown in Table 9.4. For ease of reference the various rows have been numbered. As a second example let us consider the problem introduced in Chapter 8 relating to the influence of small stress cycles. This was concerned with the

Damaging cycle per block

2

0.02879

α 10.0673

0.0673 α end

I αα 1end

8

9

Damaging cycles N=I×F

Total cycles

11

12

1 010 512 512

599 032 303

×

303 514

10 542 360

0.69512

αend

6

7

10 13 4.782 σ e3

0.0045 0.66633

αstart

5

F=

0.0024

Equiv σ of damaging stress

4

10

58.32

Damaging 0.2777

512

33.6

3

cycles σ σ max

Lowest damaging stress

1

1 010 512 2512

433 896 635

×

1 078 610

39 307 952

0.02744

0.72256

0.69512

0.0080

0.0045

37.61

0.1791

2512

25.2

1010 512 10 512 366 447 007

×

3 812 019

105 654 651

0.03608

0.75864

0.72256

0.0165

0.0080

27.05

0.1288

10 512

21.0

Table 9.5 Fracture mechanics calculation for spectrum E (see Table 8.5) with peak stress = 210 N/mm2 Spectrum E Smax = 210

1 010 512 110 512 150 772 291

×

16 488 817

338 440 417

0.04872

0.80736

0.75864

0.0416

0.0165

18.35

0.0874

110 512

16.8

135 004 735 1 685 152 971 (= 1667 blocks)

× 1.0

135 004 735

849 941 671

0.15884

0.96620

0.80736

0.6

0.0416

13.50

0.0643

1 010 512

12.6

Design for variable amplitude loading

283

life to be expected under Spectrum E (see Table 8.5), which had a very large number of small stress cycles, and for which the peak stress range was 210 N/mm2. The calculations for this problem are set out in Table 9.5, from which it will be seen that the calculated life is 1685 × 106 cycles (1667 blocks). The method of calculation is precisely the same as that set out in Table 9.4; it is only the spectrum that is different. However, unfortunately, that is not really the end of the story, because it has to be recognised that the calculated life is critically dependent on the assumptions made and particularly on the selected values of C and, particularly, on the initial defect size (ai). In a real situation neither of those values will be available at the design stage. Consequently it has to be concluded that fracture mechanics is not a suitable method by which to try to define precise fatigue strengths or lives. Hence if the objective of the calculation is to define a particular fatigue strength or life the only sensible approach is to make the most pessimistic assumptions and to accept that the only possibility is to use the method to derive a realistic lower limit estimate of life. This is not to say that fracture mechanics can never be used in the fatigue context. For example it is a very useful method for carrying out parametric studies, where the objective is to define the relative influence of a particular set of variables. In that situation all the variables, except the one under consideration, can be held constant and thereby enable its influence to be assessed. In this type of situation, however, there would be little benefit in using variable amplitude loading; constant amplitude loads would be quicker, easier and just as effective. Some other situations in which the use of fracture mechanics may be helpful are when: • assessing the fitness for purpose of a structure known to contain flaws whose size, shape and distribution are outside normally acceptable limits but which would be difficult to repair; • the effects of relatively minor variations in the geometrical or stress parameters for a given detail are being studied; • the joint detail under consideration is unusual and is not adequately represented by one of the standard joint classifications, or when a joint is subjected to the influence of another stress concentration; • defining the frequency of in-service inspections; • assessing the remaining fatigue life of a structure in which fatigue cracks already exist.

10 More on the fracture mechanics approach – the effect of stress interaction

10.1

Introduction

In Chapter 1 it was noted that the vast majority of fatigue cracks in welded joints initiate either from small, sharp crack-like intrusions at the weld toe or, in the case of load-carrying fillet welded joints, from the crack-like lack of penetration at the weld root. The existence of these defects effectively means that there is no need for a crack initiation phase in the life of welded joints and that crack propagation commences at the beginning of life and continues through the life of the joint. This means that they should be ideally suited to fracture mechanics analysis. Furthermore, it was shown that, if such an analysis were carried out for a joint subjected to constant amplitude loading, the resulting equation of the curve relating the applied stress range to the life of the joint was identical in form to that found experimentally. Subsequently, in Chapter 4, it was also shown that Miner’s rule is wholly consistent with the fracture mechanics approach provided that it is assumed that there is no stress interaction between the various stress cycles (in other words, that the rate of propagation due to the particular stress cycle under consideration is not affected by any of the stress cycles which have been applied before it). However, as we have seen in earlier chapters, Miner’s rule is not always accurate and it therefore follows that the discrepancies in Miner summations almost certainly stem from errors in the assumption that damage accumulation is necessarily linear. In fact it is now well established that that assumption is often invalidated by sequence effects, whereby the individual cycles in the loading spectrum interact with each other. This can lead both to faster or slower crack growth than predicted by the rule. In the light of this situation it is evident that the best chance of defining, at some stage in the future, ‘a better rule than Miner’, will be to consider crack propagation on a cycle by cycle basis taking account of sequence effects as they occur. This will require understanding of sequence effects and the means to quantify them. It may therefore be helpful to include here a brief review of currently available 284

More on the fracture mechanics approach

285

published information, although it has to be noted that virtually all of it refers to parent materials of several different types, rather than welded joints. If nothing else, it is hoped that this information may help to explain why Miner’s rule often gives incorrect answers. Whether it will actually lead to a more accurate approach in the future is hard to predict; one suspects that the overriding problem that will always exist is that it will rarely, or even never, be possible to know in advance the order in which successive stresses will impinge upon the structure, and that information may well be essential before accurate forecasts can be made of the expected life. So far as this particular review is concerned, it has to be accepted that, although a considerable amount of research has been carried out into fatigue loading sequence effects, the vast majority of this work has been focused on effects which reduce the rate of crack growth and hence result in values of n which are greater than 1.0. However, from the safety viewpoint, it is ΣN obviously features which have the opposite effect which are the more important. Sequence effects, resulting either from underloads or overloads, can take many different forms and they have, on occasions, been ascribed to several different causes. In general, however, overloads tend to have a more significant effect than underloads. The general form of the relationship between da/dN and crack length following an overload is as shown in Fig. 10.1. Typically, the relationship can be divided into four stages, as follows:

A

D

da/dN (mm/cycle) Log scale

erload Without ov

B C

Overload

O Crack length (mm), distance since overload

10.1 Typical effect on rate of crack propagation following an overload.

286

Cumulative damage of welded joints

1. On application of the overload there may be an immediate increase in crack growth rate, both for the overload cycle itself and for the cycles immediately following it. 2. Subsequently, the growth rate decelerates to a minimum. If stage A is present, this is usually referred to as ‘delayed retardation’, but if stage A is not present the overload may be followed by immediate, rather than delayed, retardation. 3. The growth rate then re-accelerates to return to the expected rate in the absence of any overload. 4. Sometimes the re-acceleration involves ‘overshoot’ of the expected rate to some higher rate, followed by a fall-back to the projected rate, but this stage is often absent. As noted above, several mechanisms have been put forward which may contribute to these crack growth retardation phenomena, although it may well be that no single mechanism can account for all retardation behaviour. The principal models have included: • The formation of residual compressive stresses ahead of the crack tip, leading to a slower rate of growth, e.g., Schijve (1960); Bernard, et al., (1976); Damri and Knott (1991); Matsuoka and Tanaka (1978). • Crack tip blunting, followed by re-sharpening and re-initiation e.g., Christensen (1959); Hudson and Hardrath (1961). • Strain hardening in the plastic zone at the crack tip (Jones, 1973; Knott and Pickard, 1977). • The state of stress (plane stress or plane strain) (Damri and Knott, 1991; Fleck, 1988). • Crack closure e.g., Elber (1970a,b); von Euw et al., 1972). This results from the overload causing large tensile deformations in the material ahead of the crack tip so that, when the crack propagates through that material, a wake of residual plastic deformation is left on the crack edges, which causes the crack to shut before the minimum load is reached. As a result the strain range at the crack tip is less than it otherwise might have been. Of these, the contributions of residual stresses and of crack closure have received the most attention. With regard to the former, both Willenborg et al., (1971) and Wheeler (1970) proposed models which are supposed to account for the effect of residual stresses by reducing the effective rate of fatigue crack propagation. In the case of the Wheeler model, the growth rate is reduced by a factor which incorporates an empirical shaping factor which varies according to the specific material, load spectrum and other factors; it therefore relies upon data fitting. Subsequently, Gray and Gallagher (1976) attempted to modify the Wheeler model to avoid the need for data fitting and to extend its generality. Meanwhile, the Willenborg model imposes a reduction on both the maximum and minimum values of stress intensity at the crack tip, this reduction being

More on the fracture mechanics approach

287

supposed to represent the effect of the compressive residual stresses around the crack tip due to the overload. In spite of many statements in the literature to the contrary, it is important to recognise that, whatever else it may be, the parameter derived by Willenborg certainly is not the residual stress due to the overload. That would depend only on the magnitude of the overload, whereas Willenborg’s stress reduction depends both on the overload and on the magnitude of the succeeding stresses. However, assuming that retardation ceases when the further boundary of the current plastic zone reaches the far boundary of the overload plastic zone (an important assumption, discussed in more detail below), the model gives a sensible answer (i.e. that there is no retardation) in the two limiting cases: • when the overload is immediately followed by a stress equal to the overload and • when the crack has grown sufficiently for the boundary of the current crack tip plastic zone to coincide with the boundary of the overload plastic zone. Intermediate values would appear to be much more problematical. Nevertheless, in spite of the rather curious assumptions made in deriving the magnitude of the stress reduction, the model has been used by several investigators and has been shown to give passable answers. However, in the light of the method of derivation, it might be wise to regard the model as essentially another empirical approach, in which the reduction in stress intensity is related to the position of the crack tip in the overload plastic zone. However, as has previously been noted by Schijve (1976) both these models effectively deal only with retardation and predict maximum retardation immediately after an overload. In fact, as discussed below, an overload is sometimes followed by crack growth acceleration, rather than retardation. In consequence, it is clear that the models certainly cannot be wholly correct and as Schijve and several other investigators, e.g., Carlson, et al., (1991) have pointed out, systematic agreement with test results has rarely been found. As far as crack closure is concerned, it has been noted by McEvily (1988) that plasticity induced closure (outlined above), which was the first type of closure to be considered, is essentially a plane stress effect. There are, however, other types of closure, of which the two most important are probably roughness induced closure and crack filling closure. The former results from mismatch of fracture surface asperities on unloading, while the latter can be caused by the presence of corrosion products, fretting debris, etc. Both may occur either under plane strain or plane stress conditions.

10.2

Summary of experimental evidence about stress interaction effects

At this stage, it may be helpful to consider the available experimental evidence relating to stress interaction effects. Of necessity, such evidence has been

288

Cumulative damage of welded joints

obtained in tests using relatively simple types of loading, and the main variants are shown in Fig. 10.2. Type 1 loading By far the greatest number of tests have involved the application of a single tensile overload in a constant amplitude sequence (Type 1 in Fig. 10.2). However, they have involved several different variables, in particular the value of the overload ratio (Kpeak/Kmax), the stress ratio of the constant amplitude loading (Kmin/Kmax) and the magnitude (∆K) of the constant amplitude loading. Considering first the influence of the overload ratio (OLR) it has been found in several investigations (Bernard et al., 1976; Garwood, 1978; Matsuoka et al., 1976; Voorwald et al., 1991) that there is a threshold level of ratio below which the application of an overload seems to have no effect on the rate of crack propagation. Its value has varied quite widely, from at least 1.2 for Ti6Al4V up to 1.5 for HT80 steel. Intermediate values in the region of 1.33 have been reported for other steels. (These are, incidentally, lower limit values of the ratio; the actual transition point from ‘no effect’ to ‘retardation’ may be slightly higher.) At the other end of the scale high values of OLR, in the region of 2.5–3.0, have led to crack arrest (Blom, 1989a; Garwood, 1978; Stephens et al., 1976), although if the overload is so high as to give net section yielding the retardation is reduced. For intermediate values of OLR it has been found that there is a general tendency for the number of delay cycles to increase, and for the minimum growth rate to decrease, as the value of OLR increases (Bernard et al., 1976; Chand, 1992; Kumar, 1991; Lu and Li, 1993; Matsuoka et al., 1976; Stephens et al., 1976). Indeed for 6061-T6 aluminium alloy Kumar (1991), for values of OLR between 1.75 and 2.5, found a linear loglog relationship between OLR and the number of delay cycles. Several studies have been undertaken with OLR approximately equal to 2.0 but with varying values of ∆K for the constant amplitude cycles (Dhar, 1989; Matsuoka et al., 1976; Ranganathan et al., 1984). These showed a clear difference in behaviour for differing values of ∆K; low values tended to result in immediate retardation while higher values tended to result in initial acceleration followed by delayed retardation. Some typical results reported by Ranganathan et al. (1984) are shown in Fig. 10.3. It is interesting to note that this change in behaviour corresponded to the state of stress. Tests in which the overload stress involved plane strain (i.e. with Kpeak less than σ Y

T where T is the thickness) 2.5

gave immediate retardation, while higher values of Kpeak gave delayed retardation. This corresponds to the earlier finding of Bernard et al. (1976).

More on the fracture mechanics approach Type 1. Single peak overload

∆k

289

Kpeak Kmax Kmin

2. Multiple peak overload

3. Step change down

4. Step change up

5. Low–high overload

6. High–low overload

7. Step change of Smean down

8. Step change of Smean up

9. Step change, constant Kmax

10. Single underload

11. Multiple underloads

10.2 Main types of loading used in the study of stress interaction effects.

Delayed retardation under mixed mode or plane stress conditions has also been reported by many other investigators, e.g., Chand (1992) Chand and Garg (1984); Fleck and Shercliff (1989); Jacoby et al. (1976); Kumar (1991); Lu and Li (1993); Machida et al., (1984); Nowack et al., (1979); Shin and Fleck (1987); Topper and Yu (1985); Zuidema et al., (1991). Relatively few

290

Cumulative damage of welded joints Behaviour obtained with

da/dN (mm/~)

7.0 < ∆k < 8.0 MPa m (Note: for ∆k = 5.6 MPa m total crack arrest occurred)

10–5

10–6

22

23

S

25

a(mm) Behaviour obtained with

da/dN (mm/~)

10–3

10.0 < ∆k < 13.5 MPa m Gradual deceleration followed by sudden re-acceleration

10–4

10–5 S 28

30 32 a(mm) Behaviour obtained with

da/dN (mm/~)

10–3

13.5 < ∆k < 20.2 MPa m gradual deceleration and re-acceleration

10–4

10–5 S 34

36

38

40

a(mm)

10.3 Results obtained by Ranganathan et al. (1984) for 2024 Al alloy under Type 1 loading with OLR = 2.

other results seem to exist for plane strain conditions, although a result reported by Fleck and Shercliff (1989) suggests that in that case there was a very small degree of acceleration preceding delayed retardation; even then, however, the rate of decrease of da/dN after the overload was very steep so that, for all practical purposes, retardation might be assumed to be immediate.

More on the fracture mechanics approach α /β annealed titanium

B

10–3

291

C

100%

Crack growth rate (mm/cycle)

15 MPa m

10–4

A

D

F

10–5

E

10–6

Crack growth due to overload cycle

0 0.1 0.2 0.3 Distance from start of overload (mm)

0.4

10.4 Delayed retardation under plane strain conditions as reported by Ward Close et al. (1989) for IMI 550 alloy (thickness = 10 mm, σ = 984 N/mm2). Note: A = steady-state crack growth rate, B = crack during overload cycle, C – D = accelerated crack growth, D – E = retarded crack growth and E – F gradual return to steady state.

Very similar behaviour has also been reported by Ward-Close et al. (1989) for α/β annealed titanium alloy (IMI 550) (see Fig. 10.4). Some interesting additional evidence on this subject has also been obtained by Blom (1989b). In tests on two different steels, with yield strengths of 690 N/mm2 and 350 N/mm2 respectively, he found that, with OLR = 2.0, the higher-strength material gave immediate retardation, with the minimum growth rate after only very small crack growth. In the lower-strength material, however, there was initial acceleration and delayed retardation; the maximum retardation occurred after significant crack growth and the total retardation was much greater. The difference in behaviour can be explained by the fact that, although the stress level and OLR were the same in both steels, the higher-strength material was in plane strain during overloading while the lower-strength steel was in plane stress. For welded joints the significance of welding residual stresses on retardation

Cumulative damage of welded joints

Number of cycles (N)

292

da/dN

a*

Crack length

10.5 Typical retardation behaviour following an overload in the absence of initial acceleration and delayed retardation.

effects was demonstrated by Garwood (1978). The effect of a single tensile overload repeated every 3 × 105 cycles was to cause the slope of the aswelded constant amplitude S-N curve to become less steep. Stress-relieved specimens showed larger retardation effects and these occurred at all ratios of overload to base stress. For as-welded specimens an overload exceeding 30% of the base level was required to cause noticeable increases in fatigue life. It has often been postulated, e.g., Druce et al. (1979); Lu and Li (1993), that the length (a*) of the overload affected crack growth increment (i.e. starting at the position of the overload and ending where the crack growth rate returns to that existing prior to the overload; see Fig. 10.5) must be related to the plane stress plastic zone size caused by the overload. Some investigators, e.g., Chand (1992); Lankford and Davidson (1976); Lu and Li (1993), have claimed that there is a good correlation between a* and the overload plastic zone diameter (ZOL), while others, e.g., Druce et al. (1979),

More on the fracture mechanics approach

293

ZOL Plastic zone due to overload Current crack tip Crack

Z (a)

a*

Z (b)

a* = ZOL

Z

(c)

10.6 Possible definitions of overload affected crack growth increment (a*). (a) Situation immediately following the overload. (b) Situation at the end of interaction, as proposed by Druce. (c) Situation at end of interaction if a* = Z0L.

have maintained that interaction ceases when the crack has grown sufficiently for the current plastic zone to reach the boundary of the overload plastic zone (see Fig. 10.6). In order to try to assess the validity of these two suggestions, Figs 10.7

294

Cumulative damage of welded joints 8

Length of overload affected zone

7

6

Stress Overload Pl strain Mixed Mixed Pl stress Pl stress

state CA load Pl strain Pl strain Mixed Mixed Pl stress

5 4 3 2 1 0 0 1 2 3 4 5 6 Plane stress plastic zone size due to overload (mm)

Length of overload affected zone (mm)

10.7 Relationship between length of overload affected zone and the plane stress plastic zone size due to the overload.

8 7 6

Stress Overload Pl strain Mixed Mixed Pl stress Pl stress

state CA load Pl strain Pl strain Mixed Mixed Pl stress

ZOL

5 Direction of crack propagation

4 3

ZOL – Z

Z

2 1 0

0 1 2 3 4 5 6 Difference between overload and CA plane stress plastic zone sizes

10.8 Relationship between length of overload affected zone and (Z0L – Z) (assuming plane stress throughout).

More on the fracture mechanics approach

295

and 10.8 show the relationships between the measured values of a* and the corresponding calculated values ac*, namely ac* = ZOL and ac* = ZOL – Z, where Z is the plane stress plastic zone diameter due to the current load. Unfortunately, relatively few investigators (Bernard et al. 1976; von Euw et al. 1972; Lu and Li, 1993; Ranganathan et al., 1984) have presented results in sufficient detail to enable them to be used in this way. However, the results that do exist have been divided to indicate the theoretical stress states both under the overload and under the subsequent constant amplitude loading. For the purposes of this exercise it has been assumed that plane strain exists when 2

  T ≥ 2.5  K  or K ≤ σ Y σ  y

T 2.5

[10.1]

and plane stress exists when 2

  T ≤ plastic zone size = 1  K  or K ≥ σy πT π  σy 

[10.2]

Mixed mode conditions are assumed to exist between those two limits. From Fig. 10.7 it will be seen that the results expressed in terms of a* and ZOL (derived on the basis of assumed plane stress conditions) are extremely scattered. A reasonable correlation between a* and ZOL only exists for those specimens subjected to mixed mode conditions during both overloading and subsequent constant amplitude loading. The majority of the specimens subjected to plane stress conditions during overloading and mixed mode conditions afterwards had larger-than-predicted overload affected zones, while all other combinations of stress conditions had overload affected zones which were smaller than the plane stress plastic zone size due to the overload. For these latter stress conditions it therefore did seem possible that there might well be a better correlation between a* and (ZOL – Z). Figure 10.8 shows, however, that the correlation is no better, except for the specimens subjected to mixed mode loading during the overload cycle and plane strain conditions thereafter. In general the length of the overload affected region for those specimens was only slightly smaller than predicted. However, the specimens subjected either to plane stress or mixed mode loading during overloading and followed by mixed mode loading certainly had overload affected regions which were larger than predicted, while specimens subjected to fully plane strain conditions had overload affected zones that were smaller than predicted. Since Figs 10.7 and 10.8 suggest that neither of the proposed relationships are wholly satisfactory, an attempt was also made to relate a* to the actual theoretical plastic zone size (i.e. allowing for the existence of mixed mode conditions where appropriate). This was approximated in a somewhat arbitrary

296

Cumulative damage of welded joints

manner, as follows. Given that the size of the overload plastic zone can be written as

Z OL

K  = 2 ⋅ 1  OL  d ⋅ π  σy 

2

[10.3]

where d = 2 for plane stress and 6 for plane strain, the relevant value of d was defined by linear interpolation of KOL between the relevant plane strain and plane stress values defined by eqns [10.1] and [10.2]. Then, knowing the value of d, the corresponding value of ZOL could be derived from eqn [10.3]. The results are shown in Figs 10.9 and 10.10; Fig. 10.9 compares the length of the overload affected zone to the plastic zone size due to the overload, while Fig. 10.10 compares it with ZOL – Z (both being based on the actual relevant stress conditions). Given the amount of scatter in the test results, perfect correlation between the theoretical and predicted lengths of the overload affected zones is obviously unattainable, but Fig. 10.9 is obviously a considerable advance on Figs 10.7 and 10.8. It is only the specimens subjected to plane stress conditions throughout which are very badly represented. For the other loading conditions a slight improvement in correlation can be obtained by assuming, empirically, that a* = 1.15 × ZOL. Figure 10.10 obviously shows far less satisfactory correlation, overall, than Fig. 10.9. To some extent it has to be admitted that the results summarised in Figs 10.7 to 10.10 are rather less than satisfactory due to the fact that so few investigators have presented usable results. However, the available information does not appear to support the published view that the length of the overload affected zone is related primarily to the plane stress plastic zone size. A summary of the conclusions emanating from a study of Figs 10.7 to 10.10 is shown in Table 10.1. Given that the overload affected zone is predominantly a region where the crack growth rate is reduced, any part of the table where it is noted that the actual length of the affected zone is longer than the predicted length implies that the corresponding prediction will be safe. On the other hand, when the actual length is less than the predicted length the prediction will be unsafe. The basic assumption of the Willenborg model is that the length of the retardation zone corresponds to that for which results are summarised in Fig. 10.8. It will be noted from Table 10.2 that it would be expected to give safe predictions only when the load cycles following the overload lead to mixed mode plastic zones. In a more recent study of the effect of thickness on fatigue crack growth retardation, Shuter and Geary (1995) studied thicknesses ranging from 5 to 25 mm in BS 4360 grade 50D material. Retardation effects were found to decrease with increasing specimen thickness following a single 100% tensile overload. They conclude from their studies that crack closure is the dominant mechanism for retardation effects at low R ratios. At high R ratios crack tip blunting and induced residual stresses are operative.

More on the fracture mechanics approach Stress state Overload CA load

8

Length of overload affected zone (mm)

297

Pl strain Mixed Mixed Pl stress Pl stress

7

6

Pl strain Pl strain Mixed Mixed Pl stress

5

4 3

2

1 0

0

1

2 3 4 5 Overload plastic zone size (mm)

6

10.9 Comparison of the overload affected size zone with overload plastic zone size.

Length of overload affected zone (mm)

8 7 6

Stress state Overload CA load Pl strain Mixed Mixed Pl stress Pl stress

Pl strain Pl strain Mixed Mixed Pl stress

5 4 3 2 1 0 0 1 2 3 4 5 6 Difference between overload and CA plastic zone sizes

10.10 Comparison of the length of the overload affected zone with the difference between the overload and constant amplitude plastic zone sizes.

a* a*pred

a* >> a*pred

a* < a*pred

a* a*pred

a* > a*pred

a* = apred

a* _ a*pred

Fig. 10.9

Fig. 10.8 a* > a*pred

a* >> a*pred

a* > a*pred

a* _ a*pred

Fig. 10.10

ZOL – Z (both based on actual conditions)

Note: The relationships between a* and a*pred set out in this table are necessarily subjective and approximate. For example a* = a*pred merely indicates that there is a reasonable fit between the two values, not that there is perfect agreement

a* a*pred

a* 200

12 35 182 0 0

15.1–25.5

≤51.5 51.6–102.5 102.6–153 153.1–200 >200

0 15 39 0 0

≤51.5 51.6–102.5 102.6–153 153.1–200 >200

0 0 30 0 0

≥25.6

* Only 1 valid result

Mean

Mean –2s.d.

Correlation coefficient

s.d. of ln N

0.1252 0.0938

0.958 0.933

0.2168 0.4444

4 3

2.956 2.984

* 3.330

272 232

235 171

97 84

84 62

3.151 2.771 2.803

0.4882 0.0647 0.0558

0.965 0.956 0.964

0.1809 0.2781 0.2761

3 3 36

3.139 2.481 2.370

* 3.342 3.331

275 287 255

245 235 210

106 97 88

95 80 72

2.858 2.954

0.0555 0.1233

0.979 0.977

0.2006 0.2504

2 7

2.637 2.785

* 3.394

229 251

199 212

80 91

70 77

2.959

0.1128

0.935

0.3640

5

2.446

3.324

242

189

88

69

(a) Slopes of S-N curves not pre-defined ≤10.5 ≤51.5 0 51.6–102.5 31 2.915 102.6–153 18 2.948 153.1–200 0 >200 0 10.6–15.0

No. of data sets

Fatigue strengths, N/mm2 ——————————————— 105 2 × 106

Range of values of slope (m) ——————— Min. Max.

Mean

Mean –2s.d.

Table A5 Continued

Joint geometry, mm ———————————— Plate Attachment thickness length

No. of results

Coefficients of best-fit equation ————————— C m (× 1013)

(b) Slopes of S-N curves pre-defined as m = 3.0 ≤10.5 ≤51.5 0 51.6–102.5 31 3.0 0.1954 102.6–153 18 3.0 0.1208 153.1–200 0 >200 0 10.6–15.0

≤51.5 51.6–102.5 102.6–153 153.1–200 >200

12 35 182 0 0

15.1–25.5

≤51.5 51.6–102.5 102.6–153 153.1–200 >200

0 15 39 0 0

≤51.5 51.6–102.5 102.6–153 153.1–200 >200

0 0 30 0 0

≥25.6

Correlation coefficient

s.d. of ln N

0.960 0.935

0.2178 0.4448

No. of data sets

Range of values of slope (m) ——————— Min. Max.

Fatigue strengths, N/mm2 ——————————————— 105 2 × 106 Mean

Mean –2s.d.

4 3

269 229

233 171

99 85

86 63

Mean

Mean –2s.d.

3.0 3.0 3.0

0.2257 0.2096 0.1449

0.961 0.960 0.967

0.1837 0.2880 0.2849

3 3 36

283 276 244

250 228 202

104 192 90

92 84 74

3.0 3.0

0.1057 0.1547

0.980 0.978

0.2062 0.2510

2 7

219 249

191 211

81 92

70 78

3.0

0.1381

0.937

0.3642

5

240

188

88

69

Table A6 Summary of results for transverse non-load-carrying fillet welds

Joint geometry, mm ———————————— Plate Attachment thickness thickness

No. of results

Coefficients of best-fit equation ————————— C m (× 1013)

(a) Slopes of S-N curves not pre-defined 0–10.5 0–10.5 181 2.981 10.6–15.0 168 3.064 15.1–25.5 0 ≥25.6 11 3.051

Correlation coefficient

s.d. of ln N

No. of data sets

0.536 0.359

0.825 0.974

0.5142 0.2644

17 14

2.410 2.272

0.379

0.957

0.2797

1

0.452 0.942 0.735

0.921 0.888 0.919

0.4645 0.504 0.430

9 14 30

3.047 3.051 2.333

4.183 4.426 4.619

332 285 273

245 209 209

124 113 108

91 83 83

2.589

3.547

227

185

84

69

222

176

98

78

10.6–15.0

0–10.5 10.6–12.0 12.1–15.0 15.1–25.5 ≥25.6

90 180 226 0 48

3.036 3.248 3.229 3.024

0.134

0.961

0.3111

7

15.1–25.5

0–10.5 10.6–15.0 15.1–25.5 ≥25.6

8 0 106 13

3.682

4.33

0.923

0.4228

1

2.751 3.596

0.058 2.86

0.850 0.957

0.570 0.3703

15 3

0–10.5 10.6–15.0 15.1–25.5 ≥25.6

0 16 0 33

≥25.6

* Only 1 valid result

Fatigue strengths, N/mm2 ——————————————— 105 2 × 106

Range of values of slope (m) ——————— Min. Max.

4.694 3.894

Mean

Mean –2s.d.

Mean

Mean –2s.d.

392 292

277 246

143 110

102 93

305

254

114

95

2.77 3.33

4.74 3.41

287 225

190 183

97 98

64 79

3.26

*

222

174

86

68

3 3.175

0.281

0.857

0.3895

5

Table A6 Continued

Joint geometry, mm ———————————— Plate Attachment thickness thickness

No. of results

Coefficients of best-fit equation ————————— C m (× 1013)

(b) Slopes of S-N curves pre-defined as m = 3.0 0–10.5 0–10.5 181 3.0 0.593 10.6–15.0 168 3.0 0.258 15.1–25.5 0 ≥25.6 11 3.0 0.293 10.6–15.0

15.1–25.5

≥25.6

Correlation coefficient

s.d. of ln N

No. of data sets

0.827 0.973

0.5142 0.2655

Range of values of slope (m) ——————— Min. Max.

Fatigue strengths, N/mm2 ——————————————— 105 2 × 106 Mean

Mean –2s.d.

Mean

Mean –2s.d.

17 14

390 296

277 248

144 109

102 91

0.955

0.2801

1

308

256

114

94

0–10.5 10.6–12.0 12.1–15.0 15.1–25.5 ≥25.6

90 180 226 0 48

3.0 3.0 3.0

0.375 0.261 0.229

0.919 0.870 0.907

0.4647 0.5096 0.4365

9 14 30

335 297 284

245 211 212

123 109 105

90 78 78

3.0

0.119

0.960

0.3112

7

228

185

84

68

0–10.5 10.6–15.0 15.1–25.5 ≥25.6

8 0 106 13

3.0

0.175

0.373

0.4631

1

260

191

96

70

3.0 3.0

0.203 0.150

0.867 0.925

0.5761 0.4228

15 3

273 247

186 186

100 91

68 69

0–10.5 10.6–15.0 15.1–25.5 ≥25.6

0 16 0 33

3.0

0.260

3

296

3.0

0.118

5

227

0.843

0.3912

109 175

84

65

Table A7 Summary of results for non-load-carrying stud welded attachments Coefficients of best-fit equation ————————— C m (× 1013)

s.d. of ln N

No. of data sets

(a) Slopes of S-N curves not pre-defined 61 3.215 0.6208 0.888

0.2699

9

(b) Slopes of S-N curves pre-defined as m = 3.0 61 3.0 0.2164 0.873

0.2722

9

No. of results

Correlation coefficient

Fatigue strengths, N/mm2 ————————————————— 105 2 × 106

Range of values of slope (m) Min.

Max.

Mean

Mean –2s.d.

Mean

Mean –2s.d.

2.55

4.26

265

224

104

88

279

232

103

86

Table A8 Summary of results for plates with gussets welded to the plate edge Coefficients of best-fit equation ———————— C m (× 1013)

Joint geometry

No. of results

(a) Slopes of S-N curves not Gusset on plate edge (butt welded) Attachment ≤51.5 length 51.6–102.5 102.6–153 153.1–200 ≥200.1

pre-defined

Gusset on plate edge (fillet welded) Attachment ≤51.5 length 51.6–102.5 102.6–153 153.1–200 ≥200.1

0 22 0 18 16

0 34 25 0 11

Correlation coefficient

s.d. of ln N

No. of data sets

Range of values of slope (m) Min. Max.

Fatigue strengths, N/mm2 ——————————————— 105 2 × 106 Mean

Mean –2s.d.

Mean

Mean –2s.d.

294

217

93

68

2.596

0.0256

0.905

0.3933

1

3.992 2.966

4.825 0.0644

0.970 0.936

0.2956 0.4037

2 4

3.875 2.666

4.230 3.388

150 197

129 150

71 72

61 55

3.543 2.653

0.5581 0.0117

0.874 0.991

0.5903 0.1695

4 5

3.158 3.016

4.546 3.123

154 194

110 171

66 63

47 55

2.788

0.0318

0.986

0.1912

2

2.734

2.925

215

188

73

64

Table A8 Continued

Joint geometry

No. of results

Coefficients of best-fit equation ———————— C m (× 1013)

Correlation coefficient

s.d. of ln N

No. of data sets

Range of values of slope (m) Min. Max.

Fatigue strengths, N/mm2 ——————————————— 105 2 × 106 Mean

Mean –2s.d.

Mean

Mean –2s.d.

(b) Slopes of S-N curves pre-defined as m = 3.0 Gusset on plate edge (butt) welded Attachment ≤51.5 0 length 51.6–102.5 22 3.0 0.0256 102.6–153 0 153.1–200 18 3.0 4.100 ≥200.1 16 3.0 0.0756

0.919

0.4141

1

273

207

101

76

0.905 0.937

0.4154 0.4038

2 4

160 196

121 150

59 72

45 55

Gusset on plate edge (fillet welded) Attachment ≤51.5 length 51.6–102.5 102.6–153 ≥200.1

0.826 0.987 0.985

0.6124 0.2374 0.2097

4 5 2

166 186 205

110 159 178

61 69 75

41 59 66

0 34 25 11

3.0 3.0 3.0

0.0455 0.0647 0.0356

Table A9 Summary of results for beams with welded cover plates

Joint geometry

No. of results

Coefficients of best-fit equation ———————— C m (× 1011)

(a) Slopes of S-N curves not pre-defined Beam with cover plates Ends welded 95 2.464 Ends unwelded 58 2.327

0.3773 0.2905

(b) Slopes of S-N curves pre-defined as m = 3.0 Beam with cover plates Ends welded 95 3.0 4.521 Ends unwelded 58 3.0 7.018

Correlation coefficient

s.d. of ln N

No. of data sets

Range of values of slope (m) —————— Min. Max.

0.949 0.916

0.3033 0.3535

21 11

2.883 2.616

0.951 0.926

0.3624 0.4231

21 11

3.902 4.909

Fatigue strengths, N/mm2 ——————————————— 105 2 × 106 Mean

Mean –2s.d.

Mean

Mean –2s.d.

183 223

143 164

54 61

43 45

165 191

130 144

61 71

48 53

Table A10 Summary of results for transverse load-carrying fillet welded joints (toe failure)

No. of results

Coefficients of best-fit equation ———————— C m (× 1013)

(a) Slopes of S-N curves not pre-defined Partial penetration 29 3.180 cruciform joint Transverse load-carrying fillet weld Plate ≤10.5 thickness 10.6–15.0

No. of data sets

Fatigue strengths, N/mm2 ——————————————— 105 2 × 106 Mean

Mean –2s.d.

Correlation coefficient

s.d. of ln N

0.4464

0.908

0.3621

9

3.365

3.400

254

203

99

79

2.461 4.033 No valid results 3.061 3.510 3.836 *

266 268

184 132

100 63

69 31

232 186

169 163

83 88

60 77

230 274 294

215 225 247

106 95 94

100 78 79

44 13

3.051 2.075

0.2487 0.0011

0.907 0.731

0.5618 0.7364

5

15.1–25.5 ≥25.6

36 22

2.911 3.985

0.0769 11.13

0.912 0.966

0.4643 0.2655

12 10

Lap plate joint with transverse welds Plate ≤10.5 thickness 10.6–15.0 15.1–25.5 ≥25.6

7 18 6 0

3.886 2.835 2.633

150.3 0.8118 0.3167

0.995 0.976 0.967

0.1253 0.2758 0.2298

1 4 1

* Only 1 valid result

Range of values of slope (m) ——————— Min. Max.

3.509

3.851

Mean

Mean –2s.d.

Table A10 Contined

No. of results

Coefficients of best-fit equation ———————— C m (× 1013)

(b) Slopes of S-N curves pre-defined as m = 3.0 Partial penetration 29 3.0 0.1846 cruciform joint

Correlation coefficient

s.d. of ln N

0.897

0.3649

No. of data sets

Range of values of slope (m) ——————— Min. Max.

Fatigue strengths, N/mm2 ——————————————— 105 2 × 106 Mean

Mean –2s.d.

Mean

Mean –2s.d.

9

264

207

97

76

5

184 119 168 162

99 75* 84 76

68 44 62 60

218 221 224

99 98 98

80 81 83

Transverse load-carrying fillet weld Plate ≤10.5 thickness 10.6–15.0 15.1–25.5 ≥25.6

44 13 36 22

3.0 3.0 3.0 3.0

0.1927 0.0857 0.1193 0.0878

0.904 0.813 0.916 0.901

0.5622 0.8162 0.4653 0.3630

12 10

268 205 228 206

Lap plate joint with transverse welds Plate ≤10.5 thickness 10.6–15.0 15.1–25.5 ≥25.6

7 18 6 0

3.0 3.0 3.0

0.1920 0.1909 0.1900

0.951 0.977 0.967

0.3111 0.2851 0.2601

1 4 1

268 267 267

Table A11 Summary of results for longitudinal load-carrying fillet welds

Correlation coefficient

s.d. of ln N

No. of data sets

Range of values of slope (m) ——————— Min. Max.

0.2437 0.1890

0.950 0.923

0.3440 0.4827

2 4

2.819 3.090

(b) Slopes of S-N curves pre-defined as m = 3.0 Longitudinal load-carrying fillet weld Main plate failure 10 3.0 0.1230 Cover plate failure 20 3.0 0.0398

0.954 0.929

0.3675 0.4874

2 4

No. of results

Coefficients of best-fit equation ———————— C m (× 1012)

(a) Slopes of S-N curves not pre-defined Longitudinal load-carrying fillet weld Main plate failure 10 2.669 Cover plate failure 20 2.835

3.134 3.581

Fatigue strengths, N/mm2 ——————————————— 105 2 × 106 Mean

Mean –2s.d.

Mean

Mean –2s.d.

247 164

191 116

80 57

62 40

231 158

181 114

85 58

67 42

Appendix B Fatigue loading spectra

B1

Introduction

In the various summary tables of variable amplitude fatigue test results contained in Appendix C, and also in some of the main chapters, it has only been possible to specify the forms of the loading spectra used in the various investigations in a very abbreviated manner. Further details of the spectra, so far as these are available, are therefore summarised in Table B1. In addition, several of the spectra have been used in more than one investigation, often by different investigators, and further details of them are also given in this appendix. This information may be of particular use to other workers wishing to use the same spectra for comparative tests in other work. The particular spectra for which these additional details are given are as follows: Section B3 B4 B5 B6 B7 B8 B9 B10

Two parameter Weibull distribution, including Rayleigh and Laplace distributions. Rayleigh distribution as used by Schilling et al. in the USA Gauss spectrum (with I = 0.99 and 0.7) as used by Haibach and Overbeeke. Offshore spectrum WASH Offshore spectrum C/12/20 UKOSRP II wide band spectrum S4 Spectra used in the Nordic test programme (VAMP, SPI-SP5, BROAD, PMMOD and NARROW) LBF (Block programme) spectrum.

However, before going on to consider any of these specific spectra it may be helpful to look at the Markov matrix in a little detail, since it can be a useful method of defining the general ‘shape’ of a spectrum.

325

S14 Superimposed sine waves, with 15 small cycles (p = 0.429) on each large cycle S15 As S14, 9 cycles with p = 0.154

Swensson 1 Swensson 2 Swensson 3 26 level test truck programme Rheden Bridge (shortened) Rail car bogey 16

10

0.429

0.154

850 710 99 27 99 820

524 286

0.063 0.02 0.02 0.02 0.002 0.22

524 221

0.094

    

524 287

0.063

Traffic loading

507 509

0.167

S8 S9 S10 S11 S12 S13

524 287

0.063

524 287 1000

Block length per peak (NL)

S3 Axle load spectrum (22 levels, 524 288 cycles) S4 As S3, but with all stresses above p = 0.377 omitted S5 As S3, but with an increased number of cycles at the top 2 stresses and less at the bottom level S6 As S3, but with the top 2 levels (above p = 0.667) omitted S7 Modified axle spectrum

Minimum range Maximum range (= p) 0.080 0.568

Description

S1 Wye bridge spectrum S2 US Route 301 road bridge spectrum

Ref no.

Table B1 (a) Details of loading spectra (excluding block programme loading)

0.355

1.189

1.986 3.020 1.433 0.994 8.796

7.875

9.509

9.645

12.335

7.793

9.141 5.940

Area under exceedence curve

0.239

0.465

0.0927 0.162 0.142 0.200 0.349

0.215

0.322

0.219

0.551

0.215

0.165 0.641

Mean stress S max

0.246

0.343

0.085 0.144 0.140 0.206 0.263

0.169

0.254

0.176

0.433

0.169

0.142 0.459

r.m.s. of process S max

0.469

0.515

0.133 0.250 0.271 0.377 0.398

0.261

0.390

0.280

0.664

0.260

0.242 0.658

r.m.c. of peaks S max

4.059

2.917

11.76 6.94 7.14 4.850 3.802

5.897

3.938

5.680

2.307

5.899

7.049 2.179

Clipping ratio

Description

S22 S23 S24 S25 S26 S27 S28 S29 S30 S31 S32 S33 S34 S35 S36 S37 S38

Rayleigh Rayleigh Rayleigh Rayleigh Rayleigh Rayleigh Rayleigh Rayleigh Rayleigh Rayleigh Rayleigh Rayleigh Rayleigh Rayleigh Rayleigh Rayleigh Rayleigh

(White and Lewszuk 1971)

(Fisher et al. 1983)

approximation approximation

S16 As S14, 9 cycles with p = 0.333 S17 Superimposed sine waves, with 9 cycles (p = 0.50) on each major cycle S18 As S17, 9 cycles p = 0.56 S19 As S17, 9 cycles p = 0.667 S20 As S17, 7 cycles p = 0.571 S21 As S17, 4 cycles p = 0.6

Ref no.

Table B1 (a) Continued

10 10 8 5

0.560 0.667 0.571 0.60

0.027

0.508 0.263 0.017 500 500 500 2980 1024

9 19 101 103 104 100 000 524 284 524 288 1024 1024 1024

10

0.50

0.21 0.13 0.033 0.027 0.023 0.021 0.020 0.045 0.404 0.25 0.341

10

Block length per peak (NL)

0.333

Minimum range Maximum range (= p)

4.612

5.370 4.947 4.524

9.725 4.880 4.348 4.659

1.383 1.892 3.055 4.520 6.116 7.662

1.289 1.536 1.187 0.966

1.151

0.769

Area under exceedence curve

0.336

0.705 0.557 0.409

0.568 0.503 0.421 0.336 0.291 0.261 0.245 0.276 0.517 0.391 0.465

0.604 0.700 0.625 0.680

0.550

0.400

Mean stress S max

0.504 0.409 0.324 0.25 0.268

0.436 0.392 0.328 0.268 0.233 0.209 0.196 0.220 0.371 0.288 0.337

0.437 0.500 0.453 0.494

0.403

0.316

r.m.s. of process S max

0.417

0.720 0.599 0.500

0.658 0.599 0.507 0.417 0.363 0.325 0.304 0.342 0.533 0.425 0.487

0.637 0.716 0.660 0.720

0.597

0.511

r.m.c. of peaks S max

1.985 2.445 3.082 4.0 3.727

2.10 2.43 3.03 3.72 4.29 4.790 5.11 4.541 2.694 3.466 2.970

2.287 1.999 2.208 2.024

2.480

3.163

Clipping ratio

S62 9 level block

S60 8 level block S61 9 level block

      

0.2

Convex upwards relationship ln S v. ln NE

512

128 512

512 256 128 64 16 32 8

0.06 0.08 0.10 0.15 0.27 0.03 0.12 0.1 0.2

512 106

0 0.12

Concave upwards relationship ln S v. In NE

S51 Linear relationship between S52 log stress and log number of exceedences S53 10 level block Linear relationship S54 9 level block  between log  S55 8 level block  stress and log S56 7 level block  number of S57 5 level block  exceedences S58 6 level block  S59 4 level block 

3.413

1.012 1.762

1.705 1.663 1.608 1.539 1.296 0.665 0.603

1.444 1.115

7.281

5.856 2.104 3.413 4.584 3.640 2.429

0.282

0.128 0.211

0.090 0.121 0.162 0.224 0.397 0.107 0.310

0.074 0.143

0.097 0.215 0.145 0.109 0.6184 0.237 0.190 0.542

0.218

0.110 0.152

0.082 0.108 0.142 0.188 0.314 0.152 0.299

0.071 0.126

0.094 0.199 0.144 0.109 0.442 0.209 0.161 0.384

0.208

r.m.s. of process S max

0.340

0.221 0.226

0.168 0.210 0.262 0.327 0.500 0.331 0.526

0.155 0.246

0.169 0.346 0.260 0.197 0.632 0.357 0.272 0.546

0.323 4.4

r.m.c. of peaks S max

4.59

9.08 6.57

12.18 9.26 7.05 5.32 3.18 6.58 3.34

14.10 7.94

4.796

Clipping ratio

10.609 5.03 6.93 9.20 2.26 4.78 6.20 2.60

100,022 100 1000 104 128 128 1024 25 000

0.263

Mean stress S max

0.009 0.022 0.014 0.011 0.5 0 0 0.5

7.728

Area under exceedence curve

Laplace Laplace Laplace Laplace Laplace Laplace Laplace Laplace (Sperle 1980)

S43 S44 S45 S46 S47 S48 S49 S50

131 072

Block length per peak (NL)

5.0

Minimum range Maximum range (= p) 0.030

Description

S39 Rayleigh S40 Rayleigh (Marsh et al. 1975) S41 Rayleigh

Ref no.

Table B1 (a) Continued

  

Description

k = 0.74 k = 1.11 I = 0.99 I = 0.7

C/12/20 offshore spectrum WASH WASH-WA WASH-WB UKOSRP II - S4 German offshore Narrow band spectrum NARROW 64 (I = 0.987) Truncated white noise spectral density BROAD 64 (I = 0.745) Pierson-Moscowitz wave elevation PMMOD 64 (I = 0.817) PM 32 J = 0.84 SP1 SP2 SP3 SP4 SP5 underground railway bogie VAMP (9 sea states)

S86 Marquis 1 S87 Marquis 2 S88 Marquis 3

S85

S79 S80 S81 S82 S83 S84

S78

S77

S70 S71 S72 S73 S74 S75 S76

S63 Weibull S64 S65 Gauss S66

Ref no.

Table B1 (a) Continued

0.31 0.16 0.10

0.167

0.167 0.167 0.167 0.167 0.167

0.016

0.016

3532 99 459 408 000

500 000

500 000 500 000 500 000 500 000 296 346

2.487 3.989 5.644

8.523

106

0.155 0.016

6.312

2.732 2.290

Area under exceedence curve

99 991

128 128

Block length per peak (NL)

0.014

0.04 0.02

Minimum range Maximum range (= p)

0.360 0.185 0.127

0.5

0.5

0.5 0.5 0

0.227

0.145

0.290 0.206

Mean stress S max

0.259 0.135 0.094

0369

0.295

0.363

0.537

0.126

0.249 0.194

r.m.s. of process S max

0.374 0.199 0.143

0.359

0.341

(0.40)

0.549

0.210

0.407 0.341

r.m.c. of peaks S max

3.86 7.40 10.64

1.86

7.927

4.02 5.12

Clipping ratio

0.38 0.69 0.69 0.71 0.125 0.344 0.563 0.562 0.212 0.780 0.110

P14 Stationary Gaussian P15 8 step block P16

P17 Stationary Gaussian, 8 P18 step block P19 P20 Stationary Gaussian, 8 step block

0.2 0.102

P8 Quadratic. 6 step block P9 Quadratic, 10 level block

P10 Stationary Gaussian P11 8 step block P12 P13

0.538 0.538 0.538 0.538

P4 10 level block P5 programmes P6 P7

Minimum range Maximum range (= p) 0.333 0.44 0.36

Description

P1 2 level block programme P2 8 step block P3 9 step block. Same as P2 but with further 13 880 cycles at p = 0.36

Ref no.

Table B1 (b) Details of block programme loading spectra

500 000 500 000 500 000 500 000

500 000 500 000 500 000

200 000 200 000 20 000 203 580

10 000 2851

100 1000 10 000 100 000

10 6121 20 000

Block length per peak (NL)

10.791 8.925 11.945 8.546

8.459 9.628 10.795

8.806 10.534 8.329 10.359

6.818 5.429

3.601 5.942 8.244 10.547

0.767 6.044 6.470

Area under exceedence curve

0.603 0.285 0.801 0.201

0.205 0.404 0.603

0.439 0.718 0.718 0.723

0.359 0.297

0.639 0.641 0.641 0.641

0.400 0.476 0.395

Mean stress S max

0.428 0.214 0.567 0.168

0.167 0.292 0.429

0.316 0.509 0.509 0.512

0.286 0.248

0.459 0.460 0.460 0.460

0.316 0.480 0.283

r.m.s. of process S max

0.608 0.324 0.802 0.274

0.267 0.423 0.609

0.456 0.721 0.721 0.725

0.448 0.399

0.659 0.661 0.661 0.661

0.511 0.485 0.407

r.m.c. of peaks S max

2.335 4.663 1.764 5.940

6.002 3.422 2.333

3.161 1.965 1.965 1.953

3.492 4.026

2.180 2.173 2.173 2.173

3.163 2.946 3.530

Clipping ratio

Fatigue loading spectra

B2

331

The Markov transition matrix

As discussed in Chapter 4 it is useful to consider many loading spectra in terms of ‘levels’ rather than actual stresses. The Markov matrix then becomes a convenient method of recording the numbers of half-cycles of each particular range in the spectrum and its general form is shown in Fig. B1. The two axes define the starting and finishing levels of each half-cycle and each individual cell in the diagram shows the number of half-cycles of each specific range a11

a12

Starting level, i

a21 Starting interval i

p

a1j

a1n

a22 Target interval j

Transitions from troughs to peaks j>i

ai 1

Target interval j

p

Transition from peaks to troughs i>j

q

Starting interval i

q ann

an1 Target level, j (a)

1

5

10

Target level 15 20 25

30

32

1

5

Starting level

10

8*

15

Rising ranges

20 Falling ranges 25

30

*Example. There are 8 transitions from level 11 to level 30.

32 (b)

B1 The Markov transition matrix for recording loading sequences: (a) General form of matrix; (b) Example for n = 32 levels.

332

Cumulative damage of welded joints

(see Fig. B1(b)). In this form of plot rising half-ranges appear above the diagonal and falling half-ranges below it. Some common types of loading used in fatigue testing, and the forms of the Markov matrix which correspond to them, are shown in Fig. B2. For example, constant amplitude loading (Fig. B2(a)) obviously involves all cycles having the same upper and lower limits so that there will be (say) half-cycles starting at level (i) and rising to level (j) and an equal number of falling halfcycles starting at level (j) and finishing at level (i). However, if the spectrum involves variable amplitudes with a constant minimum stress the form of the matrix will be as shown in Fig. B2(b). This shows that all the cycles start at level 1 and rise to higher levels while the corresponding falling half-cycles then return to level 1. The equivalent situation when all cycles have the same top limit stress (‘stalactitic’ loading) is illustrated in Fig. B2(c). An important and commonly occurring type of loading involves a constant mean stress and a smooth variation of stress ranges (i.e. narrow band loading) and this is shown in Fig. B2(d). The important feature of the Markov matrix in this situation is that all the non-zero entries are located on or close to the diagonal; the results of a real cycle count for narrow band loading are shown in Fig. B7. In contrast, wide band loading (Fig. B2(e)) normally results in non-zero entries appearing over a large proportion of the diagram, since associated peaks and troughs may occur anywhere. From the fatigue point of view it is certainly arguable that narrow band loading can best be identified by the fact that the Markov matrix is zero except along a diagonal bend. The purist, of course, would no doubt argue that the definitions of narrow and wide band loading have to go back to the shape of the power spectral density (psd), but for someone interested in fatigue that is often not very helpful. For example, many structures, (e.g. a bridge) are not loaded by something that can readily be comprehended as being defined by a psd. Another long-used and relatively simple distinction between narrow and wide band loading has been provided by the value of the ratio. I=

number of mean level crossings number of peaks

and if that exceeded 0.99 it was narrow band. However, that criterion can break down. For example, Fig. B2(f) satisfies that requirement but it would be difficult to accept it as being typical of narrow band loading. Similarly Fig. B2(g) can satisfy it, but in that instance the value of the ratio can depend on the magnitude of the minimum stress range in the spectrum (see Fig. B3). It will be noted, however, that even if the loadings shown in Fig. B2 ((f) and (g)) do satisfy the requirement for being called narrow band loading because I ≥ 0.99, it is immediately obvious from the form of the Markov matrix that they are not. Incidentally, it is interesting to note that, with the majority of

Fatigue loading spectra

333

Starting level

Finishing level 1 2 3 4 …

n

1 2 3 4 ⯗

a b c d e f g a b c d e f g (b)

(a) g f e d c b a

j a k b p l c q d r m e n f s o g t h u

g f e d c b a (d)

(c) a b c d a h j k b′ h′ o p c′ j ′ o′ t d ′ k′ p ′ t ′ e′ l′ q′ u′ x′ f ′ m′ r′ v′ y′ g′ n′ s′ w′ z′

e l q u x

f m r v y α

α′ β′ γ

a e j n

g n s w z β γ

b f k o

c g l p

d h m q

r s t u (f)

(e)

a b c d e f g a b c d e f g (g)

B2 Common types of fatigue loading: (a) constant amplitude; (b) variable amplitude, random order with constant minimum; (c) variable amplitude, random order with constant maximum (i.e. stalactitic); (d) variable amplitude, random order with constant mean stress (often r = –1); (e) variable amplitude, with constant mean stress and smooth variation in stress ranges, (typical of ‘narrow band’); (f) as (e) but with constant minimum stress: (g) variable amplitude, with individual cycles varying in mean stress, (typical of ‘wide band’)

334

Cumulative damage of welded joints

Mean

(a)

Mean

(b)

B3 Effect of magnitude of smallest ranges: (a) number of mean level crossings/number of peaks 0.99.

the loading types shown in Fig. B2, the Markov matrix does not alter if the signal is played backwards. With Type f loading, however, the matrix is reversed about the NW-SE diagonal.

B3

Two-parameter Weibull distribution

A number of investigations have made use of the two-parameter Weibull distribution, particularly those by Gurney. This distribution provides a convenient means of representing a family of stress spectra of different shapes. It is usually expressed in the form

( ) 

 P (S) = exp  – s  θ

a

[B1]

where P(S) a θ

is the probability that a peak exceeds stress, S, is the Weibull slope or modulus (shape parameter), is the characteristic value or scale parameter.

It has been suggested by Pook (1984) that this can more conveniently be written as

( )

( ) 

 P S = exp  – b s σ  a σ

a

[B2]

Fatigue loading spectra

335

where σ is the r.m.s. of the process, assumed to be sinusoidal; b is then a function of a. Pook also showed that, to a close approximation, over the range 0.71 ≤ a ≤ 2.36, b = 1.0 – 0.076 (a2 – 3a + 2)

[B3]

It will be noted that, with a = 2.0 or 1.0, we get b = 1.0 so that, as special cases, eqn [B2] gives the Rayleigh and Laplace distributions respectively. For other values of the parameter a, Pook has tabulated the corresponding values of b. Over the range 0.5 ≤ a ≤ 0.71 it appears that b can be adequately represented by b = 0.833 + 0.1952a

[B4]

In fact a is more usually written as 1/k, with the stress expressed in terms of stress range, ∆S. Then the probability of load exceedence becomes:

( )

( )

 P ∆S = exp  – kb ∆S σ σ 

1/k

  

[B5]

In other words, in a block of NL cycles, the number of cycles which exceed a range ∆S/σ is given by P(∆S/σ) × NL. It also follows that the peak stress range in the spectrum, ∆Smax, which occurs only once in each block, will be related to NL. Thus, writing β = ∆Smax/σ (where β is the clipping ratio) we get, from eqn [B5] P 

∆S max  = 1 = exp[– kbβ 1/k ] σ  NL

[B6]

so that ln (NL) = kbβ1/k

[B7]

Thus, the family of spectra represented by eqn [B5] can easily be varied both in terms of shape, by varying k, and clipping ratio. The influence of the value of k on spectrum shape is shown for a range of k values in Fig. B4. From eqn [B5] it is easy to deduce that the Rayleigh and Laplace distributions are:

( )

( ) 

 Rayleigh P ∆S = exp  – 1 ∆S σ  2 σ

( )

( )

Laplace P ∆S = exp  – ∆S   σ σ 

2

[B8]

[B9]

336

Cumulative damage of welded joints 1.4

1.2

Relative stress

1.0 k = 0.5 (Rayleigh)

0.8

0.6

k = 0.8

0.4 k = 1.0 (Laplace) 0.2

0 1

Block length Number of exceedences, log scale

B4 Schematic illustration of spectrum shapes for different values of k.

A simple method of deriving the corresponding numbers of cycles for various stress ranges, as required for testing purposes, is illustrated in Table B2. In this table the first two columns show the probability of exceedence for a range of values of ∆S derived from eqn [B8]. The corresponding approximate σ block spectrum can then be derived by subtracting successive values of P ∆S from each other to calculate the number of peaks occurring at the σ corresponding midpoint stress level (columns 3 and 4). Thus at a stress level equal to

( )

( )

( ) + ( ∆σS )

1  ∆S 2  σ

( )

 P ∆S  σ  block.

i

i+1

  the number of peaks is ι +1 

( )

– P ∆S  × NL, where NL is the number of cycles per σ i

As shown previously (see eqn [B7]), for a Rayleigh distribution (Table B2), NL is related to the peak stress in the spectrum by ∆S max  ln (NL) = 1  2 σ 

2

Fatigue loading spectra

337

Table B2 Basic Rayleigh type distribution and derivation of corresponding stress spectrum Stress

∆S σ

Probable exceedence  exp  – 1  ∆S   2  σ 

0

1.0

0.2

0.9802

0.4

0.92312

0.6

0.83527

0.8

0.72615

1.0

0.60653

1.2

0.48675

1.4

0.37531

1.6

0.27804

1.8

0.19790

2.0

0.13534

2.25

0.07956

2.5

0.04394

2.75

0.02279

3.0

0.01111

3.25

0.00509

3.5

0.00219

3.75

0.00088

4.0

0.00034

4.25

0.00012

4.5

0.000040

4.75

0.000013

5.0

0.000004

5.25

0.000001

2

  

Number of peaks

At average stress

0.0198

0.1

0.05708

0.3

0.08785

0.5

0.10912

0.7

0.11962

0.9

0.11978

1.1

0.11144

1.3

0.09727

1.5

0.08014

1.7

0.06256

1.9

0.05578

2.125

0.03562

2.375

0.02114

2.625

0.01169

2.875

0.00602

3.125

0.00290

3.375

0.00131

3.625

0.00054

3.875

0.00022

4.125

0.000080

4.375

0.000027

4.625

0.00009

4.875

0.000003

5.125

338

Cumulative damage of welded joints

Table B2 Continued Stress

∆S σ

Probable exceedence

Number of peaks

2    exp  – 1  ∆S     2 σ  

At average stress

From eqn [B5] the required values of peak stress for various block lengths are: Block length ∆S max σ

9

19

100

103

104

105

524 284

2.10

2.43

3.03

3.72

4.29

4.80

5.13

Hence, knowing the block length, the peak stress is defined and the number of cycles at each lower stress level can be calculated so as to give one cycle at the peak stress level. Table B3 shows some calculated Rayleigh spectra for a range of block lengths.

B4

Rayleigh distribution used by Schilling et al. (USA)

This loading was simulated by a block of 500 individual loads which satisfied a standard Rayleigh probability density curve. They were then applied in a random sequence and the block was repeated throughout the test. The actual stress ranges applied (Srn) can be expressed as a proportion of the maximum stress range (S rmax) and in terms of the modal stress range (Srm) and the distance (Srd) between the starting point and the modal value (see Fig. B.6) by the following equation:

S rn = S rmax

1 + (x n ′ – 1) 1+2

S rd S rm

S rd S rm

where: x′ = {–2 ln [1 – 0.001978 (n – 0.5)]}1/2 which gives values of x ′n (for n ranging from 1 to 500) varying from slightly S rn more than 0 to slightly less than 3.0. Thus the values of varied from S rmax a minimum to a maximum value which approaches 1.0. The minimum value S is a function of rd but is always positive. S rm This formulation implies that the Rayleigh distribution was always clipped at high stress ranges and, except for those situations in which Srd = Srm, was

2 6 9 11 12 12 11 10 8 6 6 4 2 1 1

0.033 0.099 0.165 0.231 0.297 0.363 0.429 0.495 0.561 0.627 0.701 0.784 0.866 0.949 1.0 0.027 0.081 0.134 0.188 0.242 0.296 0.349 0.403 0.457 0.511 0.571 0.638 0.706 0.773 0.840 0.907 1.0

p

3.72

3.03

N

103

101

p

Block length ∆S max σ

20 57 88 109 120 120 111 97 80 63 56 36 21 12 6 3 1

N 0.023 0.070 0.117 0.163 0.210 0.256 0.303 0.350 0.396 0.443 0.495 0.554 0.612 0.670 0.728 0.787 0.845 0.903 0.962 1.0

p

4.29

104

198 571 879 1091 1196 1198 1115 973 801 626 558 356 211 117 60 29 13 5 2 1

N

Table B3 Rayleigh type stress distributions for block lengths exceeding 100 cycles

0.021 0.063 0.104 0.146 0.188 0.229 0.271 0.313 0.354 0.396 0.443 0.495 0.547 0.599 0.651 0.703 0.755 0.807 0.859 0.911 0.964 1.0

p

4.80

105

1980 5708 8785 10 912 11 962 11 978 11 144 9727 8014 6256 5578 3562 2114 1169 602 290 131 54 22 8 3 1

N 0.020 0.059 0.098 0.137 0.176 0.215 0.254 0.293 0.332 0.371 0.415 0.463 0.512 0.561 0.610 0.659 0.707 0.756 0.805 0.854 0.902 0.951 1.0

p

5.13

524 284

10 381 29 926 46 059 57 210 62 715 62 799 58 427 50 997 42 016 32 799 29 245 18 675 11083 6129 3156 1520 687 283 115 42 14 5 1

N

340

Cumulative damage of welded joints

Idealised vehicle loading

Smax

11 10 9 8 7 6

Sr

Stress, ksi

14 13 12

1 0

Smin

4 3 2 20 16 12 8 4 0 Occurrence, %

Deod load

5

Time

Probability density, l/ksi

0.25 0.20 0.15 0.10 0.05

0 Conversion factor: 0 1 ksi = 6.895 MPa

1 2 3 4 5 6 7 8 9 10 Stress range, ksi

B5 Method of deriving Rayleigh load distributions for tests in the USA.

also clipped at the bottom end. The form of the load distribution was therefore as shown in Figs B5 and B6. The American nomenclature can be translated into a form similar to that employed by LBF, who define the minimum stress range in the spectrum by the value of p=

S rmin (see Table B4). Thus, we have: S rmax

S rmax = Srm + 2S rd S rmin = Srm – Srd

p=

S rmin S – S rd = rm S rmax S rm + 2 S rd

or, writing x =

S rd S rm

Non-dimensional probability density, p′

Fatigue loading spectra

1.378

341

(x′RMS)

1.230

(x′mean)

1.168 1.000

(x′median) (x′modal)

0.6 p′ = 1.011 x ′e–1/2(x ′) x′ = (Sr – Srmin)/Srd

0.4

2

0.2 0 0

1 2 3 Non-dimensional stress range, x ′

2Srd

Probability density, p, l/ksi

Srm (modal) Sr

0.168Srd (median)

Srd

min.

0.230Srd (mean) 0.378Srd (RMS) 0.607/Srd p = p′/Srd

Sr max

Probability density, p, l/ksi

Stress range, Sr, ksi

Conversion factor 1 ksi = 6.895 MPa

Srd /Srm = 0 Srd /Srm = 0.25 Srd /Srm = 0.50 Srd /Srm = 1.00 Srm Stress range, ksi

B6 Characteristics of Rayleigh probability curves as used in variable amplitude tests in the USA.

342

Cumulative damage of welded joints Table B4 Stress spectra used by Schilling (Block length = 500 cycles) Number of cycles p=

S rd = 0.25 S rm

Sr S rmax

0.025 0.075 0.125 0.175 0.225 0.275 0.325 0.375 0.425 0.475 0.525 0.575 0.625 0.675 0.725 0.775 0.825 0.875 0.925 0.975

23 61 84 91 84 63 45 26 15 8

S rd = 0.5 S rm

S rd = 1.0 S rm

10 29 44 57 59 62 55 50 40 32 23 16 12 7 4

5 17 27 35 41 44 46 45 43 39 34 30 24 20 15 12 9 6 5 3

p= 1–x 1 + 2x

so that the corresponding values of p and x are: x=

S rd S rm

p

0

0.25

0.5

0.75

1.0

1.0

0.5

0.25

0.1

0

Similarly, since the r.m.s. value of the distribution is at 1 (Srm + 0.378Srd), 2 we have clipping ratio =

S rmax (S + 2S rd ) 2 (1 + 2x) 2 = rm = r.m.s. S rm + 0.378S rd 1 + 0.378x

so that the corresponding values of x and clipping ratio are: x=

S rd S rm

Clipping ratio

0

0.25

0.5

0.75

1.0

1.41

1.94

2.38

2.76

3.08

Fatigue loading spectra

B5

343

Gauss spectrum used by Haibach and Overbeeke

The following description is a shortened version of that originally given by Haibach et al., (1976). This spectrum represents a frequency distribution of level crossings equal to that of a stationary Gaussian process. It is based upon the equation: h (m, a) =

  a N1 –m 2  –a 2  exp exp    2 2 2 2 2 2 {2 πσ 2 (1– I 2 )}1/2  2σ I   2 σ (1– I )  a I

which defines the joint frequency distribution of means (m) and amplitudes (a). In this formula: σ = r.m.s. value No = number of zero level crossings with positive slope N1 = number of peaks = number of troughs I

= irregularity factor =

No N1

In many of the investigations carried out by Haibach and by Overbeeke the block length was No = 106 so that the clipping ratio (= maximum peak value/ r.m.s.) = (2 ln No)1/2 = 5.2562. The matrix elements aij of the Markov transition matrix (see Fig. B1) can then be calculated from the integral: aij =

∫∫ h(m, a) dm da

and the relevant matrices for I = 0.99 and 0.7 (taken from Haibach et al., 1976) are reproduced in Figs B7 and B8. It will be noted that these are 32 × 32 matrices, with the mean stress (= zero) situated at the boundary between intervals 16 and 17. In deriving the relevant values of aij in the matrices it is important to note that Haibach assumed that the peak and trough stresses in the spectrum were situated at the centre of elements 1 and 32 and not at the boundary of the matrix. Thus the peak and trough levels were effectively each distributed uniformly over 151/2 elements rather than 16 elements. This has quite a large effect on the calculated distribution of peaks and troughs. Subsequently, the peak and trough sequences were generated from the Markov matrices by means of a pseudo-random number sequence derived by the mixed congruential method. The rainflow counts resulting from the generated time histories are summarised in Tables B5 and B6.

344 1

2

3

Cumulative damage of welded joints 4

5

6

7

8

9

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

1

1

2 3

1

4 5

7

6 7 2

9

3

10

5

11

6

12

6

13

6

14

3

15

3

16

5

17

6

18

6

19

5

20

3

21

2

22

1

23 24

20

25 26

102

6325 5409

241

2255

2472 2255

27

1

25

267

850

826

241

28

7

76

258

267

83

7

2

2

19

69

76

25

4

16

19

7

1

3

4

2

31 32

1

617

6516 26771 25136

2820

5

4651 32962 69658 44600 8474

1387 11364 27861 20809 4651

826

6

8596 44938 66393 26771 2820

2725 20809 47494 32962 6826

5409 14199 11364 2725

7

455

406

1387

102

49

1

187

297

402

455

406

242

242

16

4 2

2

25

76

69

19

83

267

258

76

7

1

241

826

850

267

25

2472

826

83

2

1387

5409

6325

2255

241

7

2725 11364 14199 5409

617

20

102

8596 50340 86420 44600 6826

25136 26771 6516

19

2255

8474 44600 69658 32962 4651

6516 44938 87615 50340 8474

7

20

4651 20809 27861 11364 1387

2820 26771 66393 44938 8596

8474 50340 87615 44938 6516

49

83

30

402

406

406

297

3

617

6826 32962 47494 20809 2725

2820

242

6826 44600 86420 50340 8596

617

2

29

187

297

455

242

455

402

187

4

49

1

8

2

402

297

187

102

49 1

2

3

5

6

3 3

5

6

6

5

3

2

20

1

B7 Transition matrix for generating a random process of Gaussian type defined by N0 = 106 and I = N0/N1 = 0.99.

B6

The WASH spectrum – North Sea Wave Action Standard History

WASH (W) is a wide band load spectrum consisting of 500 000 cycles distributed between eight sea states (r.m.s. levels). The eight sea states are randomly distributed throughout the time history using a Markov chain technique with the result shown in Fig. B9. The irregularity factor (I) decreases from 0.83 for sea state 1 (comparatively narrow band) to 0.58 for sea state 8,

1

Fatigue loading spectra 1

2

3

4

5

6

7

8

9

345

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

1

1

2 3 4

1

1

1

2

1

1

1

1

1

2

3

4

5

6

7

6

5

4

1 2

1

1

1

1

3

6

10

16

21

25

27

25

21

15

10

6

3

1

1

5

1

2

4

10

21

36

56

76

91

97

91

76

57

37

22

11

5

2

1

6

1

4

12

30

61

110

174

238

290

313

296

250

187

124

73

38

18

7

3

7

2

9

28

73

158

293

470

663

817

887

853

725

546

365

217

114

53

22

8

3

1

13

52

151

349

675

1120

1608 2082

2230

2170

1864

1417

955

570

303

143

60

22

7

2

66

247

645

1334

2308

3420 4396

4941

4876

4241

3258 2216

1335

713

338

143

53

18

5

1

286

945

2204

4077

6307 8358

9609

9657

8524

6630 4556

2770

1493

713

303

114

38

11

3

1

973

2889

6024

9963 13785 16338 16797 15096 11917 8290

5094

2770

1335

570

217

73

22

6

1

2651

7060 13166 19474 24096 25541 23485 18875 13325 8290

4556

2216

955

365

124

37

10

2

1

5781 13799 23014 30444 33689 31934 26260 18875 11917 6630

3258

1417

546

187

57

15

4

1

8

1

1

2

9

1

2

4

9

13

10

1

4

12

28

52

66

11

1

3

10

30

73

151

247

286

12

2

6

21

61

158

349

645

945

973

3

10

36

110

293

675

1334

2204

2889

2651 7060

13

1

14

1

4

16

56

174

470

1120

2308

4077

6024

15

1

5

21

76

238

663

1608

3420

6307

9963 13166 13799 10088

16 17

1

5781

4241

1864

725

250

76

21

5

1

14091 26971 35977 38064 33689 25541 16797 9657

4876

2170

853

296

91

25

6

1

2

6

25

91

290

817

2022

4396

8358 13785 19474 23014 21571 14091

7

27

97

313

887

2230

4941

9609 16338 24096 30444 32175 26971 15751

15751 26971 32175 30444 24096 16338 9609

4941

2230

887

313

97

27

7

1

14091 21571 23014 19474 13785 8358

4396

2022

817

290

91

25

6

2

18

1

6

25

91

296

853

2170

4870

9657 16797 25541 33689 38064 35977 26971 14091

19

1

5

21

76

250

725

1864

4241

8524 15096 23485 31934 37670 38064 32175 21571 10088

10088 13799 13166 9963

6307

3420

1608

663

238

76

21

5

1

5781 7060

6024

4077

2308

1120

470

174

56

16

4

1

2651

2889

2204

1334

675

293

110

36

10

3

1

973

945

645

349

158

61

21

6

2

286

247

151

73

30

10

3

1

66

52

28

12

4

1 1

20

1

4

15

57

187

546

1417

3258

6630 11917 18875 26260 31934 33689 30444 23014 13799 5781

21

1

2

10

37

124

365

955

2216

4556

22

1

6

22

73

217

570

1335

23

1

3

11

38

114

303

713

8290 13325 18875 23485 25541 24096 19474 13166 7060

2651

2770

5094

8290 11917 15096 16797 16338 13785 9963

6024

2889

973

1493

2779

4556

4077

2204

945

8524

9657 9609

8358

6307

286

24

1

5

18

53

143

338

713

1335

2216

3258

4241

4876 4941

4396

3420

2308

1334

645

247

66

25

1

2

7

22

60

143

303

570

955

1417

1864

2170 2230

2022

1608

1120

675

349

151

52

13

1

3

8

22

53

114

217

365

546

725

853

887

817

663

470

293

158

73

28

9

1

3

7

18

38

73

124

187

250

296

313

290

238

174

110

61

30

12

4

1

1

2

5

11

22

37

57

76

91

97

91

76

56

36

21

10

4

2

1

1

1

3

6

10

15

21

25

27

25

21

16

10

6

3

1

1

1

1

2

4

5

6

7

6

5

4

3

2

1

1

1

1

1

1

2

1

1

1

26 27 28 29 30 31 32

1

10088 21571 32175 38084 37670 31934 23485 15096 8524

1

6630

1

13

9

4

2

2

1

1

2

1

B8 Transition matrix for generating a random process of Gaussian type defined by N0 = 106 and I = N0/N1 = 0.7.

a typical section of which is shown in Fig. B10. In real time the average wave frequency is approximately 0.24 Hz. The exceedence distribution of the WASH spectrum is very close to Laplace but in some investigations the spectrum has been shortened by truncating the relatively small number of high stress ranges and also omitting the very small ranges. Figure B11 shows the exceedence diagram for a spectrum in which ranges larger than 75% of the maximum and smaller than 15% were omitted. This reduces the block length from 500 000 to 153 750 cycles and hence reduces testing time (obviously at the expense of accuracy).

1

346

Cumulative damage of welded joints

Table B5 Summary of cycle count for Gauss spectrum with I = 0.99 Count of ranges for 1 006 393 reservoir cycles Stress range (levels)

Rainflow count

Range count

No. of peaks

No. of troughs

Stress level

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

41 872 43 872 93 984 64 334 132 047 69 212 134 851 56 277 111 468 42 255 77 607 25 408 46 519 14 001 24 289 6520 11 177 2520 4544 952 1612 271 523 83 140 18 35 6 7 0 1 0

61 563 108 052 158 870 181 376 209 638 203 180 206 760 180 008 166 640 133 036 113 604 83 984 66 630 45 864 33 950 21 832 15 118 9100 5908 3332 2032 1076 616 308 166 76 40 16 6 4 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 3067 58 835 149 982 198 535 197 493 160 995 111 327 66 423 34524 15 728 6305 2230 698 193 47 10 1

1 10 47 193 698 2230 6305 15 728 34 524 66 423 111 327 160 995 197 493 198 535 149 982 58 835 3067 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1.5 2.5 3.5 4.5 5.5 6.5 7.5 8.5 9.5 10.5 11.5 12.5 13.5 14.5 15.5 16.5 17.5 18.5 19.5 20.5 21.5 22.5 23.5 24.5 25.5 26.5 27.5 28.5 29.5 30.5 31.5

No. of crossings

1 11 58 251 949 3179 9484 25 212 59 736 126 159 237 486 398 481 595 974 794 509 944 485 1 000 253 944 485 794 509 595 974 398 481 237 486 126 159 59 736 25 212 9484 3179 949 251 58 11 1

Mean stress based on peaks = 16.5

B7

C/12/20 spectrum (in the reports on UKOSRP it is referred to as Spectrum S1)

The C/12/20 spectrum was intended to be representative of wave loading on structures in the North Sea and a useful summary of some of the factors that affect such loading has been given by Pook (1987). A theoretical analysis (WASH (W) Time History User Documentation, Technical Software Consultants) of such loading showed that, for cylinders immersed near the water surface, the peak load distribution approached the Rayleigh distribution.

Fatigue loading spectra

347

Table B6 Summary of cycle count for Gauss spectrum with I = 0.7 Count of ranges for 1 418 119 reservoir cycles Stress range (levels)

Rainflow count

Range count

No. of peaks

No. of troughs

Stress level

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

148 091 210 440 207 353 183 375 149 698 115 149 94 759 69 247 59 260 41 374 37 775 25 227 24 457 13 470 14 854 6359 7854 2631 3437 967 1339 309 431 78 116 27 29 5 7 0 1 0

167 305 294176 381 602 415 168 399 552 348 304 278 516 205 884 141 330 90 412 54 032 30 208 15 830 7780 3592 1556 632 240 84 28 6 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 4 29 163 801 3160 10 206 27 091 59 355 107 942 163 888 209 105 225 790 207 828 164 229 112 160 66 599 34 554 15 734 6307 2230 695 191 47 10 1

1 10 47 192 696 2230 6306 15 733 34 554 66 599 112 160 164 229 207 828 225 790 209 105 163 888 107 942 59 355 27 091 10 206 3160 801 163 29 4 0 0 0 0 0 0 0

1.5 2.5 3.5 4.5 5.5 6.5 7.5 8.5 9.5 10.5 11.5 12.5 13.5 14.5 15.5 16.5 17.5 18.5 19.5 20.5 21.5 22.5 23.5 24.5 25.5 26.5 27.5 28.5 29.5 30.5 31.5

No. of crossings

1 11 58 250 946 3176 9482 25 211 59 736 126 172 237 531 398 600 596 222 794 921 944 671 1 000 617 944 671 794 921 596 222 398 600 237 531 126 172 59 736 25 211 9481 3174 944 249 58 11 1

Mean stress based on peaks = 16.5

For deeply immersed cylinders (greater than 18 m) peak loads approached the Laplace distribution, and at intermediate depths the peak load distribution was between these two extremes. The set of results for a 3.66 m (12 ft) diameter cylinder immersed at a depth of 6.10 m (20 ft) was selected as being representative of intermediate data and designated the C/12/20 distribution. For testing purposes the spectrum was synthesised from a sequence of six Rayleigh distributions with different stress levels with an overall block length of 105 cycles, (see Fig. B12), and the long-term peak distribution is shown in Fig. B13. For the purposes of the analysis contained in Table B7

348

Cumulative damage of welded joints 9

Sea state

7

5

3

1 0

1.0 × 106

0.5 Turning points

B9 Occurrence of sea states within the WASH-W sequence.

100

Amplitude (WASH units)

60

20

–20

–60

–100 0

30

60

90

120

150

Time (secs)

B10 Typical section of WASH-W loading sequence, sea state 8.

the spectrum was approximated by the 27 level block programme shown in that table. In all the tests using this spectrum the maximum stress was limited to an arbitrary upper limit value which was chosen to be the nominal material flow

Fatigue loading spectra

349

100

Load range, non-dimensional

80

60

40

20

0 100

101

102

103 Exceedence

104

105

106

B11 Load range exceedence distribution for WASH-W.

stress (i.e. the average of the minimum specified yield stress and the tensile strength). This corresponded to 440 N/mm2 and all peak stresses which would have been greater than that value were reduced to it. Some tests were conducted at high positive mean stress such that σmean + 3σLT = σy where σLT is the long term rms stress amplitude (of the process and σy is yield stress (assumed to be 345 N/mm2). Hence, in terms of the r.m.s. stress range (σr) of the process, where σr = 2σLT, we have σmean = σy – 1.5σr Now, if the clipping ratio = c, the nominal peak range is c.σr and the nominal range of level p is pcσr. Hence the nominal maximum stress of that range is given by (σy – 1.5σr) + 0.5pcσr so that, since clipping occurred when the peak level was 440 N/mm2, the value of p at which clipping started is given by (σy – 1.5σr) + 0.5pcσr = 440 or p =

440 – ( σ y – 1.5 σ r ) 0.5cσ r

350

Cumulative damage of welded joints 100 VAMP

S/Smax (%)

80 60 40 20 0 1

10

102

103 Cycles

104

105

106

100 WASH (W)

S/Smax (%)

80 60 40 20 0 1

10

102

103 Cycles

104

105

106

100 Modified Wash

S/Smax (%)

80 60 40 20 0 1

10

102

103 Cycles

104

105

106

B12 Comparison of exceedence distributions for the three offshore spectra VAMP, WASH(W) and modified WASH

In principle the same applied at the troughs, but in fact no specimens were tested at a sufficiently high stress range for any clipping to occur at a trough.

B8

UKOSRP II Wide Band Spectrum S4

This spectrum was defined in terms of the twin peaked power spectral density (psd) shown in Fig. B14. The raw data had to be compressed by extracting the peak and trough values from the ‘continuous’ signal to give 100 000 pairs of peak and trough values. In this process cycles smaller than about 5% of the maximum range were eliminated. The various peak/trough pairs were reconstituted by fitting half cosine waves.

Fatigue loading spectra

351

Overall block length 100,000 cycles

26 000 cycles r.m.s., 0.347 σ LT

20 000 cycles r.m.s. 0.799 σ LT

15 000 cycles r.m.s. 1.366 σ LT

S1 = Spectrum distribution σ LT = Overall block r.m.s.

15 000 cycles 20 000 cycles r.m.s. 1.366 r.m.s. 0.799 σ LT σ LT

4000 cycles r.m.s. 1.960σ LT

B13 Breakdown of non-stationary narrow band random load history to form spectrum C/12/20.

B9

Spectra used in the Nordic Test Programme

Details of these spectra are as follows: Spectrum name

Mean stress

Stress ratio

p

I

Block length

VAMP SP1 SP2 SP3 SP4 SP5

0.5 0.5 0.5 0 – 0.5

01.01

S3

Axle with incr./decr. peaks (A1)

5.89

0.063 524 287 11 915

2 1U/B

6E Fmean

3.89 3.32

0.90 0.87

S7

Modified axle with gradual decr. after peaks (A2)

5.89

0.063 524 287 11 915

1

6E Fmean

3.33 2.71

0.77 0.71

S7

Modified axle with gradual decr. after peaks (A3)

5.89

0.063 524 287 11 915

1

6E Fmean

2.49 2.03

0.62 0.58

Smax = 200

S3

Axle with const. max. stress = 200

5.89

0.063 524 287 11 915

4

6E Fmean

0.65 0.53

0.58 0.47

0.61 0.48

0.15 0.14

0.13 0.12

0.14 0.13

Smax = 100

S3

Axle with const. max.

5.89

0.063 524 287 11 915

3

6E Fmean

0.63 0.49

0.45 0.36

0.54 0.43

0.15 0.13

0.10 0.09

0.12 0.11

Table C1/5 Longitudinal non-load-carrying fillet weld (continued) Reference Joint dimensions, mm

Maddox 1983 (cont)

Material yield stress, N/mm2

Stress ratio or Smax

Constant amplitude results ————————————— Ref m F.S. at no. C 2 × 106 N/mm2

Smax = 50

S3

R=0

S3

R=0 (comparison) Blom 1995

B = 80 T = 12 l = 140

Variable amplitude loading spectrum ——————————————————— Ref Form Clip p Block no. ratio length, cycles (a) actual (b) per peak

R=0

6F

2.40 7.726 E10

81

R = –1 R=0

81

stress = 100 Axle with const. max. stress = 50

Variable amplitude results —————————————————————————————— Σn No. of S-N RG N speci- curve mens used in ————————–— ————————–— analysis Max. Min. Mean Max. Min. Mean

5.89

0.063 524 287 11 915

2

6E Fmean

0.62 0.51

0.54 0.44

0.58 0.47

0.14 0.13

0.13 0.11

0.13 0.12

Axle (A4)

5.89

0.063 524 287 11 915

5

6E Fmean

9.12 7.91

2.74 2.31

4.74 4.02

2.11 2.07

0.63 0.61

1.10 1.05

S1

Wye Bridge

7.05

0.08

524 287 12 193

5

6E Fmean

1.55 1.35

1.16 0.96

1.37 1.16

1.77 1.71

1.32 1.21

1.57 1.46

S29

Rayleigh E

4.54

0.045 524 287 19 418

6

6E Fmean

1.23 1.01

0.96 0.78

1.08 0.89

0.89 0.80

0.70 0.62

0.79 0.71

S29

Rayleigh E

4.54

0.045 524 287 19 418

3

6E Fmean

1.65 1.37

0.80 0.65

1.14 0.95

1.20 1.09

0.58 0.52

0.83 0.75

1/6

11

6F

4.10

1.86

2.20

1/6

5

6F

2.69

1.96

2.31

0

3

6F

2.89

1.24

2.17

1/6

4

6F

1.43

0.64

1.10

11

6F

4.41

1.63

2.44

5

6F

2.33

1.26

1.79

1/6 0

25 000

25 000

Table C1/5 Longitudinal non-load-carrying fillet weld (continued) Reference Joint dimensions, mm

Material yield stress, N/mm2

Stress ratio or Smin

Constant amplitude results ————————————— Ref m F.S. at no. C 2 × 106 N/mm2

Variable amplitude loading spectrum ———————————————————––– Ref Form Clip p Block no. ratio length, cycles (a) actual (b) per peak 1/6

Overbeeke B = 90 611 et al. 1990 T = 6 l = 150

8

6F

2.81

1.22

2.06

8

6G

2.06

0.91

1.30

Gauss 1 = 0.99 Gauss l = 0.99

5

6H

2.15

1.26

1.28

9

6J

2.66

0.79

1.53

S65

Gauss l = 0.99

7

6K

1.97

0.25

1.15

S80

SP1

5 × 105

6

6L

2.05

0.72

1.20

SP2

5 × 10

5

18

6L

3.97

0.48

1.46

SP3

5 × 10

5

3

6L

1.23

0.83

S83

SP4

5 × 10

5

3

6L

5.07

4.41

S84

SP5

296 346

2

6L

0.67

0.62

S85

VAMP

5 × 105

27

6L

1.33

0.52

R = –1 6G

4.01 4.417 E14

120

S65

Gauss I = 0.99

6H

4.30* 2.480 E15 2.71 2.322 E11

130

S65

74

S65

119

6J

Variable amplitude results —————————————————————————————— Σn No. of S-N RG N speci- curve mens used in ————————–— ————————–— analysis Max. Min. Mean Max. Min. Mean

106

Note: TIG dressed*

Bogren and Martinez 1993

W = 80 370 T = 12 L = 140

0 VA al const. Smax

6K

3.97* 3.484 E14

6L

3.0 1.426 E12

S81 S82

5.0

0.92

Table C1/5 Longitudinal non-load-carrying fillet weld (continued) Reference Joint dimensions, mm

Petersen et al. 1995

Fisher et al. 1983

Material yield stress, N/mm2

Stress ratio or Smin

W = 80 1010 T=6 L = 140 W = 80 1010 T = 12 L = 140 T = 10 499 l = 305 t = 25 h = 102 W = 16

Constant amplitude results ————————————— Ref m F.S. at no. C 2 × 106 N/mm2

Variable amplitude loading spectrum ———————————————————––– Ref Form Clip p Block no. ratio length, cycles (a) actual (b) per peak

Variable amplitude results —————————————————————————————— Σn No. of S-N RG N speci- curve mens used in ————————–— ————————–— analysis Max. Min. Mean Max. Min. Mean

6M

5.256 5.087 E17

S76

BROAD 64

17

6M

1.99

0.45

0.89

6N

3.745 1.309 E14

S76

BROAD 64

16

6N

3.41

0.25

1.08

S31

Rayleigh Srmax – Srmin = 4.75 Srd Rayleigh Srmax – Srmin = 3.75 Srd Rayleigh Srmax – Srmin = 4.0 Srd

R=0

S30

S32

Booth 1982

B = 100 370 T = 25 l = 150 t = 13 h = 50 w = 10

Tilly and B = 125 365 Nunn 1980 T = 12.7 I = 150

R = –1 6P

2.643 0.295 E12

90

R = 0.5

R=0

6Q

0.107

90

S3

3.46

0.25

1024

24 5 U/B

F2mean

0.77

0.12

0.49

0.76

0.12

0.47

2.69

0.40

1024

12 6 U/B

F2mean

0.29

0.15

0.21

0.25

0.13

0.18

2.97

0.34

1024

12 2 U/B

F2mean

0.21

0.08

0.14

0.19

0.07

0.14

C/12/20

7.98

0

105

5

6P Fmean

2.46 3.40

1.62 1.71

2.08 2.44

0.97 2.03

0.64 1.02

0.82 1.45

Rayleigh narrow band

4.6

39 340

7

6Q Fmean

2.00 1.46

0.81 0.63

1.08 0.82

Full axle

5.68

7

6Q

4.54

0.48

3.38

0.063 524 287

Table C1/5 Longitudinal non-load-carrying fillet weld (continued) Reference Joint dimensions, mm

Material yield stress, N/mm2

Stress ratio or Smin

Constant amplitude results ————————————— Ref m F.S. at no. C 2 × 106 N/mm2

Variable amplitude loading spectrum ———————————————————––– Ref Form Clip p Block no. ratio length, cycles (a) actual (b) per peak

E13 Tilly and Nunn 1980 (cont)

S4

S5

Martinez and Blom 1997

350

6R

3.0 1.03 E12

590 900 Marquis 1996

W = 68 420 T = 10 L = 150

6S

3.0 2E12

spectrum (A) Axle spectrum 2.31 truncated at 0.377 (B) Enlarged axle spectrum (D)

298 0.167 507 509 11.5

Variable amplitude results —————————————————————————————— Σn No. of S-N RG N speci- curve mens used in ————————–— ————————–— analysis Max. Min. Mean Max. Min. Mean

3 U/B 3

Fmean 6Q Fmean

3.97 0.90 0.73

0.40 0.72 0.60

2.91 0.82 0.68

2

6Q Fmean

1.26 1.08

1.17 0.98

1.22 1.03

S81

SP2

6

6R

4.46

1.46

2.24

S81 S82

SP2 SP3

8 6

6R 6R

0.87 1.84

0.61 1.23

0.74 1.47

S13

Rail car bogie 10% omission l = 0.86 16% omission l = 0.88 31% omission l = 0.99

5 5 5

6S 6S 6S

1.64 1.66 1.21

0.51 0.71 0.53

1.10 1.10 0.80

408 × 103 99 459 3532

Table C1/6 Transverse non-load-carrying fillet weld Reference Joint dimensions, mm

Material Stress yield ratio stress, N/mm2

Constant amplitude results ————————————— Ref m F.S. at no. C 2 × 106 N/mm2

Variable amplitude loading spectrum ——————————————————— Ref Form Clip p Block no. ratio length, cycles (a) actual (b) per peak

Variable amplitude results —————————————————————————————— Σn No. of S-N RG N speci- curve mens used in ————————–— ————————–— analysis Max. Min. Mean Max. Min. Mean

Agerskov and lbso 1993a

400– 409

7A

S78

PM32 l = 0.84

10

7A

1.17

0.50

0.73

S76

BROAD 64 l = 0.745

14

7A

0.92

0.20

0.46

S77

PMMOD 64 l = 0.817 NARROW 64 l = 0.987

13

7A

1.00

0.30

0.53

13

7A

0.99

0.44

0.65

S78

PM32 l = 0.84

23

7B

0.76

0.22

0.52

S76

BROAD 64 l = 0.745 PMMOD 64 l = 0.817 NARROW 64 l = 0.987

12

7B

0.80

0.33

0.55

16

7B

2.96

0.38

1.18

17

7B

1.32

0.61

0.97

B = 90 T = 16 t = 10

–1

3.102 8.054 E12

S75

B = 90 T=8 t=5

393– 411

7B

2.851 3.162 E12

S77 S75 Petersen et al. 1993

B = 90 T = 16 t = 10

810

B = 90 T=8 t=5

810

7C

7D

2.741 1.164 E12

2.935 9.162 E12

S76

BROAD 64

12

7C

0.87

0.44

0.65

S77

PMMPOD 64

10

7C

1.07

0.48

0.72

S76

BROAD 64

21

7D

1.83

0.29

0.58

S77

PMMPOD 64

21

7D

2.52

0.21

0.78

Table C1/6 Transverse non-load-carrying fillet weld (continued) Reference Joint dimensions, mm

Sarkani et al. 1995

Fisher et al. 1993

Material Stress yield ratio stress, N/mm2

W = 102 638 T = 11 t = 11

Constant amplitude results ————————————— Ref m F.S. at no. C 2 × 106 N/mm2

7E

Variable amplitude loading spectrum ———————————————————––– Ref Form Clip p Block no. ratio length, cycles (a) actual (b) per peak

3.210 3.622 E9

Unimodal Gaussian Kurtosis = 3 Unimodal, nonGaussian Kurtosis = 5 Bimodel Gaussian Kurtosis = 3 Bimodal nonGaussian Kurtosis = 5

Haibach B = 50 et al. 1980 T = 10 t = 10

345

Smin= 2ksi

7F

R = –1 7G

7E

2.50

0.55

1.0

12

7E

0.98

0.33

0.63

12

7E

1.36

0.66

0.91

12

7E

1.19

0.27

0.64

103–104

8

Class F mean25D

>9.7 (mainly unbroken)

Truncated Rayleigh

500

10

7F

1.67

0.64

1.10

S65

Gauss l = 0.99

106

54

7G

0.80

0.15

0.39

S66

Gauss l = 0.7

106

14

7G

0.84

0.19

0.39

4.062

3.75

12

Rayleigh in random order

Beam 312 stiffeners web 864 × 19 Stiffs. 9.5

Klippstein B = 64 and T = 10 Schilling t = 10 1989 w= 6

Variable amplitude results —————————————————————————————— Σn No. of S-N RG N speci- curve mens used in ————————–— ————————–— analysis Max. Min. Mean Max. Min. Mean

Table C1/6 Transverse non-load-carrying fillet weld (continued) Reference Joint dimensions, mm

Material Stress yield ratio stress, N/mm2

Tubby and B = 120– 317 Razmjoo 200 1991 T = 12.5 t =8

Constant amplitude results ————————————— Ref m F.S. at no. C 2 × 106 N/mm2

Tensile 7H

3.19 4.904 E12

Variable amplitude loading spectrum ———————————————————––– Ref Form Clip p Block no. ratio length, cycles (a) actual (b) per peak Narrow band Broad band Random order Constant mean

484 879 1776 457 982 5871 457 982 5371

Variable amplitude results —————————————————————————————— Σn No. of S-N RG N speci- curve mens used in ————————–— ————————–— analysis Max. Min. Mean Max. Min. Mean

11 15 18 2U/B

7H Fmean 7H Fmean 7H Fmean

2.38 3.20 1.38 1.85 2.48 3.08

0.78 0.90 0.67 0.79 0.80 0.89

1.01 1.21 1.04 1.30 1.29 1.57

Table C1/8 Transverse load-carrying fillet weld Reference Joint dimensions, mm

Material yield stress, N/mm2

Stress ratio or Smin

Marsh et al 1975

530

R= 0.33

3.743 0.114 E14

109

Rayleigh Narrow band

4.4

16 000

12 2 U/B

9A F2mean

1.49 0.59

0.19 0.06

0.76 0.26

2.32 0.53

0.29 0.06

1.20 0.24

250 minm

R = –1

4.310 0.587 E16

±79

Rayleigh Narrow band

4.0

2 980

38 1 U/B

9B F2mean

3.38 0.30 17.18 0.99

1.46 6.20

8.95 0.80 20.32 1.17

3.88 7.33

Mean stress = 75 Mean stress = 154

3.302 0.137 E14

116

Rayleigh Narrow band

4.0

2 980

32 1 U/B

9C F2mean

2.77 8.44

0.50 0.86

1.12 2.66

4.01 9.98

0.72 1.02

1.63 3.13

3.342 0.100 E14

101

Rayleigh Narrow band

4.0

2 980

34 1 U/B

9D F2mean

3.87 4.97

0.42 0.77

1.35 1.93

5.76 5.88

0.62 0.92

2.02 2.29

Lap joint B = 51 T=6 Cover plates b = 51 t=6 w= 6

White and T joint Lewszuk B = 19 1971 T = 10 l = 10 b = 19 h = 35 w = 10

Constant amplitude results ————————————— Ref m F.S. at no. C 2 × 106 N/mm2

Variable amplitude loading spectrum ———————————————————––– Ref Form Clip p Block no. ratio length, cycles (a) actual (b) per peak

Variable amplitude results —————————————————————————————— Σn No. of S-N RG N speci- curve mens used in ————————–— ————————–— analysis Max. Min. Mean Max. Min. Mean

Table C1/10 Longitudinal non-load-carrying fillet weld on plate edge Reference Joint dimensions, mm

Tilly and Nunn 1980

Material Stress yield ratio stress, N/mm2

B = 125 365 T = 12.7 l = 150

R=0

Constant amplitude results ————————————— Ref m F.S. at no. C 2 × 106 N/mm2

Variable amplitude loading spectrum ——————————————————— Ref Form Clip p Block no. ratio length, cycles (a) actual (b) per peak

Variable amplitude results —————————————————————————————— Σn No. of S-N RG N speci- curve mens used in ————————–— ————————–— analysis Max. Min. Mean Max. Min. Mean

11A 3.016 0.640 E12

S3

Axle spectrum (A)

5.90

0.063 524 287 11 915

4

11A Gmean

7.69 8.14

4.32 4.59

6.48 6.85

2.05 2.13

1.15 1.20

1.73 1.79

S4

Axle spectrum 2.30 truncated at 0.377 (B) Axle spectrum 3.94 truncated at 0.667 (C)

0.167 507 509 11.5

4

11A Gmean

1.10 1.16

0.64 0.68

0.84 0.89

1.69 1.78

0.99 1.05

1.30 1.37

0.094 524 221 18 722

4

11A Gmean

3.91 4.11

1.81 1.91

2.50 2.64

1.72 1.79

0.80 0.83

1.10 1.15

0.063 524 2871 11 915 0.063 524 2871 11 915 0.094 524 2211 18 722

    

No relevant S-N curve

Random order

6

11B

1.57

1.02

1.36

2.13

1.39

1.84

Simulated narrow band Wide band Random order

8

11B

2.03

1.17

1.60

2.76

1.60

2.17

7 8

11B 11C

0.85 1.35

0.60 0.69

0.73 0.95

1.15 1.30

0.82 0.73

0.99 0.91

67

S6

Maddox 1979

B = 125 355 T = 12.7 l = 152

R = –1 R=0 (Comp.)

S3 III-defined S-N curve

S3 S6

Gurney 1993

B = 125 349–395 0 T = 12 l = 150

–1

11B 3.0 0.5585 E12

11C 2.483 0.5462 E11

65

61

S38

S38

Axle 5.89 spectrum (A) Axle 5.89 spectrum (A) Axle spectrum 3.94 truncated at 0.667 (C) Rayleigh

1024 1024

1024 1024

Table C1/10 Longitudinal non-load-carrying fillet weld on plate edge (continued) Reference Joint dimensions, mm

Material Stress yield ratio stress, N/mm2

Constant amplitude results ————————————— Ref m F.S. at no. C 2 × 106 N/mm2

Variable amplitude loading spectrum ———————————————————––– Ref Form Clip p Block no. ratio length, cycles (a) actual (b) per peak

Gurney 1993 (cont)

Simulated narrow band Wide band 0

11B 3.0 0.5585 E12

65

S39

Random order

131 072 131 072

Simulated narrow band Wide band –1

11C 2.483 0.5462 E11

61

S39

Random order

131 072 131 072

Simulated narrow band Wide band Gurney 1985

B = 125 370–435 0 T = 12 l = 150

11D 3.0 0.568 E12

66

Variable amplitude results —————————————————————————————— Σn No. of S-N RG N speci- curve mens used in ————————–— ————————–— analysis Max. Min. Mean Max. Min. Mean

7

11C

1.19

0.87

0.99

0.16

0.87

0.98

9

11C

1.09

0.45

0.80

1.05

0.49

0.79

7

11B

4.60

2.50

3.45

2.37

1.29

1.77

6

11B

5.59

2.50

4.04

2.88

1.29

2.08

8

11B

0.92

0.50

0.75

0.47

0.26

0.38

6

11C

1.34

0.82

1.07

0.43

0.31

0.34

6

11C

1.28

0.89

1.02

0.48

0.33

0.38

7

11C

1.03

0.74

0.94

0.39

0.28

0.35

6

11D

2.09

0.84

1.17

3.24

1.30

1.81

5

11D

1.33

1.05

1.14

2.18

1.54

1.87

4

11D

1.29

0.84

1.05

2.03

1.36

1.69

3

11D

1.62

1.20

1.43

2.06

1.52

1.82

3

11D

2.08

1.56

1.78

1.97

1.48

1.68

Rayleigh in Random order S22

0.21

S23 S24

3.03

S25

3.72

S26

4.29

9 9 0.13 19 19 0.033 101 101 0.027 103 103 0.023 104 104

Table C1/10 Longitudinal non-load-carrying fillet weld on plate edge (continued) Reference Joint dimensions, mm

Material Stress yield ratio stress, N/mm2

Constant amplitude results ————————————— Ref m F.S. at no. C 2 × 106 N/mm2

Gurney 1985 (cont)

Gurney 1988b

B = 125 385–408 0 T = 12 l = 150

11D 3.0 0.568 E12

66

Variable amplitude loading spectrum ——————————————————— Ref Form Clip p Block no. ratio length, cycles (a) actual (b) per peak

544 S45 S46

Laplace Laplace Laplace

5.03 6.93 9.20

0.021 105 105 0.020 524 284 524 284 99 103 104

S63

Weibull k = 0.74 Stalactitic from peak

4.02

0.04

S27

4.80

S28

5.13

11E 2.56 0.910 E11

66

0

11D 3.0 0.568 E12

66

Random order

S64

Wide band Weibull k = 0.11 Stalactitic from peak

128 128

11E 2.56 0.910 E11

66

Random order Wide band

11D

3.09

1.83

2.42

1.92

1.14

1.50

4

11D

2.75

1.66

2.20

1.21

0.73

0.97

3 3 2

11D 11D 11D

1.13 1.63 2.25

0.95 1.25 2.25

1.05 1.49 2.25

2.19 2.81 2.88

1.84 2.17 2.87

2.03 2.56 2.88

2

11D

0.57

0.50

0.53

1.01

0.88

0.94

8

11D

0.82

0.70

0.76

1.45

1.24

1.34

2

11E

0.75

0.73

0.74

1.04

1.02

1.03

18

11E

1.03

0.50

0.75

1.43

0.69

1.04

2

11D

0.58

0.54

0.56

1.12

1.05

1.08

12

11D

1.02

0.66

0.83

1.99

1.28

1.62

2

11E

0.82

0.71

0.76

1.25

1.07

1.16

9

11E

0.86

0.60

0.76

1.30

0.91

1.15

128 128

Wide band –1

4

128 128

Wide band –1

Variable amplitude results —————————————————————————————— Σn No. of S-N RG N speci- curve mens used in ————————–— ————————–— analysis Max. Min. Mean Max. Min. Mean

128 128

Table C1/10 Longitudinal non-load-carrying fillet weld on plate edge (continued) Reference Joint dimensions, mm

Gurney 1992b

Material Stress yield ratio stress, N/mm2

B = 125 347–385 0 T = 12 l = 150

805

Constant amplitude results ————————————— Ref m F.S. at no. C 2 × 106 N/mm2

3.0 0.568 E12

66

–1

2.72 0.433 E12

91

0

2.62 0.101 E12 2.22 8.356 E9

62

Stalactitic from 500

43

Variable amplitude loading spectrum ——————————————————— Ref Form Clip p Block no. ratio length, cycles (a) actual (b) per peak Weibull k = 0.74 Random order Wide band Random order Wide band Rayleigh Random order Random order

Variable amplitude results —————————————————————————————— Σn No. of S-N RG N speci- curve mens used in ————————–— ————————–— analysis Max. Min. Mean Max. Min. Mean

128 128

128 128 104 104 104 104

4 6 4

11D 11D 11F

1.44 0.88 1.31

1.04 0.68 0.59

1.28 0.77 0.97

2.56 1.57 2.01

1.86 1.21 0.90

4 5

11F 11G

1.02 2.37

0.64 1.18

0.79 1.83

1.56 1.67

0.98 0.83

5

11H

0.79

0.58

0.69

0.40

0.29

Table C1/11 Transverse single lap load-carrying fillet Reference Joint dimensions, mm

Material Stress yield ratio stress, N/mm2

Overbeeke B = 90 et al. 1990 T = 6

611

Constant amplitude results ————————————— Ref m F.S. at no. C 2 × 106 N/mm2

R = –1 12A 3.16 1.817 E12

77

Variable amplitude loading spectrum ——————————————————— Ref Form Clip p Block no. ratio length, cycles (a) actual (b) per peak

Variable amplitude results —————————————————————————————— Σn No. of S-N RG N speci- curve mens used in ————————–— ————————–— analysis Max. Min. Mean Max. Min. Mean

S65

Gauss l = 0.99

106

9

12A

4.53

0.25

1.17

S66

Gauss l = 0.7

106

7

12A

2.68

0.60

1.29

Table C1/12 Tubular T joints (axial loading) Reference Joint dimensions, mm

Material Stress yield ratio stress, N/mm2

Constant amplitude results ————————————— Ref m F.S. at no. C 2 × 106 N/mm2

Variable amplitude loading spectrum ——————————————————— Ref Form Clip p Block no. ratio length, cycles (a) actual (b) per peak

Variable amplitude results —————————————————————————————— Σn No. of S-N RG N speci- curve mens used in ————————–— ————————–— analysis Max. Min. Mean Max. Min. Mean

5

No relevant S-N curve T curve not usable because hot spot stress not defined

3

Mean 0.30 16 mm T Curve 206 424

Martin and McGregor 1977

Chord 312 D = 114 T = 5.4 Brace d = 60 t = 4.9

R = –1

S40

Rayleigh narrow band

Dover et al. 1992

D = 457 381 T = 16 d = 324 t = 16

0.62 to 0.69

S72

CP = –850 WASHWA (Sea states 5-7)

S73

WASHWB (Sea states 3-7) CP = –850

4.4

62 756

1

0.22

0.26

0.17

Table C1/13 Plain welded beams Reference Joint dimensions, mm

Material yield stress, N/mm2

Schilling Flange 824 et al. 1975 B = 171

Stress ratio or Smin

Constant amplitude results ————————————— Ref m F.S. at no. C 2 × 106 N/mm2

Variable amplitude loading spectrum ——————————————————— Ref Form Clip p Block no. ratio length, cycles (a) actual (b) per peak

Variable amplitude results —————————————————————————————— Σn No. of S-N RG N speci- curve mens used in ————————–— ————————–— analysis Max. Min. Mean Max. Min. Mean

R=0

21A 3.398 0.565 E14

S35

156

Rayleigh

2.44

0.25

500 500

9

21A Cmean

12.73 0.55 11.81 0.49

2.61 2.44

19.58 0.85 18.90 0.79

4.02 3.91

3.08

0

500 500

9

21A Cmean

11.58 0.70 10.71 0.64

2.41 2.23

20.38 1.23 19.67 1.18

4.25 4.10

1.985 0.5

500 500

8

21B Cmean

1.40 1.70

0.53 0.64

0.90 1.07

1.92 2.26

0.72 0.85

1.23 1.43

2.44

0.25

500 500

3 (275)

21B Cmean

1.29 1.53

0.50 0.59

0.92 1.09

2.14 2.45

0.83 0.95

1.52 1.74

1.98

0.5

500 500

3 (309)

21C Cmean

1.14 1.21

0.81 0.86

0.99 1.05

1.31 1.61

0.93 1.14

1.14 1.40

2.44

0.25

500 500

3 (274)

21C Cmean

1.02 0.97

0.86 0.81

0.92 0.88

1.34 1.55

1.12 1.30

1.21 1.40

S rd = 0.5 S rm

T = 10 Web d = 332

S34

Rayleigh

S rd = 1.0 S rm

t=7 w=5 Sub-arc 291

R=0

21B 3.594

171

S34

Rayleigh

S rd = 0.25 S rm

0.210 E15 S35

Rayleigh

S rd = 0.5 S rm Smin = –69

21C 2.993 0.729 E13

156

S34

Rayleigh

S rd = 0.25 S rm S35

Rayleigh

S rd = 0.5 S rm

Table C1/14 Beam with butt welded splice Reference Joint dimensions, mm

Yamada and Albrecht 1977

Material Stress yield ratio stress, N/mm2

B = 125 250 T1 = 10 minm T2 = 19 Weld overfill dressed

R=0

Constant amplitude results ————————————— Ref m F.S. at no. C 2 × 106 N/mm2

Only 4 results

Variable amplitude loading spectrum ——————————————————— Ref Form Clip p Block no. ratio length, cycles (a) actual (b) per peak

Variable amplitude results —————————————————————————————— Σn No. of S-N RG N speci- curve mens used in ————————–— ————————–— analysis Max. Min. Mean Max. Min. Mean

S2

9

US route 301 road bridge

2.18

0.57

1000 27

Cmean

5.45

1.80

3.25

8.69

2.88

5.18

Table C1/15 Beams with welded cover plates Reference Joint dimensions, mm

Material Stress yield ratio stress, N/mm2

Schilling Flange 824 et al. 1975 B = 170 T = 10 Cover plate b = 115 t = 14 Cover plate welded all round

R=0

Smin = 69

Constant amplitude results ————————————— Ref m F.S. at no. C 2 × 106 N/mm2

Variable amplitude loading spectrum ——————————————————— Ref Form Clip p Block no. ratio length, cycles (a) actual (b) per peak

Variable amplitude results —————————————————————————————— Σn No. of S-N RG N speci- curve mens used in ————————–— ————————–— analysis Max. Min. Mean Max. Min. Mean

Only 3 results

Approx. 58

S36

2.542 0.607 E11

58

S35

Rayleigh

3.08

0

500 500

3 (206)

23B Gmean

2.16 1.61

1.70 1.26

1.92 1.43

3.66 2.37

2.88 1.86

3.25 2.10

2.44

0.25

500 500

9

23A Gmean

1.69 2.34

0.91 1.00

1.35 1.53

1.83 3.07

0.98 1.31

1.46 2.01

3.08

0

500 500

12 1 U/B

23A Gmean

3.58 2.92

0.67 0.66

1.64 1.27

4.21 4.30

0.79 0.97

1.94 1.87

1.98

0.5

500 500

6

23B Gmean

1.15 0.81

0.60 0.52

0.87 0.67

1.45 0.94

0.75 0.60

1.10 0.78

2.44

0.25

500 500

9

23B Gmean

1.66 1.13

0.45 0.47

0.89 0.73

2.46 1.48

0.67 0.61

1.33 0.95

2.44

0.25

500 500

3 (138)

Gmean

0.58

0.50

0.53

0.76

0.66

0.70

S rd = 1.0 S rm Rayleigh

S rd = 0.5 S rm S36

Rayleigh

S rd = 1.0 S rm 291

R=0

3.309 0.190 E13

64

S34

Rayleigh

S rd = 0.25 S rm S35

Rayleigh

S rd = 0.5 S rm Smin = 69

Only 3 results

Approx. 53

S35

Rayleigh

S rd = 0.5 S rm

Table C1/15 Beams with welded cover plates (continued) Reference Joint dimensions, mm

Fisher et al. 1983

Yamada and Albrecht 1977

Material Stress yield ratio stress, N/mm2

B = 190 449 T = 14 Cover plate t = 125

B = 150 250 T = 10 minm l = 1550 b = 125 h = 10 w=6

Schilling, Flange 825 et al. 1975 B = 170 T = 10 Cover plate b = 115 t = 14 No trans. welds

Constant amplitude results ————————————— Ref m F.S. at no. C 2 × 106 N/mm2

R=0

Variable amplitude loading spectrum ——————————————————— Ref Form Clip p Block no. ratio length, cycles (a) actual (b) per peak

Variable amplitude results —————————————————————————————— Σn No. of S-N RG N speci- curve mens used in ————————–— ————————–— analysis Max. Min. Mean Max. Min. Mean

S32

Rayleigh

2.97

0.34

1.06

0.16

0.41

0.94

0.14

0.36

Rayleigh

2.69

0.40

4 (≈ 48) 4 (≈ 42)

Gmean

S30

1024 1024 1024 1024

Gmean

0.57

0.37

0.46

0.48

0.32

0.39

S31

Rayleigh

3.46

0.25

1024 1024

8

Gmean

1.74

0.76

1.28

1.70

0.75

1.25

R=0

23C 3.453 0.286 E14

118

S2

US route 301 road bridge

2.18

0.57

1000 27

8

23C Gmean

1.08 6.94

0.16 0.95

0.44 2.74

1.70 9.24

0.25 1.27

0.68 3.65

R=0

23D 4.188 0.682 E15

109

S35

Rayleigh

2.44

0.25

500 500

9 3 U/B

23D Gmean

3.23 5.60

1.18 2.82

2.28 4.32

6.63 7.33

2.42 3.69

4.70 5.66

3.08

0

500 500

6

23D Gmean

2.86 6.69

0.65 2.27

1.68 3.83

6.88 9.86

1.57 3.35

4.03 5.64

2.44

0.25

500 500

6

23E Gmean

1.56 3.07

0.90 1.55

1.23 2.30

1.93 4.02

1.12 2.03

1.53 3.01

S rd = 0.5 S rm S36

Rayleigh

S rd = 1.0 S rm Smin = 69

23E 2.875 0.554 E12

78

S35

Rayleigh

S rd = 0.5 S rm

Table C1/15 Beams with welded cover plates (continued) Reference Joint dimensions, mm

Material Stress yield ratio stress, N/mm2

Schilling CP et al. 1975 welded (cont) to flange before flange to web

Smin

Constant amplitude results ————————————— Ref m F.S. at no. C 2 × 106 N/mm2

2.875 0.554 E12

78

Smin = 276

2.977 0.440 E12

62

Smin = 69

3.281 0.214 E13

69

= 69

Variable amplitude loading spectrum ——————————————————— Ref Form Clip p Block no. ratio length, cycles (a) actual (b) per peak S36

Rayleigh

3.08

0

500 500

9

23E Gmean

1.87 3.66

0.66 1.05

1.29 2.33

2.60 5.40

0.91 1.55

1.80 3.44

2.44

0.25

500 500

6

23F Gmean

1.89 1.67

1.11 0.96

1.32 1.15

2.45 2.19

1.45 1.26

1.71 1.51

1.98

0.5

500 500

6

23G Gmean

1.31 1.45

0.98 0.90

1.11 1.14

1.62 1.67

1.23 1.04

1.39 1.31

2.44

0.25

500 500

3 (138)

23G Gmean

1.13 1.20

0.80 0.85

0.99 1.05

1.66 1.57

1.17 1.11

1.45 1.38

3.08

0

500 500

3 (206)

Gmean

2.08

1.65

1.80

3.07

2.44

2.66

103 – 104

16 (7U/B)

Class G 5.43 mean-2SD

1.07

S rd = 1.0 S rm S35

Rayleigh

S rd = 0.5 S rm S34

Rayleigh

S rd = 0.25 S rm S35

Rayleigh

S rd = 0.5 S rm Ditto, 825 but web welded before CP Fisher et al. 1993

Flange 312 305 × 25 Web 864 × 9.5

R=0

Only 3 results

Approx. 68

S36

Rayleigh

S rd = 1.0 S rm Rayleigh in random order

Variable amplitude results —————————————————————————————— Σn No. of S-N RG N speci- curve mens used in ————————–— ————————–— analysis Max. Min. Mean Max. Min. Mean

Table C1/17 Transverse K butt in bending Reference Joint dimensions, mm

Material Stress yield ratio stress, N/mm2

Holmes B = 125 371 and Booth T = 38 1981 l = 38 h = 200 w = 18

Constant amplitude results ————————————— Ref m F.S. at no. C 2 × 106 N/mm2

R = –1 31A 3.367 0.2015 E14 31B 3.05 0.2246 E13 31C 4.577 0.1529 E17 31D 3.187 0.3693 E13 High mean stres Smax = 440

Variable amplitude loading spectrum ——————————————————— Ref Form Clip p Block no. ratio length, cycles (a) actual (b) per peak

Variable amplitude results —————————————————————————————— Σn No. of S-N RG N speci- curve mens used in ————————–— ————————–— analysis Max. Min. Mean Max. Min. Mean

120

S70

C/12/20 Air

7.98

0

105

9

31A

1.45

0.83

1.13

1.28

0.74

1.01

96

S70

C/12/20 Free corrosion C/12/20 CP = –0.85V C/12/20 Intermittent immersion C/12/20 Free corrosion

7.98

0

105

3

31B

1.51

1.21

1.35

0.95

0.75

0.85

0

10

5

3

31C

2.41

1.29

1.81

6.70

3.59

5.04

5

3

31D

1.83

1.13

1.48

1.34

0.83

1.09

144 93

S70 S70

S70

S70

S70

S27 S43

C/12/20 Cathodic protection C/12/20 Intermittent immersion Rayleigh Free corrosion Laplace Free corrosion

7.98 7.98

0

10

7.98

0

105

3

31B

1.34

1.16

1.25

1.14

1.00

1.07*

7.98

0

105

3

31C

1.05

0.62

0.77

2.93

1.83

2.24*

7.98

0

105

3

31D

1.34

0.90

1.17

1.32

0.85

1.13*

4.80

0

105

3

31B

1.38

0.98

1.13

0.90

0.64

0.77*

11.51 0

105

3

31B

1.19

0.97

1.08

1.11

0.94

1.00*

*NB. These results are based upon S-N curves obtained at a different stress ratio. They should be treated with caution.

Table C1/17 Transverse K butt in bending (continued) Reference Joint dimensions, mm

UKOSRP II (Series, B, C, D) 1986

Slind 1993

B = 125 T = 38

W = 75 T = 30

Material Stress yield ratio stress, N/mm2

Constant amplitude results ————————————— Ref m F.S. at no. C 2 × 106 N/mm2

Variable amplitude loading spectrum ——————————————————— Ref Form Clip p Block no. ratio length, cycles (a) actual (b) per peak

Variable amplitude results —————————————————————————————— Σn No. of S-N RG N speci- curve mens used in ————————–— ————————–— analysis Max. Min. Mean Max. Min. Mean

R = –1 31E 4.575 1.089 E16

134

S70

C/12/20 CP = –0.85V

105

8 1 U/B

31E

3.77

0.79

2.17

R = –1 31F 3.358 1.930 E13 R = –1 31G 4.575 1.089 E16

120

S74

Wide band S4 Air Wide band S4 CP = –0.85V

106

7

31F

1.44

0.65

1.04

7 3 U/B

31G

0.69

0.52

0.61

WASH Wide band l = 0.58-0.83 Ditto in sea water CP = –1020 [NB 2 highest and lowest sea states omitted]

2 × 105

Tension

134

S74

10

6

14

1.07

0.67

0.83

11

1.14

0.53

0.72

Table C18 Partial penetration K butt in bending Reference Joint dimensions, mm

Joehnk 1982

Zwerneman 1983

Material Stress yield ratio stress, N/mm2

B = 76 400 T = 25 l = 25 h = 200 w = 17

B = 76 400 T = 25 l = 25 h = 200 w = 17

Constant amplitude results ————————————— Ref m F.S. at no. C 2 × 106 N/mm2

Variable amplitude loading spectrum ——————————————————— Ref Form Clip p Block no. ratio length, cycles (a) actual (b) per peak

R=0

Smin = 34

*Includes 2 results from Swensson

32A 3.465 0.756 E14

154

S21

Superimposed sine waves with n small stresses of size p in each main cycle n=4 2.02

0.6

S20

n=7

2.21

0.571

S15

n=9

4.06

0.154

S16

n=9

3.16

0.333

S17

n=9

2.48

0.50

S19

n=9

2.0

0.667

S18

n=9

2.28

0.56

S14

n = 15 p = 0.429

2.92

0.429

S11

26 level test truck programme

4.85

0

5 5 8 8 10 10 10 10 10 10 10 10 10 10 16 16 27

Variable amplitude results —————————————————————————————— Σn No. of S-N RG N speci- curve mens used in ————————–— ————————–— analysis Max. Min. Mean Max. Min. Mean

2 (163)

32A

0.45

0.40

0.43

0.70

0.63

0.66

2 (230)

32A

0.66

0.57

0.62

01.08 0.93

1.01

2 (229)

32A

0.68

0.37

0.53

0.96

0.52

0.74

2 (196)

32A

0.31

0.28

0.30

0.56

0.51

0.53

4 (196)

32A

0.42

0.38

0.40

0.74

0.66

0.70

2 (196)

32A

0.61

0.50

0.55

0.88

0.72

0.80

2 (164)

32A

0.69

0.54

0.62

1.14

0.89

1.02

2 (229)

32A

0.32

0.30

0.31

0.59

0.56

0.57

3 + 2*

32A

0.54

0.50

0.52

1.13

1.05

1.10

Table C18 Partial penetration K butt in bending (continued) Reference Joint dimensions, mm

Material Stress yield ratio stress, N/mm2

Constant amplitude results ————————————— Ref m F.S. at no. C 2 × 106 N/mm2

Variable amplitude loading spectrum ——————————————————— Ref Form Clip p Block no. ratio length, cycles (a) actual (b) per peak

Zwerneman 1983 (cont)

Swensson B = 76 400 1984 T = 25 l = 25 h = 200 w = 17

see Zwerneman

S8

2 level block 3.16 programme Smean of minor stress = 69 2 level block 3.16 but Smean of minor stress = 138 2 level block 316 programme but Smean of minor stress = 207 Traffic load (1)

S9

Variable amplitude results —————————————————————————————— Σn No. of S-N RG N speci- curve mens used in ————————–— ————————–— analysis Max. Min. Mean Max. Min. Mean

0.33

10 10

4 (207)

32A

0.68

0.47

0.61

1.22

0.85

1.10

0.33

10 10

2 (207)

32A

0.55

0.48

0.51

0.99

0.86

0.92

0.33

10 10

2 (207)

32A

0.38

0.37

0.38

0.69

0.67

0.68

0

850

2 (241)

32A

1.06

0.98

1.02

3.14

2.90

3.02

Traffic load (2) 6.96

0

710

3

32A

2.17

0.70

1.37

5.57

1.80

3.51

S10

Traffic load (3) 7.11

0

99

2

32A

0.93

0.71

0.82

2.39

1.82

2.10

S36

Rayleigh

0

500

2

32A

0.71

0.64

0.68

1.29

1.16

1.23

S rd = 1.00 S rm

3.08

Table C1/19 Transverse non-load-carrying fillet weld in bending Reference Joint dimensions, mm

Zwaans et al. 1986

Material Stress yield ratio stress, N/mm2

B = 218 408 T = 40 l = 40 b = 2.8

Constant amplitude results ————————————— Ref m F.S. at no. C 2 × 106 N/mm2

Variable amplitude loading spectrum ——————————————————— Ref Form Clip p Block no. ratio length, cycles (a) actual (b) per peak

R = –1

Variable amplitude results —————————————————————————————— Σn No. of S-N RG N speci- curve mens used in ————————–— ————————–— analysis Max. Min. Mean Max. Min. Mean

Rayleigh Gaussian

5.4 5.4

6 4

33A*

2.15

1.07

1.40

1.07

0.53

0.69

Rayleigh Gaussian

5.4 5.4

4 4

33B*

1.24

0.74

1.02

0.23

0.14

0.19

Tests in sea water R = –1 Fisher et al. 1983

B = 31 T = 16 l = 16 h = 31 w = 12

690 minm

Mean stress = 215

Only 4 results

–

S33

Rayleigh 8 level block (random) with

243

S rd = 0.5 S rm Srmax – Srmin = 3Srd

* These results are based upon constant amplitude S-N curves for R = 0.1. They should therefore be treated with caution. Relevant curves were: Air (33A) m = 3.438, C = 0.1378 E14; sea water (33B) m = 2.458, C = 0.627 E11.

No relevant curve

Table C1/20 Transverse load-carrying fillet welds in bending Reference Joint dimensions, mm

Material Stress yield ratio stress, N/mm2

Suhr 1970

267

B = 50 T = 25 l = 76 b = 50 h = 89 w = 19

R = –1

Constant amplitude results ————————————— Ref m F.S. at no. C 2 × 106 N/mm2

Variable amplitude loading spectrum ——————————————————— Ref Form Clip p Block no. ratio length, cycles (a) actual (b) per peak

Variable amplitude results —————————————————————————————— Σn No. of S-N RG N speci- curve mens used in ————————–— ————————–— analysis Max. Min. Mean Max. Min. Mean

S41

12

Rayleigh Narrow band

5.0

No relevant curve

Table C1/21 Longitudinal non-load-carrying fillet weld in bending (in artificial sea water) Reference Joint dimensions, mm

Material Stress yield ratio stress, N/mm2

Holmes B = 125 370 and Booth T = 38 1981 l = 150 t = 13 h = 50 w = 10

Constant amplitude results ————————————— Ref m F.S. at no. C 2 × 106 N/mm2

Variable amplitude loading spectrum ——————————————————— Ref Form Clip p Block no. ratio length, cycles (a) actual (b) per peak

Variable amplitude results —————————————————————————————— Σn No. of S-N RG N speci- curve mens used in ————————–— ————————–— analysis Max. Min. Mean Max. Min. Mean

R = –1 36A 2.884 0.1702 E13

114

S70

C/12/20 Free corrosion

7.98 0

105

3

35A

1.32

0.75

1.11

0.69

0.39

0.58

36B 3.559 0.8058 E14

137

S70

C/12/20 Cathodic protection

7.98 0

105

3

35B

2.11

1.28

1.61

2.29

1.39

1.74

36C 2.483 0.3174 E12

124

S70

C/12/20 Intermittent immersion

7.98 0

105

3

35C

1.16

0.59

0.84

0.38

0.19

0.27

S70

C/12/20 Free corrosion

7.98 0

105

3

35A

0.91

0.76

0.83

0.65

0.56

0.60*

S70

C/12/20 Cathodic protection

7.98 0

105

3

35B

0.88

0.41

0.58

1.13

0.55

0.77*

S70

C/12/20 Intermittent immersion

7.98 0

105

3

35C

0.75

0.35

0.48

0.36

0.17

0.23*

S43

Laplace 11.51 0 Free corrosion

105

3 1 U/B

35A

1.51

0.51

1.03

0.95

0.32

0.68

S27

Rayleigh Free corrosion

105

3

35A

2.94

1.01

1.84

3.61

0.58

1.68

High mean stress Smax = 440

R = –1

4.80 0

* NB. These results are based upon S-N curves obtained at a different stress ratio. They should be treated with caution.

Table C1/22 Tubular T joints (IPB) Reference Joint dimensions, mm

Material Stress yield ratio stress, N/mm2

Tubby and D = 914 335 Skallerud T = 32 1992 d = 457 t = 16

Constant amplitude results ————————————— Ref m F.S. at no. C 2 × 106 N/mm2

Variable amplitude loading spectrum ——————————————————— Ref Form Clip p Block no. ratio length, cycles (a) actual (b) per peak S71

WASHW

Variable amplitude results —————————————————————————————— Σn No. of S-N RG N speci- curve mens used in ————————–— ————————–— analysis Max. Min. Mean Max. Min. Mean

499 962 (8 sea states)

Air

4

Sea water CP = 1030

10

0.71 Tmean for 32 mm Ditto 0.55

0.19

0.45

0.13

0.27

Table C1/23 Tubular Y joint (OPB) Reference Joint dimensions, mm

Material Stress yield ratio stress, N/mm2

Constant amplitude results ————————————— Ref m F.S. at no. C 2 × 106 N/mm2

Vinas D = 457 367 Pich et al. T = 16 1995 d = 324 t = 12.5 θ = 35°

* Hot spot equivalent stress range at mean stress = 572 N/mm2

Variable amplitude loading spectrum ——————————————————— Ref Form Clip p Block no. ratio length, cycles (a) actual (b) per peak S71

WASHW 8 sea states Air

Variable amplitude results —————————————————————————————— Σn No. of S-N RG N speci- curve mens used in ————————–— ————————–— analysis Max. Min. Mean Max. Min. Mean

1 (200)

Sea water CP = –850

1 (180)

CP = –1000 CP = –1100

1 (180) 1 (180)

16 mm Tcurve Mean 16 mm Tcurve

1.74 0.31

0.29 0.17

Table C1/24 Orthotropic deck, trough to cross girder joints Reference Joint dimensions, mm

Material Stress yield ratio stress, N/mm2

Beales 1990

355 minm

Constant amplitude results ————————————— Ref m F.S. at no. C 2 × 106 N/mm2

Variable amplitude loading spectrum ——————————————————— Ref Form Clip p Block no. ratio length, cycles (a) actual (b) per peak S12

Rheden Bridge measured stresses in random order omitting all below 50 N/mm2

Variable amplitude results —————————————————————————————— Σn No. of S-N RG N speci- curve mens used in ————————–— ————————–— analysis Max. Min. Mean Max. Min. Mean

Class E

0.73

0.20

Table C1/25 K butt in bending (toe ground) Reference Joint dimensions, mm

Material Stress yield ratio stress, N/mm2

Suhr 1970

267

B = 51 T = 25 l = 76 h = 89 w = 19

Constant amplitude results ————————————— Ref m F.S. at no. C 2 × 106 N/mm2

R = –1

B = 51 T = 25 l = 76 h = 89 w = 19 Profile ground UKOSRP II B = 125 (Series A) T = 38 1986

R = –1 51A 2.807 1.696 E12

129

Variable amplitude loading spectrum ——————————————————— Ref Form Clip p Block no. ratio length, cycles (a) actual (b) per peak

Variable amplitude results —————————————————————————————— Σn No. of S-N RG N speci- curve mens used in ————————–— ————————–— analysis Max. Min. Mean Max. Min. Mean

S41

Rayleigh Narrow band

5.0

11

No relevant curve

S41

Rayleigh Narrow band

5.0

16

No relevant curve

S70

C/12/20 Sea water CP = –0.85V

8 3 U/B

51A

4.71

0.37

1.85

Table C1/26 Transverse load-carrying fillet welds in bending (toe ground) Reference Joint dimensions, mm

Material Stress yield ratio stress, N/mm2

Jones and B = 45 310 Higgins T = 9.5 1970 l = 25 b = 45 h = 76 w=7

Lewszuk B = 45 324 and White T = 9.5 1970 l = 25 w=7 B = 35 T = 9.5 l = 9.5 b = 35 h = 45 w=7

Constant amplitude results ————————————— Ref m F.S. at no. C 2 × 106 N/mm2

R = –1

R = –1

Variable amplitude loading spectrum ——————————————————— Ref Form Clip p Block no. ratio length, cycles (a) actual (b) per peak

Variable amplitude results —————————————————————————————— Σn No. of S-N RG N speci- curve mens used in ————————–— ————————–— analysis Max. Min. Mean Max. Min. Mean

S41

Narrow band Rayleigh (100 Hz)

5.0

8

S41

Rayleigh (500 Hz)

5.0

14

S41

Rayleigh (1000 Hz)

5.0

20

S41

Rayleigh (2500 Hz)

5.0

9

S37

Rayleigh (room temp.)

4.0

16

S37

4.0

16

4.0

2980

26

2980 1 U/B

20

R = –1 52A 3.763 0.394 E 16

295

S37

Rayleigh (350°C) Rayleigh

Mean stress = 230 Mean stress = 154

227

S37

Rayleigh

4.0

S37

Rayleigh

4.0

52B 4.032 0.632 E16

21

    

No relevant curve

52A

4.75

0.28

1.37

9.18

0.54

2.65

52B

1.84

0.37

1.05

4.16

0.84

2.40

Table C1/26 Transverse load-carrying fillet welds in bending (toe ground) (continued) Reference Joint dimensions, mm

Suhr 1970

B = 50 T = 25 l = 76 b = 50 h = 89 w = 19

Joint B = Nuclear T = 10 Power w=7 Committee (ex-White and Lewszuk) 1971

Material Stress yield ratio stress, N/mm2

Constant amplitude results ————————————— Ref m F.S. at no. C 2 × 106 N/mm2

Variable amplitude loading spectrum ——————————————————— Ref Form Clip p Block no. ratio length, cycles (a) actual (b) per peak

Variable amplitude results —————————————————————————————— Σn No. of S-N RG N speci- curve mens used in ————————–— ————————–— analysis Max. Min. Mean Max. Min. Mean

Mean stress = 77

S37

Rayleigh

4.0

19

267 R = –1 (stressrelieved)

S41

Rayleigh Narrow band

5.0

4

267

S41

Rayleigh Narrow band

5.0

31

S37

Rayleigh Narrow band

4.0

25

324

R = –1

    

No relevant curve

Table C2.1 Summary of test results obtained under block programme loading Table C2/1 Longitudinal butt weld Reference Joint dimensions, mm

Material Stress yield ratio stress, 2 N/mm

Plumier 1977

393

B = 80 T = 10

R = –1

Constant amplitude results ————————————— Ref m F.S. at no. C 2 × 106 N/mm2

Only 4 results

Variable amplitude loading spectrum ———————————————————––– Ref Form Clip p Block no. ratio length, cycles (a) actual (b) per peak

Approx. ± 100 F17

R = 0.1 F17

8 step block programme Stationary Gaussian

2.33

8 step block programme Stationary Gaussian

2.33

0.5

0.5

Variable amplitude results —————————————————————————————— Σn RG No. of S-N N speci- curve mens used in ————————–— ————————–— analysis Max. Min. Mean Max. Min. Mean

500 009 500 009

19 5 in grips

500 009 500 009

11

Dmean

3.45

0.70

1.53

1.49

0.30

0.66

Dmean

1.70

0.75

1.19

0.73

0.32

0.51

Table C2/2 Transverse butt weld (automatic) Reference Joint dimensions, mm

Material Stress yield ratio stress, N/mm2

Bastenaire B = 60 395 and T = 12 Lieurade Sub. arc 1978b

Benoit et al. 1977

B = 60 T = 12 Sub. arc square butt

395

B = 60 395 T = 12 Sub. arc double V prep.

R=0

R=0

R=0

Constant amplitude results ————————————— Ref m F.S. at no. C 2 × 106 N/mm2

Variable amplitude loading spectrum ———————————————————––– Ref Form Clip p Block no. ratio length, cycles (a) actual (b) per peak

Variable amplitude results —————————————————————————————— Σn No. of S-N RG N speci- curve mens used in ————————–— ————————–— analysis Max. Min. Mean Max. Min. Mean

–

P11

lncr/Decr 8 step block

1.96

0.67

400 000 200 000

15 (250)

Emean

2.31

0.80

1.62

1.16

0.40

0.82

P12

8 step block random order

1.96

0.67

200 000 20 000

15 (250)

Emean

3.07

0.82

1.71

1.70

0.45

0.94

P13

lncr/Decr 4 step block

1.95

0.67

407 160 203 580

15 (250)

Emean

2.89

1.36

1.82

1.45

0.68

0.91

P13

4 step block, 1.95 lowest following highest

0.67

203 580 203 580

15 (250)

Emean

5.94

1.72

2.86

2.97

0.86

1.43

P10

8 step block Stationary Gaussian

3.16

0.33

200 001

21 7 U/B

(a) Emean

6.97 4.09

2.63 0.95

4.72 2.30

8.87 1.44

3.44 0.33

6.01 0.81

P11

8 step block Stationary Gaussian 8 step block Stationary Gaussian

1.96

0.67

200 001

18 6 U/B

(a) Emean

3.59 2.84

1.01 0.62

2.01 1.51

3.33 1.43

0.94 0.31

1.86 0.76

3.16

0.33

200 001

20 5 U/B

Emean

0.92

0.08

0.30

0.32

0.03

0.11

Emean

9.46

3.58

7.42

4.75

1.80

3.72

(a)

m = 4.921 127 c = 0.446 E17

P10

P11

8 step block Stationary Gaussian

1.96

0.67

200 001

6 3 U/B (360)

Table C2/3 Transverse butt weld (manual) Reference Joint dimensions, mm

Plumier 1977

Material Stress yield ratio stress, N/mm2

B = 100 363 T = 10

393

Constant amplitude results ————————————— Ref m F.S. at no. C 2 × 106 N/mm2

R = 0.1 (b)

3.267 0.190 E14

137

R = –1 (c)

2.207 0.172 E12

172

R = 0.1

R = –1

Only 3 results

–

Variable amplitude loading spectrum ——————————————————— Ref Form Clip p Block no. ratio length, cycles (a) actual (b) per peak

Variable amplitude results —————————————————————————————— Σn No. of S-N RG N speci- curve mens used in ————————–— ————————–— analysis Max. Min. Mean Max. Min. Mean

F17

8 step block programme Stationary Gaussian

2.33

0.5

500 009 500 009

10

(c) Dmean

3.04 9.31

0.48 1.61

1.17 4.03

0.89 4.02

0.14 0.69

0.34 1.74

F17

8 step block programme Stationary Gaussian

2.33

0.5

500 009 500 009

15

(b) Dmean

2.34 2.88

0.61 0.77

1.41 1.72

1.15 1.24

0.30 0.33

0.69 0.74

F17

8 step block programme Stationary Gaussian

2.33

0.5

500 009 500 009

13

(c) Dmean

2.29 7.65

0.57 2.03

1.14 3.98

0.67 3.30

0.17 0.88

0.33 1.72

Table C2/4 Transverse butt welds with deliberate slag inclusions Reference Joint dimensions, mm

Anon. 1963

Material Stress yield ratio stress, N/mm2

Constant amplitude results ————————————— Ref m F.S. at no. C 2 × 106 N/mm2

Variable amplitude loading spectrum ——————————————————— Ref Form Clip p Block no. ratio length, cycles (a) actual (b) per peak

Variable amplitude results —————————————————————————————— Σn No. of S-N RG N speci- curve mens used in ————————–— ————————–— analysis Max. Min. Mean Max. Min. Mean

With 6.3 mm slag

R=0

(d)

3.390 0.2752 E14

128

P20

8 step block programme Stationary Gaussian

5.94

0.11

500 000 500 000

5

(d)

3.45

1.56

2.48

2.42

1.09

1.74

With cont. slag

R=0

(e)

3.266 0.4872 E13

90

P20

8 step block programme Stationary Gaussian

5.94

0.11

500 000 500 000

8

(e)

3.66

0.72

2.02

2.31

0.46

1.28

Table C2/5 Longitudinal non-load-carrying fillet welds Reference Joint dimensions, mm

Harrison 1974a

Material yield stress, N/mm2

Stress ratio or Smin

Constant amplitude results ————————————— Ref m F.S. at no. C 2 × 106 N/mm2

Variable amplitude loading spectrum ———————————————————––– Ref Form Clip p Block no. ratio length, cycles (a) actual (b) per peak

Variable amplitude results —————————————————————————————— Σn No. of S-N RG N speci- curve mens used in ————————–— ————————–— analysis Max. Min. Mean Max. Min. Mean

R=0

(g)

P9

10 level block programme Quadratic random order

4.03

0.1

2851 2851

10

(g) Fmean

3.18 3.09

0.64 0.66

1.91 1.81

3.69 3.90

0.74 0.83

2.21 2.29

P9

10 level block programme Quadratic random order

4.03

0.1

2851 2851

8

(g) Fmean

2.61 2.57

1.35 1.25

1.70 1.61

3.03 3.24

1.57 1.58

1.97 2.03

436

P9

10 level block programme Quadratic random order

4.03

0.1

2851 2851

9

(g) Fmean

2.14 1.97

1.29 1.17

1.64 1.52

2.49 2.48

1.49 1.48

1.90 1.92

388

P9

10 level block programme Quadratic random order

4.03

0.1

2851 2851

7

(g) Fmean

1.56 1.43

1.05 1.01

1.31 1.20

1.81 1.80

1.21 1.27

1.52 1.51

365

P9

10 level block programme Quadratic random order

4.03

0.1

2851 2851

8

(g) Fmean

2.30 2.14

0.74 0.70

1.44 1.33

2.66 2.69

0.85 0.88

1.67 1.67

B = 102 726 T = 13 l = 152 t = 13 h = 38 w=8 544

2.86 89 7.62 E11 m = 2.86 C = 7.62 E11

Table C2/5 Longitudinal non-load-carrying fillet welds (continued) Reference Joint dimensions, mm

Berge 1981

Material Stress yield ratio stress, N/mm2

B = 75 Approx. T = 12 300 l = 150 b = 12 h = 100

R=0

Constant amplitude results ————————————— Ref m F.S. at no. C 2 × 106 N/mm2

Variable amplitude loading spectrum ———————————————————––– Ref Form Clip p Block no. ratio length, cycles (a) actual (b) per peak

Variable amplitude results —————————————————————————————— Σn No. of S-N RG N speci- curve mens used in ————————–— ————————–— analysis Max. Min. Mean Max. Min. Mean

(h)

P3

9 step block 3.53 programme Linear p.v. Ln NE

0.36

20 000 20 000

5 (177)

(h) Fmean

1.12 0.86

0.68 0.52

0.93 0.71

0.56 0.41

0.34 0.25

0.46 0.34

P2

8 step block programme

2.95

0.44

6121 6121

(h) Fmean

4.83 3.65

4.09 3.10

4.46 3.37

3.00 2.20

2.54 1.87

2.77 2.03

(i) Fmean

2.80 2.52

1.82 1.64

2.23 2.02

1.86 1.52

1.21 0.99

1.49 1.22

R=0 (i) (stress relieved) R = –1 (j)

3.041 0.157 E13

87

3.143 0.299 E13

92

P2

8 step block programme

2.95

0.44

6121 6121

5 3 U/B (177) 5 (191)

3.020 0.157 E13

89

P3

9 step block programme

3.53

0.36

20 000 20 000

5 (184)

(j) Fmean

1.33 1.11

1.02 0.85

1.13 0.94

0.65 0.53

0.50 0.41

0.55 0.45

P2

8 step 2.95 programme ending tensile

0.44

6121 6121

5 (184)

(j) Fmean

1.80 1.49

1.22 1.01

1.40 1.16

1.10 0.90

0.74 0.61

0.86 0.70

P2

8 step block programme ending compressive

2.95

0.44

6121 6121

5 (184)

(j) Fmean

1.15 0.96

0.72 0.60

0.99 0.82

0.71 0.58

0.44 0.36

0.60 0.50

P2

8 step programme with mean stress = 25

2.95

0.44

6121 6121

5 (184)

(j) Fmean

1.38 1.15

0.91 0.76

1.12 0.93

0.84 0.69

0.56 0.46

0.69 0.56

Table C2/5 Longitudinal non-load-carrying fillet welds (continued) Reference Joint dimensions, mm

Material Stress yield ratio stress, N/mm2

Constant amplitude results ————————————— Ref m F.S. at no. C 2 × 106 N/mm2

Berge 1981 (cont) R = –1 (k) (stress relieved)

3.481 0.325 E14

118

Variable amplitude loading spectrum ———————————————————––– Ref Form Clip p Block no. ratio length, cycles (a) actual (b) per peak

Variable amplitude results —————————————————————————————— Σn No. of S-N RG N speci- curve mens used in ————————–— ————————–— analysis Max. Min. Mean Max. Min. Mean

P2

Randomised 8 step programme

2.95

0.44

6121 6121

7 (184)

(j) Fmean

1.52 1.27

0.75 0.63

1.09 0.91

0.93 0.76

0.46 0.38

0.67 0.55

P2

8 step block programme

2.95

0.44

6121 6121

10 (252)

(k) Fmean

1.20 2.19

0.48 0.88

0.78 1.42

1.00 1.32

0.40 0.53

0.65 0.86

Table C2/6 Transverse non-load-carrying fillet welds Reference Joint dimensions, mm

Material Stress yield ratio stress, N/mm2

Constant amplitude results ————————————— Ref m F.S. at no. C 2 × 106 N/mm2

Albrecht and Yamada 1979

420

(γ)

Benoit et al. 1977

B = 26 T = 10 l=6 b = 26 h = 50 w=6

B = 60 395 T = 12 l = 12 b = 60 h = 20 w=8 Manual weld B = 60 T = 12 l = 12 b = 60 h = 20 w=7 Automatic weld

R=0

R=0

R=0

(l)

3.232 0.159 E14

3.749 0.303 E15

(m) 4.608 0.189 E17

Variable amplitude loading spectrum ———————————————————––– Ref Form Clip p Block no. ratio length, cycles (a) actual (b) per peak

136

152

146

10 level block programme

2.18

10 level block programme

2.17

10 level block programme

2.17

10 level block programme

2.17

P11

8 step block Stationary Gaussian

P10

0.53

Variable amplitude results —————————————————————————————— Σn No. of S-N RG N speci- curve mens used in ————————–— ————————–— analysis Max. Min. Mean Max. Min. Mean

100 62

9 Fmean

1.64 4.36

0.58 1.46

0.98 2.68

2.29 5.59

0.81 1.87

1.37 3.44

1000 62

9 Fmean

1.39 4.03

0.79 2.09

1.06 2.90

1.99 5.32

1.13 2.76

1.52 3.81

10 000 62

9 Fmean

2.00 5.82

0.99 2.57

1.31 3.56

2.87 7.67

1.42 3.39

1.88 4.69

0.5

105 62

9 1 U/B

(γ) Fmean

4.16 1.39 12.08 3.69

2.17 5.88

5.95 1.99 15.91 4.86

3.10 7.74

1.96

0.69

200 001

24 8 U/B

(l) Fmean

1.64 6.44

0.91 2.91

1.14 4.16

1.05 3.23

0.58 1.46

0.73 2.24

8 step block Stationary Gaussian

3.16

0.38

200 001

9 7 U/B

(l) Fmean

2.91 9.76

2.84 9.36

2.87 9.56

1.74 3.44

1.70 3.30

1.72 3.37

P11

8 step block Stationary Gaussian

1.96

0.69

20 001

17 3 U/B

(m) Fmean

2.43 4.20

0.84 1.82

1.30 3.40

2.03 2.11

0.70 0.91

1.09 1.71

P10

8 step block Stationary Gaussian

3.16

0.38

200 001

18 5 U/B

(j) Fmean

3.06 7.55

1.05 2.48

2.23 4.93

3.27 2.66

1.12 0.87

2.39 1.73

0.53

0.5

Table C2/6 Transverse non-load-carrying fillet welds (continued) Reference Joint dimensions, mm

Gassner et al. 1964

Material Stress yield ratio stress, N/mm2

B = 60 T =4 l=8 b = 60 h = 20 w=5

Constant amplitude results ————————————— Ref m F.S. at no. C 2 × 106 N/mm2

R = –1 (n)

3.388 0.275 E14

128

Variable amplitude loading spectrum ——————————————————— Ref Form Clip p Block no. ratio length, cycles (a) actual (b) per peak P16

8 step block Stationary Gaussian

2.33

8 step block Stationary Gaussian

3.42

8 step block Stationary Gaussian

6.00

P16

8 step block Stationary Gaussian

2.33

0.56

P15

8 step block Stationary Gaussian

3.42

P15

P14

Haibach 1968b

B = 50 T = 10 l = 10 b = 50 h = 15 w=6

265

383

R = –1 (o)

R=0

(p)

R = –1 (q)

4.820 0.316 E18

±105

0.56

500 009

Variable amplitude results —————————————————————————————— Σn No. of S-N RG N speci- curve mens used in ————————–— ————————–— analysis Max. Min. Mean Max. Min. Mean

30 2 U/B

(n) Fmean

3.28 8.26

0.45 1.12

1.10 2.41

1.71 3.57

0.23 0.49

0.57 1.04

20 1 U/B

(n) Fmean

3.00 7.25

0.39 0.86

1.56 3.64

1.62 2.90

0.21 0.35

0.84 1.45

20 4 U/B

(n) Fmean

2.13 4.78

0.84 1.77

1.55 3.41

1.52 2.36

0.60 0.87

1.10 1.68

500 009

9 (470)

(o)

4.03

1.31

2.58

4.09

1.33

2.63

0.34

500 009

9 (470)

(o)

7.72

1.48

3.73

11.64 2.24

5.62

500 009 0.34

500 009 500 009

0.12

500 009 500 009

3.929 0.124 E16

173

P16

8 step block Stationary Gaussian

2.33

0.5

500 009

9 (353)

(p)

2.06

0.80

1.30

1.38

0.54

0.87

4.232 0.1515 E17

±108

P16

8 step block Stationary Gaussian

2.33

0.56

500 009

18

(q)

2.46

0.55

1.29

1.91

0.43

1.00

Table C2/6 Transverse non-load-carrying fillet welds (continued) Reference Joint dimensions, mm

Material Stress yield ratio stress, N/mm2

Constant amplitude results ————————————— Ref m F.S. at no. C 2 × 106 N/mm2

Haibach 1968b (cont)

Lieurade 1975

B = 60 395 T = 12 l = 12 b = 60 h = 20 w=8 Manual weld

R=0

B = 60 280 T = 12 l = 12 b = 60 h = 20 w=7 sub merged arc weld

R=0

(r)

(s)

3.051 0.265 E13

4.567 0.110 E17

101

136

Variable amplitude loading spectrum ——————————————————— Ref Form Clip p Block no. ratio length, cycles (a) actual (b) per peak

Variable amplitude results —————————————————————————————— Σn No. of S-N RG N speci- curve mens used in ————————–— ————————–— analysis Max. Min. Mean Max. Min. Mean

P15 8 step block Stationary Gaussian

3.42

0.34

500 009

9 (470)

(q)

8.53

1.43

2.70

8.56

1.43

2.71

P14

8 step block Stationary Gaussian

6.00

0

500 009

9 (626)

(q)

2.82

1.65

2.16

4.09

2.39

3.13

P11

8 step block programme Stationary Gaussian

1.96

0.69

200 001

14 3 U/B

(r) Fmean

1.21 1.45

0.54 0.66

0.76 0.91

0.62 0.73

0.28 0.33

0.39 0.46

P10

8 step block programme Stationary Gaussian

3.16

0.38

200 001

4 1 U/B

(r) Fmean

2.71 3.22

1.46 1.75

2.02 2.42

0.99 1.13

0.53 0.62

0.74 0.85

P11

8 step block programme Stationary Gaussian

1.96

0.69

200 001

18 5 U/B

(s) Fmean

2.80 5.21

1.14 1.97

1.81 3.89

2.31 2.61

0.94 0.99

1.50 1.94

P10

8 step block programme Stationary Gaussian

3.16

0.38

200 001

12 5 U/B

(s) Fmean

3.18 6.47

2.07 4.16

2.32 4.76

3.31 2.28

2.15 1.46

2.42 1.67

Table C2/6 Transverse non-load-carrying fillet welds (continued) Reference Joint dimensions, mm

Material Stress yield ratio stress, N/mm2

Constant amplitude results ————————————— Ref m F.S. at no. C 2 × 106 N/mm2

Abtahi et al. 1976

421

α

B = 25 T = 10 l=6 b = 25 h = 50 w= 6

R=0

3.243 0.1686 E14

137

Variable amplitude loading spectrum ——————————————————— Ref Form Clip p Block no. ratio length, cycles (a) actual (b) per peak

Variable amplitude results —————————————————————————————— Σn No. of S-N RG N speci- curve mens used in ————————–— ————————–— analysis Max. Min. Mean Max. Min. Mean

Constant amplitude with overloads of 1.67 × nominal stress at intervals of: 10 cycles 2.19

0.6

11 11

9

α Fmean

0.21 0.66

0.08 0.24

0.13 0.38

0.30 0.88

0.12 0.32

0.18 0.51

100 cycles

2.34

0.6

101 101

9

α Fmean

0.38 1.26

0.09 0.29

0.18 0.56

0.30 0.89

0.07 0.21

0.14 0.40

103 cycles

2.35

0.6

1001 1001

9

α Fmean

0.48 1.58

0.13 0.42

0.29 0.90

0.16 0.46

0.04 0.12

0.09 0.26

104 cycles

2.35

0.6

10 001 10 001

9 3 U/B

α Fmean

1.11 3.51

0.29 0.93

0.48 1.44

0.15 0.41

0.04 0.11

0.08 0.17

105 cycles

2.35

0.6

105 + 1 105 + 1

7 1 U/B

α Fmean

1.04 3.26

0.20 0.64

0.54 1.64

0.05 0.15

0.01 0.03

0.03 0.08

106 cycles

2.35

0.6

106 + 1 106 + 1

3 2 U/B (177)

α Fmean

0.26

0.01

Table C2/7 Partial penetration K butt (weld throat failure) Reference Joint dimensions, mm

Lieurade 1975

Material Stress yield ratio stress, N/mm2

B = 60 395 T = 12 l = 12 b = 60 h = 52 w=8 Sub-arc 3 mm lack of pen

R=0

Constant amplitude results ————————————— Ref m F.S. at no. C 2 × 106 N/mm2

Variable amplitude loading spectrum ——————————————————— Ref Form Clip p Block no. ratio length, cycles (a) actual (b) per peak

Variable amplitude results —————————————————————————————— Σn No. of S-N RG N speci- curve mens used in ————————–— ————————–— analysis Max. Min. Mean Max. Min. Mean

(f)

P11

8 step block

1.96

0.69

200 001

18 5 U/B

(f)

1.39

0.65

0.99

0.85

0.40

0.61

P10

8 step block

3.16

0.38

200 001

17 3 U/B

(f)

2.72

1.15

1.78

1.50

0.63

0.98

3.627 0.646 E14 Based on plate stresses

118

Table C2/8 Transverse load-carrying fillet weld Reference Joint dimensions, mm

Material Stress yield ratio stress, N/mm2

Constant amplitude results ————————————— Ref m F.S. at no. C 2 × 106 N/mm2

Variable amplitude loading spectrum ——————————————————— Ref Form Clip p Block no. ratio length, cycles (a) actual (b) per peak

Variable amplitude results —————————————————————————————— Σn No. of S-N RG N speci- curve mens used in ————————–— ————————–— analysis Max. Min. Mean Max. Min. Mean

Haibach 1968b

265

(t)

3.520 0.154 E14

90

P15

Stationary Gaussian

3.42

0.34

500 009

9 (235)

(t)

2.32

0.63

1.13

1.38

0.38

0.67

R = –1 (u)

3.609 0.433 E14

±54

P14

Stationary Gaussian

6.00

0.12

500 009

9 (352)

(u)

2.17

1.02

1.56

1.88

0.88

1.35

R=0

3.576 0.213 E14

92

P16

Stationary Gaussian

2.33

0.56

500 009

18

(v)

2.34

0.62

1.29

1.33

0.35

0.73

P15

Stationary Gaussian

3.42

0.34

500 009

18

(v)

2.37

0.66

1.41

1.48

0.41

0.88

P16

Stationary Gaussian

2.33

0.56

500 009

27

(w)

3.31

0.63

1.06

1.97

0.37

0.63

P15

Stationary Gaussian

3.42

0.34

500 009

27

(w)

1.41

0.46

0.95

0.95

0.31

0.64

P14

Stationary Gaussian

6.00

0.12

500 009

18 1 U/B

(w)

2.89

0.49

0.97

2.65

0.45

0.89

B = 50 T = 10 l = 10 b = 50 h = 60 w=8

383

R=0

(v)

R = –1 (w)

3.676 0.657 E14

Note: All the above results relate to stress in plate, but failures were in weld. σweld = 1.16 × σplate

±55

Table C2/9 Longitudinal load-carrying fillet weld Reference Joint dimensions, mm

Plumier 1977

Material Stress yield ratio stress, N/mm2

B = 100 393 T = 10 b = 110 t = 10 w = 5.7

363

Constant amplitude results ————————————— Ref m F.S. at no. C 2 × 106 N/mm2

Variable amplitude loading spectrum ——————————————————— Ref Form Clip p Block no. ratio length, cycles (a) actual (b) per peak

Variable amplitude results —————————————————————————————— Σn No. of S-N RG N speci- curve mens used in ————————–— ————————–— analysis Max. Min. Mean Max. Min. Mean

P18

8 step block Stationary Gaussian

4.66

0.21

500 009

16

(x) Gmean

3.44 1.30 1.99 0.74

2.14 1.23

1.31 0.49 0.88 0.33

0.81 0.55

P17

8 step block Stationary Gaussian

2.33

0.56

500 009

14

(x) Gmean

2.44 1.70 1.33 0.91

2.04 1.15

0.97 0.68 0.57 0.39

0.79 0.50

P19

8 step block Stationary Gaussian

1.76

0.78

500 009

6

(x) Gmean

20.26 4.62 13.11 2.99

12.29 7.91

11.67 2.66 7.82 1.78

7.08 4.72

R = 0.5

P17

8 step block Stationary Gaussian

2.33

0.56

500 009

10

(x) Gmean

3.91 1.37 2.24 0.78

1.95 1.09

1.56 0.55 0.97 0.34

0.77 0.47

R = 0.1

P17

8 step block Stationary Gaussian

2.33

0.56

500 009

14 1 U/B

(x) Gmean

3.49 1.57 1.93 0.90

2.40 1.36

1.39 0.63 0.83 0.39

0.95 0.59

P19

8 step block Stationary Gaussian

1.76

0.78

500 009

6 2 U/B

(x) Gmean

24.93 6.98 15.44 4.38

15.95 9.93

14.35 4.02 9.21 2.62

9.18 5.92

P17

8 step block Stationary Gaussian

2.33

0.56

500 009

9 3 U/B

(x) Gmean

1.80 1.19 0.99 0.67

1.46 0.81

0.72 0.48 0.43 0.29

0.59 0.35

R = 0.1 (x)

R = 0.5

2.836 0.154 E12

53

Table C2/10 Longitudinal non-load-carrying fillet weld on plate edge Reference Joint dimensions, mm

Material Stress yield ratio stress, N/mm2

Whitman T = 7 and Aldare l = 75 1960

Approx. 355

R∞0

Constant amplitude results ————————————— Ref m F.S. at no. C 2 × 106 N/mm2

Variable amplitude loading spectrum ———————————————————––– Ref Form Clip p Block no. ratio length, cycles (a) actual (b) per peak

Variable amplitude results —————————————————————————————— Σn No. of S-N RG N speci- curve mens used in ————————–— ————————–— analysis Max. Min. Mean Max. Min. Mean

(y)

P8

15 2 U/B

3.086 0.2197 E12

43

Block loading quadratic spectrum

3.49

0.2

10 001

(y) Gmean

4.98 1.29

2.60 0.66

3.72 0.95

5.27 1.30

2.75 0.67

3.94 0.96

Table C2/15 Beams with welded cover plates Reference Joint dimensions, mm

Cover plate b = 125 t = 10

Material Stress yield ratio stress, N/mm2

R=0 0.33

Constant amplitude results ————————————— Ref m F.S. at no. C 2 × 106 N/mm2

β

2.943 0.3836 E12

62

Variable amplitude loading spectrum ———————————————————––– Ref Form Clip p Block no. ratio length, cycles (a) actual (b) per peak 10 stop block programme

2.39

0.5

400 19

Variable amplitude results —————————————————————————————— Σn No. of S-N RG N speci- curve mens used in ————————–— ————————–— analysis Max. Min. Mean Max. Min. Mean

5 (144)

β Gmean

1.03 0.90

0.84 0.74

0.94 0.82

1.41 1.27

1.15 1.03

1.29 1.18

Table C2/16 Beams with welded stiffeners (transverse fillet welds) Reference Joint dimensions, mm

Woodley 1965

Maddox 1967

Material Stress yield ratio stress, N/mm2

Constant amplitude results ————————————— Ref m F.S. at no. C 2 × 106 N/mm2

Variable amplitude loading spectrum ——————————————————— Ref Form Clip p Block no. ratio length, cycles (a) actual (b) per peak

Variable amplitude results —————————————————————————————— Σn No. of S-N RG N speci- curve mens used in ————————–— ————————–— analysis Max. Min. Mean Max. Min. Mean

(z)

P8

6 step block programme (quadratic)

3.49

0.2

10 000 10 000

5

(z) Fmean

Flange 381 B = 127 T = 13

R=0

3.118 0.3774 E13

103

Stiffener t = 13 w = 18

293

R=0

P8

6 step block programme (quadratic)

3.49

0.2

10 000 10 000

1 (197)

(z) Fmean

Flange 381 B = 127 T = 13 Stiffener t = 13 w=8

R=0

P8

6 step block programme (quadratic)

3.49

0.2

10 000 10 000

11

(z) Fmean

1.77 2.15

0.81 0.97

1.42 1.71

1.91 2.18

0.87 0.99

2.58 3.20

3.13 3.80

1.52 1.88

2.57 3.07

1.52 1.73

2.78 3.24

3.37 3.85

1.64 1.90

2.77 3.11

Appendix D Probability density curves

Probability curves The objective of this section is to present, in a very simplified manner, the development of probability density curves to represent the frequency of occurrence of, for example, stress ranges in a load spectrum, as discussed in Chapter 4. One way in which the frequency of occurrence of an event can be represented is by means of a histogram, in which the height of each bar represents the percentage of occurrences within the interval represented by the width of the bar. Figure D1(a) shows a typical histogram giving the percentage of occurrences falling within each interval, ∆x, of a variable x. Alternatively, frequency density, namely the frequency of occurrence (f) divided by the corresponding interval (∆x), i.e., y = f/∆x, can be plotted as ordinate rather than f itself. In many practical applications the total number of events is quite large and the interval ∆x can be made relatively small, so that the histogram can be approximated by a smooth curve. When the ordinate represents frequency density and the original histogram is scaled to have unit area this curve is called a probability density curve. Thus, the area under the curve (y/dx) between any two values x = a and x = b is the percentage of occurrences between those two values of x, and the total area under the curve from x = –∞ to x = +∞ is unity, since it represents 100% of the occurrences. Since the dimensions of y are 1/x, the area is dimensionless. The probability density curve for the data in Fig. D1(a) is shown in Fig. D1(b). In some instances it can be useful to present probability density curves in non-dimensional form by plotting the deviation of x from the mean (x – xm) rather than the absolute value of x, and expressing that deviation in terms of the standard deviation, σ. In other words the horizontal axis is plotted in terms of x′ = (x – xm)/σ, and since the area under the curve must be kept dimensionless the probability density, y, is multiplied by σ to obtain the non-dimensional probability density (y′ = yσ) (see Fig. D1(c)). 413

Cumulative damage of welded joints ∆x

10

5 0

Variable x (units of x) (a)

0.10

0.05

Non-dimensional probability density, y ′ = yσ

Probability density, y = f/∆x (units of 1/x)

Frequency of occurrence, f (%)

414

0

Variable x (units of x) (b)

0.4 0.3 0.2 0.1 0

–2

–1 0 1 2 Deviation from mean = x ′ Standard deviation (c)

D1 Derivation of probability density curves.

In the case of the Gaussian or normal distribution the equation of the nondimensional probability density curve is y′ = {1/√(2π)} exp{–1/2(x′)2} where the factor 1/√(2π) is required to make the total area under the curve equal to unity and for the Rayleigh distribution (see Fig. D.2) it is y′ = x′ exp{–1/2(x′)2}.

Gaussian narrow-band distribution For Gaussian narrow-band loading at frequency f0 the expected number of crossings, cs, of the stress level S with positive slope in unit time is cs = f0 exp(–S2/2σ2)

Non-dimensional probability density, y ′

Probability density curves

0

1

2 3 x ′ (non-dimensional)

4

415

5

D2 The Rayleigh distribution: y ′ = x ′ exp{–1/2(x′)2}.

where σ is the r.m.s. stress. Hence the expected number of peaks in the interval between S and (S + dS) will be dcs = f0 (S/σ 2) exp(–S2/2σ 2) dS Hence the probability density function of a peak of magnitude S occurring is p(S) = (S/σ 2) exp(–S2/2σ 2) which is the Rayleigh probability density function.

Appendix E Transformation of units in the Paris fatigue crack propagation equation

As discussed in Chapter 2, the rate of fatigue crack propagation is usually expressed in terms of the range of stress intensity by the equation da/dN = C(∆K)m

[A3.1]

Unfortunately, there is at the moment no agreed set of units for the definition of da/dN and ∆K so that, although the value of m is independent of the choice of units, the value of C is not. One cannot help feeling that if stresses are to be expressed in terms of N/mm2 then the logical units to use are N/mm3/2 and mm/cycle for ∆K and da/dN respectively. Owing to the presence of the index m in the basic equation, the calculation of the value of C in any preferred set of units when results are expressed in a different set of units is not straightforward. If we express the same set of results in terms of two different sets of units denoted by the subscripts 1 and 2, then the basic equation can be rewritten as (da/dN)1 = C1(∆K1)m

[A3.2]

(da/dN)2 = C2(∆K2)m

[A3.3]

and Now, if it is only the units of da/dN which differ, ∆K1 and ∆K2 being the same, we can write, for a particular value of ∆K (da /dN )1 (da /dN ) 2 = C2 C1

[A3.4]

so that, if (da/dN)1 = b(da/dN)2 we obtain C1 = bC2 416

[A3.5]

Transformation of units in the Paris fatigue crack

417

Similarly, if ∆K1 = d∆K2 and the units of da/dN are the same we can write, for a particular value of da/dN C1(∆K1)m = C2(∆K2)m

[A3.6]

C1 = C2/d m

[A3.7]

so that

Hence, combining eqns [A3.5] and [A3.7] we get, for the situation in which both da/dN and ∆K are expressed in different units C1 = C2 b/d m

[A3.8]

This relation enables the value of C( = C1) corresponding to the system of units 1 to be derived from the value of C(= C2) relating to some other system of units. The relevant values of b and d are shown in the following table. Units 1

Units 2

Factor

mm/cycle

m/cycle in/cycle MN/m3/2 ksi √ in kg/mm3/2

b b d d d

N/mm3/2

= = = = =

1000 25.4 31.62 34.75 9.82

References and further reading

Note: This list includes the sources of constant amplitude test data referred to in Appendix A (marked by an asterisk). It also includes the work on fracture mechanics referred to in Chapter 10. *Abtahi A, Albrecht P and Irwin G R: ‘Fatigue of a periodically overloaded stiffener detail’, Proceedings ASCE J. Struct. Div. Vol. 102, No. 11, 1976. Agerskov H and Ibso J B: ‘Fatigue of plate elements with welded transverse attachments subjected to stochastic loading’, Proceedings, Conference ‘Fatigue Under Spectrum Loading and in Corrosive Environments’, Lyngby, pp. 41–60, August 1993a. Agerskov H and Ibso J B: ‘Fatigue life of repair welded tubular joints in offshore structures’, Proceedings, ISOPE 93, Singapore, June 1993b. *Albrecht P: ‘Fatigue behaviour of weathered steel bridge components’, FHWA/MD-81/ 02. Albrecht P and Yamada K: ‘Simulation of service fatigue loads for short span highway bridges’, ASTM STP 671, 1979. *Anon: IIW Commission XIII Joint test programme, 1962a (documents 253, 272, 279, 289). *Anon: IIW Commission XV Survey, 1962b. Anon: ‘Fatigue tests carried out abroad on butt welded specimens containing defects’, Welding Institute Report D2/11/63, 1963. *Anon: IIW Commission XII-Test programme (XIII-385-65), 1965. *Anon: Stranski Institute of Metals, 1969. *Anon: ‘Fatigue investigation of higher strength structural steels in notched and in welded condition’, ECSC Report EUR 5357, 1975a. *Anon: ‘The properties of flame cut edges’, ECSC Report 6210-45/6/601, October 1975b. *Anon: ‘Fatigue investigation of typical welded joints in steel FeE460 as compared to Fe E355’, ECSC Report F4.1/73, April 1977. Austin J A and Dover W D: ‘Variable amplitude corrosion fatigue of tubular welded joints under axial loading’, Fatigue crack growth in offshore structures Published EMAS, pp. 17–76, 1995. *Backman T: ‘Fatigue testing of a weldable fine grain structural steel with UTS of approxiumately 50 kg/mm2’, Svetsen 20 3 1961 45–53. *Bardell G R and Kulak G L: ‘Fatigue behaviour of steel beams with welded details’, University of Alberta Report 72 1978. *Baron H G and Brine F E: ‘Fatigue properties of welded joints in a carbon manganese steel plate over the range 104–107 cycles’, RARDE Memo (M), 21/63, May 1963.

418

References and further reading

419

Barsom J M: ‘Fatigue behaviour of pressure vessel steels’, WRC Bull. 194, May 1974. Bastenaire F and Lieurade H P: ‘Programmed fatigue – influence of the sequence of loads on lives’, Circ. d’Inf. Tech. Centre de Documentation Sidergique, Vol. 35 No. 12, 1978a. Bastenaire F and Lieurade K P: IRJID Report RE531, May 1978b. *Baxter C F G and Booth G S: ‘Improved fatigue strength by plasma dressing’, TWI Research Bulletin September 1979. Beales C: ‘Measurement and interpretation of dynamic loads in bridges Phase 3: Fatigue behaviour of orthotropic steel decks of road bridges’, Report EUR 12792 EN, 1990. *Becker G and Rieger R: ‘Fatigue strength of butt welds made with different electrodes’, Schweisstechnik (Berlin) Vol. 3, Dec. 1953, pp. 363–4, English abstract in Weld. Res. Suppl. Vol. 19, No. 6, 1954, p. 284s. Bell R and Vosikovsky O: ‘Fatigue life prediction of welded joints for offshore structures under variable amplitude loading, Offshore Mechanics and Arctic Engineering’, Proceedings, 10th International Conference, Stavanger, Norway, June 1991. *Benoit D, Lieurade H P and Truchon M: ‘Fatigue behaviour under programmed loading of welded cruciform and butt joints in steel E355’, ECSC Report 6210 – KD/3/302 November 1977. Berge S: ‘Residual stress and stress interaction in fatigue testing of welded joints’, University of Trondheim Report SK/R55, 1981. *Berge S: ‘The effect of plate thickness in fatigue of welded joints’, ECSC Offshore WG2/138, October 1982. Berge S: ‘The plate thickness in fatigue – predictions and results’, OMAE 1989, The Hague. Bernard, P J, Lindley, T C and Richards, C E: ‘Mechanisms of overload retardation during fatigue crack propagation’, ASTM STP 595, 1976, pp 78–97. Bignonnet A, Lieurade H P and Vallet C: ‘Fatigue of welded joints in high strength steel for offshore structures’, Report EUR 13398, 1992. Blom, A F: ‘Overload retardation during fatigue crack propagation in steels of different strengths’, Scand. J. Metal, Vol. 18, 1989a, pp. 197–202. Blom A F: ‘Fatigue strength of welded joints subjected to spectrum loading’, Scandinavian Journal of Metallurgy, Vol. 18, No. 4 1989b, pp. 181–184. Blom A F: ‘Spectrum fatigue behaviour of welded joints’, International Journal of Fatigue, Vol. 17, No. 7, pp. 485–491, October 1995. Bogren J and Martinez L L: ‘Spectrum fatigue testing and residual stress measurements on non-load-carrying fillet welded test specimens’, Proceedings, Conference Fatigue Under Spectrum Loading and in Corrosive Environments, Lyngby, pp. 77–90, August 1993. Booth G S: ‘Constant amplitude fatigue tests performed on welded steel joints in air’, Proc. Evi. Offshore Steels Research Seminar, Cambridge, 1978. Booth G S: ‘Corrosion fatigue of welded steel joints under narrow band random loading’, Welding Institute Report 3614/2/82 (Part of UKOSRP 1). Bouchard R, Vosikovsky O and Rivard A: ‘Fatigue life of welded plate T joints under variable amplitude loading’, International Journal of Fatigue, Vol. 13, No. 1, January 1991, pp. 7–15. *Braithwaite A B M: ‘Fatigue tests on large welded plate girders’, BWRA Report D7/ 36A/1965. Bronsted P, Slend T and Solin J P: ‘Fatigue testing using offshore load spectrum’, Steel in Marine Structures, Elsevier, 1987.

420

Cumulative damage of welded joints

Bucak O and Mang F: ‘Investigation into the fatigue behaviour of hollow sections (CHS) joints subjected to spectrum loading’, Proceedings First ISOPE Conference, Edinburgh, p. 434, August 1991. Carlson, R L, Kardomateas, G A and Bates, P R: ‘The effect of overloads in fatigue crack growth’, Int. J. Fat., Vol. 13, No. 6, 1991, pp. 453–460. *Castagna M and Pezzato G: ‘Fatigue strength of microalloyed hot rolled high strength steels’, ECSC Final Report EUR 8063, 1982. Chand, S: ‘Crack closure and propagation studies to determine the effects of simple load interaction’, J. Eng. Mat. Tech., July 1992, p. 229. Chand, S and Garg, S B L: ‘Variation of effective stress range ratio under simple variable amplitude loading’, Proc. ICF6, 1984, p. 1711. Chen, D and Nisitani, H: ‘Analysis of the delaying effects of overloads on fatigue crack propagation’, Eng. Frac. Mech., Vol. 39, No. 2, 1991, pp. 287–298. Christensen, R H: Metal fatigue, McGraw Hill, New York, 1959. Cottrell C L M: Private communication. Dahle T: ‘Spectrum fatigue of welded specimens in relation to the linear damage rule’, Proceedings, Conference Fatigue Under Spectrum Loading and in Corrosive Environments, Lyngby, pp. 133–147, August 1993. Dahle T: ‘Long life spectrum fatigue tests of welded joints’, Fatigue, Vol. 16, August 1994, pp. 392–6. Dahle T and Larsson B: Intl Symp. On Fatigue Design, Helsinki 1992, pp. 99–109. Damri, D and Knott, J F: ‘Transient retardations in fatigue crack growth following a single peak overload’, Fat. Frac. Eng. Mat. Struct., Vol. 14, No. 7, 1991, pp. 709–719. de Back J and Vaessen G: ‘Fatigue and corrosion fatigue behaviour of offshore steel structures’, ECSC Final Report 7210 – KB/6/602, 1981. De Back J, van Delft D R V and Noordhock C: ‘The effect of plate thickness on the fatigue life of welded tubular joints and flat specimens’, OMAE 1989, The Hague. *Denny A K and Jubb J E M: ‘Fatigue crack propagation in submerged-arc cruciform welded joints’, Welding Research International 7 1 1977, 57–75. Dhar, S: ‘Influence of multiple overload on fatigue crack retardation in high strength low alloy structural steel’, Proc. ICF7, Houston, 1989, p. 1395. Dijkstra O D and Hartog J: ‘Dutch part of the large scale tubular joint fatigue test programme’, Proc. Eur. Offshore Steels Research Seminar, Cambridge, 1978. *Disselmeyer H: Schweissen und Schneiden 4 2 1973. Donato L F et al.: ‘Sugli acciai tipo TI e relative giunti saldati’, Centro Studi Construzioni Metalliche of University of Pisa, Publication no. 72, 1972. Dover W D, Austin J A and Kam J C P: ‘VA corrosion fatigue of tubular welded T joints under axial loading’, UCL Report, May 1992. Druce, S G, Beevers, C J and Walker, E F: ‘Fatigue crack growth retardation following load reductions in a plain C.Mn steel’, Eng. Frac. Mech., Vol. 11, 1979, pp. 385–395. Dubois H and Hirt M A: ‘Effectiveness of improvement methods for welded connections, subjected to variable amplitude loading’, Proceedings, VTT Symposium 157, Vol. 3, pp. 21–32, Helsinki, 1995. Elber, W: ‘The significance of fatigue crack closure’, ASTM STP 486, 1970a, pp. 230– 242. Elber, W: ‘Fatigue crack closure under cyclic tension’, Eng. Frac. Mech., Vol. 2, 1970b, pp. 37–47. Ferguson H B: ‘Strength of welded T joints for ships’ bulkhead plates’, Trans. INA Pap. 11, 1942.

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Index

A Class 22, 27 AASHTO bridge rules 244–5 acceleration of crack growth 287, 288, 290–2 acceleration forces 1 alternating loading 6, 7, 9, 70–4 and residual stresses 80, 81, 84–6, 92 prior overloading 95, 96, 97, 98 aluminium alloys 17 anchor ring 75–6 apparent thickness 68–9 area under exceedence curve 326–30 area rule 132–4, 147, 190–2 concave upwards spectra 148–50, 191 design for variable amplitude loading 273–4, 275, 276 influence of block length and clipping ratio combined 180–2 influence of block length and spectrum shape combined 183, 184, 187–8 influence of spectrum shape and clipping ratio combined 177–80 modified Miner’s rule 274–5, 276, 277 narrow band loading 219–21, 242 Rayleigh and Laplace loading spectra 155–71 service loading tests 236 tests under two and three level loading 136–46 variable amplitude loading test results 360–412 Weibull spectra 172–4 wide band loading 225, 226, 228, 243 attachments length 49–50, 51, 64, 68 longitudinal gussets on a plate edge 63–5 on surface or edge of a stressed member 31–3 thickness 50–5 see also edge attachments austenitic steels 15 automatic welds continuous fillet welds 28, 38, 308–9

434

influence of weld shape on fatigue strength 41–2, 43, 44, 45 transverse butt welds 28, 397 Avonmouth Bridge 234–5 axial loading 6–7, 8, 65 narrow band loading of transverse load– carrying fillet welds 207–14 axle load design spectrum 231–2, 233 B class 22, 23, 24, 25, 26, 27, 28 backing bar 22, 42–3 backing strips 28, 31 beams in bending butt welded splices 379 cover plates 380–2 plain welded 378 transverse stiffeners 412 with welded cover plates 189, 190, 380–2, 411 ends not welded 22, 321 ends welded 22, 321 wide plates, ends not welded 22 with welded flange cover plates 62–3 bending 6–7, 8, 19 narrow band tests on toe ground joints 207–10, 214–15 plate thickness 65, 66 residual stresses and 96–7 of transverse plate for K butt weld 46 bent S-N curves 244–6, 258–61 design for variable amplitude loading 263–5 testing for different types of stress 267–73 blanketing effect 197–200 block length (sequence length) 122, 147–92 comparing loading types 196–207 fatigue tests under concave upwards spectra 148–50 fatigue tests under Rayleigh and Laplace loading spectra 150–71

Index influence of block length and clipping ratio combined 180–2 influence of block length and spectrum shape combined 183–9, 190 loading spectra 326–30 tests under Weibull stress spectra 171–7 variable amplitude test results 358, 360–412 wide band loading 226, 227, 228 block programme loading 118–19, 196–207, 208, 242 early test results 122–4 spectra 330 variable amplitude test results 356–7, 396–412 block testing 118 low stresses 247–51, 252 bottom limit stresses same in main and subsidiary cycles 138, 143, 144 small stress range cycles 250–1 box columns 78–9 bridges 102, 151, 152 tests under service loading spectra 228, 229–36 Britain Bridge Design Standard 231–2, 233 orthotropic deck bridges 230 railway bridges 102 British design rules S-N curve 244, 245 and predicting fatigue life 258–61 variable amplitude loading 264–5 British offshore test programme 236–9, 240 brittle fracture 4 BROAD (Nordic) spectrum 238, 239–41, 351–2, 354 butt welds on backing bar 22 constant amplitude tests and residual stresses 85–6 cruciform joints see cruciform joints K butt welds see K butt welds manual continuous 308–9 parallel to direction of applied stress 28–9, 396 plates with gussets welded to plate edge 319–20 residual stresses around 77–8 submerged arc 22, 24 T butt welds 33–5 transverse see transverse butt welds C (material constant) 12, 15–17, 21, 23, 358, 360–412 constant amplitude test data 305, 308–24 transformation of units in Paris fatigue crack propagation equation 416–17

435

C Class 22, 23, 24, 25, 26, 27, 28, 30 C/12/20 spectrum (spectrum S1) 236, 346–50, 351, 352 cathodic protection 236–9, 240 Class G design S-N curve 244, 258–9, 261–2, 279 clipping ratio 152, 154, 164–6, 191–2, 326–30, 335 influence of block length and clipping ratio combined 180–2 influence of spectrum shape and clipping ratio combined 177–80 variable amplitude test results 358, 360–412 Weibull stress spectra 172–7 closed stiffeners 229, 230 Cologne-Muelheim Bridge 230 comparative tests on stress relieved joints 251–8 complex cycles 125–7, 150 see also three level loading; two level loading compressive loading and residual stresses 83, 92–5 stress relieved joints 90–1 compressive residual stresses 76–7, 79 concave upwards spectra 148–50, 190–1, 265 bent S-N curve 246, 247–61 constant amplitude database 8, 19–74 influence of mean stress 70–4 influence of plate thickness 65–9 method of analysis and joint design classification 21–65 constant amplitude fatigue behaviour 10–18 constant amplitude life under maximum stress (Nc) 127, 247, 265, 273–4 area rule 132–4 constant amplitude loading 5–6, 7–8, 127, 358 intermittent overloading 168, 169 Markov matrix 332, 333 tests and residual stresses 83–95 prior overloading 95–9 statistical analysis of data 305–24 transverse load carrying fillet welds 210, 211, 212, 214, 215 constant minimum/maximum stress 119–21 continuous fillet welded joints 38, 47 continuous longitudinal welds 28–9, 38 statistical analysis of constant amplitude test data 308–9 continuous toe cracks 52–5 convex loading spectra see Rayleigh loading spectra convex upwards spectra 261, 265 cope holes 37, 310 correlation coefficient 305–6, 308–24

436

Index

counting methods 102–8 level-crossing 103, 107–8 rainflow 103–6 reservoir 103, 106–7 statistical interpretation of count data 109–11 cover plates beams with 22, 189, 190, 321, 380–2 beams with welded flange cover plates 62–3 failure 324 longitudinal load-carrying fillet welds 61–2 crack closure 286, 287 stress interaction effects 302–3, 304 crack filling closure 287 crack initiation 10–11, 214, 284 fillet joints 48–9 crack length 278–81, 282 crack mouth at free surface, correction factor for (Ms) 11–12 crack propagation 3, 10, 214, 284 design for variable amplitude loading 278–83 experimental results for stress interaction evidence 287–302 sequence effects 284–7 crack propagation rate 11–18 crack tip blunting 286 crack tip plasticity, correction factor for (Mp) 11–12 cracking 1 characteristics of fatigue cracking 3–5 critical stress intensity 13 cruciform joints 43–6, 313 transverse load-carrying fillet welds 55–61 cumulative cycle ratio 112 cumulative frequency distribution 109–10 cycle counting methods see counting methods cycles to failure (N) see S-N curve cyclic loading (fatigue loading) 3 sources 1–2 D Class 22, 23, 24, 25, 26, 29, 30 deck to longitudinal box-web joint 231, 234–6 delayed retardation 286–7, 288–91 Department of Energy Guidance Notes 67 design S-N curves 10, 21, 113–17, 263–5 design standards 2, 8 design for variable amplitude loading 263–83 area rule 273–4, 275, 276 fracture mechanics approach 275–83 possible modifications to Miner’s rule 274–5 testing for different types of stress 265–73

deterministic stress 109 discontinuous fillet welded joints 47–62 ductile fracture 4, 5 E Class 22, 23, 24, 25, 26, 30, 36, 37 eccentricity 29 edge attachments 63–5 block length and 160–8 comparing loading types for plate with fillet welded longitudinal edge attachments 196–207, 208 fillet welds narrow band testing 220, 221 wide band loading 222–6, 227, 228 predicted life using fracture mechanics 275–83 edge distance 28, 31, 32, 35, 36 egg-box-type longitudinal load-carrying fillet welds 61–2 elastic instability 78–9 environment 16–17 environmental loading 2 European design S-N curve 244, 259–61 variable amplitude loading 264–5 exceedence diagrams 148, 149 block testing of low stresses 247, 248 bridges 234–5, 236 Rayleigh and Laplace spectra 151 wide band loading 223 excursions fatigue tests using stress sequences with excursions of two sizes 134, 135, 136 stress sequences with excursions of two sizes 127–9, 131–2 stress sequences with a single size of excursion 125–7, 130–1 extrapolated linear S-N curve 244–6, 258–61, 261–2 F Class 22, 23, 24, 25, 26, 31, 32, 33, 34, 36, 37 F2 Class 22, 23, 24, 25, 26, 31, 32, 33, 34, 35 face shovel, working in a quarry 101 fatigue, defining 1 fatigue cracking 1 characteristics of 3–5 fatigue life 17–18 comparing loading types 196–207 predicting and cycles of small stress range 258–61 upper and lower bounds 126–7 fatigue limit 244, 263, 264 fatigue loading conditions 78–9 fatigue strength 2 constant amplitude test data 308–24

Index factors influencing for transverse butt welds 39–40 influence of mean stress 70–4 influence of plate thickness 65–9 S-N curve and 8–10 tests under narrow band testing 210–15 fatigue testing 5–8 ferrite pearlite steels 15 fillet welds 39, 44, 47–62 constant amplitude tests and residual stresses 84–5 continuous 38, 47, 308–9 automatic fillet welds 28, 38, 308–9 gussets welded to plate edge 319–20 intermitted longitudinal fillet welds 22, 38, 310 lap joints and narrow band testing 216–19, 220, 221 load-carrying 33–5, 47–9 longitudinal load-carrying welds see longitudinal load-carrying fillet welds longitudinal non-load-carrying welds see longitudinal non-load-carrying fillet welds mean stress 71–4 parallel to direction of applied stress 28–9 prior overloading 97 transverse load-carrying welds see transverse load-carrying fillet welds transverse non-load-carrying welds see transverse non-load-carrying fillet welds flame-cut edges 27 flange cover plates, beams with 62–3 fluctuating live loads 1 Forth road bridge 230, 232, 233 fracture mechanics 2 assessment of constant amplitude fatigue behaviour 10–18 design stresses for weld throat failure in fillet welds 56–61 design for variable amplitude loading 275–83 effect of stress interaction 284–304 Miner’s rule 112–13 suitable applications 283 fracture surfaces 3–4, 5 Frank’s K calibration 56 free corrosion 236–9, 240 frequency density 413 full penetration welds butt welds 28, 38 cruciform joints 44–6 G Class 22, 23, 24, 25, 26, 32, 33, 35, 36, 37, 63

437

Gaussian loading spectra 110–11, 193–4, 195, 414 narrow band loading 414–15 used by Haibech and Overbeeke 343–4, 345, 346, 347 generalised stress parameter 18, 56–7 Germany 230 girders, details in welded 36–7 grabbing crane slewing 101 gussets longitudinal on a plate edge 63–5 transverse non-load-carrying fillet welds 50–5 welded to plate edge 22, 319–20 Haibach, E., et al 215–16 Gaussian loading spectrum 343–4, 345, 346, 347 half tensile loading 6, 7, 9, 70–4 and residual stresses 80, 81 heat affected zone 39, 40 high-low overload 289, 295, 300–1 high strength steels 15 high stresses 4–5 highway bridges 151, 152 Hinkley Point Nuclear Power Station 208 histograms 413, 414 intermittent immersion 236–9, 240 intermittent longitudinal fillet welds 22, 38, 310 intermittent overloading 168, 169 intermittent web-to-flange welds 37 inverted spectra see ‘upside down’ stress cycles irregularity 57–9, 119–21 narrow band and wide band loading 193–4, 195 Nordic test programme 239, 241 joint design classification 21–65 joint dimensions 357, 360–412 K butt welds 22 British offshore testing programme 239, 240 with load causing bending of the transverse plate 46 partial penetration 385–6, 407 toe ground in bending 393 transverse K butt welds 67, 69, 383–4 K calibration, Frank’s 56 Kurpfalz Bridge 230 L/T ratio 50–5 lap joints 22, 24–5, 55–61, 376 narrow band loading 216–19, 220, 221

438

Index

Laplace loading spectra 150–71, 184, 191, 335–6 design for variable amplitude loading 265, 270–3 LBF spectrum 353–5 level-crossing counting 103, 107–8 levels, and cycle counting 103 linear cumulative damage rule see Miner’s rule linear loading spectra see Laplace loading spectra live loads, fluctuating 1 load-carrying fillet welds 33–5, 47–9 longitudinal see longitudinal load-carrying fillet welds transverse see transverse load-carrying fillet welds load-carrying T butt welds 33–5 loading spectra 101–2, 111–12, 325–55 C/12/20 spectrum 346–50, 351, 352 Gaussian spectra 193–4, 195, 343–4, 345, 346, 347 LBF spectrum 353–5 Markov transition matrix 331–4 Nordic test programme 351–3, 354 Rayleigh loading spectra see Rayleigh loading spectra two-parameter Weibull distribution 334–8 UKOSRP II wide band spectrum S4 350 WASH spectrum 344–5, 348, 349 see also spectrum form; spectrum shape locked-in stresses see residual stresses longitudinal butt welds 28–9, 396 longitudinal gussets, on a plate edge 63–5 longitudinal load-carrying fillet welds 47–9, 61–2, 73, 324 variable amplitude loading test results 409 weld end not on plate edge 22 weld end on plate edge 22 longitudinal non–load-carrying fillet welds 22, 47–50, 51, 72, 189, 190, 314–15 narrow band testing 216–19, 220, 221 offshore structures 239, 242 residual stresses 86–95 prior overloading 97, 98–9 variable amplitude test results in bending 389 block programme 400–2 plate edge 372–5, 410 random order 360–7 longitudinal residual stresses 77–8 low-high overload 289, 295, 300–1, 303 low stresses 4–5, 244–62 block testing 247–51, 252 comparative tests on stress relieved joints 251–8

predicting fatigue life 258–61 lower bound of fatigue life 126–7 m (material constant) 12, 15–17, 21, 23, 26, 358, 360–412 constant amplitude test data 305, 308–24 transformation of units in Paris fatigue crack propagation equation 416–17 main plate failure 324 manual metal arc (MMA) welding 219, 311–12 manual welds continuous butt welds 308–9 fillet welds 29, 38 continuous 38, 308–9 transverse butt welds 29, 398 influence of weld shape on fatigue strength 41–2, 43, 44, 45 Markov transition matrices 331–4 Gauss spectrum 343, 344, 345 martensitic steels 15 material constants see C; m maximum stress 6, 8–9 design for variable amplitude loading 266, 267, 268, 269, 272–3 maximum stress intensity (Kmax) 13–14 step change of constant Kmax 289, 295, 301 mean life 21, 22 mean life minus 2 standard deviations 21, 22, 23, 24, 25, 26 mean stress 6, 16–17, 70–4, 79, 81, 110 fillet welded joints 71–4 loading spectra 326–30 mean stress same in main and subsidiary cycles 138, 142, 143 step change down 289, 295, 301, 302 step change up 289, 295, 301, 302 tests under narrow band loading 210–14 transverse butt welds 70–1 variable and level-crossing counting 108 mechanical vibrations 2 Miner’s rule 102, 111–18, 128, 147, 190–2, 263, 284–5 comparing loading types 197–207 concave upwards spectra 148–50, 190–1 design for variable amplitude loading 265, 271–2 early test results 122–4 experimental S-N curve compared with Class G design curve 261–2 influence of block length and clipping ratio combined 180–2 influence of block length and spectrum shape combined 183, 184–7, 189, 190

Index influence of spectrum shape and clipping ratio combined 177–80 and nature of S-N curve 244–6 offshore testing 236–42 possible modifications to 274–5, 276, 277 Rayleigh and Laplace loading spectra 155–71, 191 service loading tests 235–6 stress sequences with a single size of excursion 130 tests under narrow band loading 210–14, 215, 216, 217, 219–21, 242–3 tests under two and three level loading 134, 137, 138, 144–6 tests under wide band loading 223–8, 242, 243 variable amplitude test results 359, 360–412 Weibull stress spectra 172–7, 178, 191–2 minimum stress 6, 8–9 minimum stress range/maximum stress range 326–30 variable amplitude test results 358, 360–412 modified Rayleigh loading spectra 164–8, 169, 171 ‘most damaging’ spectra 148–50 multiple peak overload 289, 295, 298–300, 303 multiple underloads 289, 301–2 narrow band loading 119–21, 164, 193–243, 274 comparing loading types 196–207, 242 Gaussian 414–15 Markov matrix 332, 333 offshore structures 238, 239–41 tests under 207–21 NARROW Nordic spectrum 351–2, 354 non-dimensional crack length (a) 278–81, 282 non-dimensional probability density curves 413–14 non-load-carrying fillet joints 47–9 longitudinal 47–50, 51 transverse 47–9, 50–5 non-propagating cracks 13 Nordic offshore test programme 236, 238, 239–42 loading spectra 351–3, 354 North Sea 229 wave loading 236–42, 346–50, 351, 352 North Sea Wave Action Standard History (WASH) 344–5, 348, 349 number of mean level crossings/number of peaks 332–4 offshore structures 228–9, 236–42 loading spectra 344–53, 354

439

offside rear wheels 232–3, 234 open stiffeners 229, 230 orthotropic decks 229–36, 392 Overbeeke, J.L. 216–19 Gaussian spectrum 343–4, 345, 346, 347 overload affected zone 292–8, 299 overload ratio 288–91 overloads 166–8, 169, 171 and crack propagation rate 285–7 high-low 289, 295, 300–1 low-high 289, 295, 300–1, 303 multiple peak 289, 295, 298–300, 303 prior overloading 95–9 single peak 288–98, 299, 303, 304 tests under Rayleigh and Laplace loading spectra 164–8, 169 pad plates 52 parallel S-N curves 116–17 parametric studies 283 Paris crack propagation equation 11–15 transformation of units 416–17 partial penetration welds 44, 322–3 K butt weld 385–6, 407 peak stress 126–9, 274 comparing loading types 197–207 tests under Rayleigh and Laplace loading spectra 152–5, 157, 159, 166–7, 170 single additional peak stress 164–8, 169, 171 Pierson-Moscowitz wave elevation spectrum 238, 239–41 plain steel 27 plain welded beams 189, 190, 378 plane strain 14–15, 286, 289–91, 295–8, 299 plane stress 14–15, 286, 296–8, 299 plastic collapse 78–9 plastic deformation crack closure 286, 287 residual stresses 76, 77, 303 plastic zone diameter due to overload 292–8, 299 plate thickness 40, 50 correction factor for K allowing for 11–12 influence on fatigue strength 65–9 L/T ratio 50–5 plate width 64, 65 PMMOD spectrum 351–2, 354 predicted life 210–12 area rule 132–4, 147, 219–20, 273 design for variable amplitude loading 265–9, 272–3 Miner’s rule 112–13, 147, 219–20 pressure changes 2 prior overloading 95–9

440

Index

probabilistic stress 109 probability density curves 109–11, 413–15 probability of load exceedence 151–4, 171–2 compared with a peak of amplitude S 194 Weibull distribution 335–8 proportional stresses 265 pulsating tension loading 6, 7, 8–9, 70–4 crack propagation 15–16 and residual stresses 80–1, 91 prior overloading 97, 98 pulsed MIG welding 219 railway bridges 102 rainflow counting 103–6 random order loading 119, 120, 162–4, 274 comparing loading types 196–207, 208 variable amplitude test results 356–7, 360–95 Rayleigh loading spectra 111, 261, 335–8 influence of spectrum shape and block length 150–71, 184, 185–6, 187–8, 190, 191 used for variable amplitude testing in the USA 338–42 Rayleigh probability density function 194, 415 reaction stresses 75 formation 76–7 see also residual stresses re-entrant corners 27 reinforcement angle 41, 42, 43 repeated block tests see block testing reservoir counting 103, 106–7 residual stresses 20, 71, 74, 75–99, 162, 191 approximate theoretical analysis 79–83 bridges 236 comparing loading types 197–200 comparison between static and fatigue conditions 78–9 formation of 75–8 prior overloading 95–9 stress interaction effects 291–2, 302–3, 304 crack growth retardation 286 tests under constant amplitude loading 83–95 tests under two and three level loading 134–46 retardation of crack propagation 285–7 experimental evidence 288–302 reverse order wide band spectra 222, 224, 225 Rheden Bridge 234–5, 236 root mean cube (rmc) of peaks stress 154–5, 158, 159, 164–6, 170, 326–30 root mean square (rms) of peaks stress 154–5, 156, 157, 164–6, 170, 209–10 root mean square (rms) of process 155, 210, 326–30

rotating bending 7–8 roughness induced closure 287 S-N curve 17–18, 19, 261–2 bent see bent S-N curves calculating design stresses using a bent curve 267–9 using a linear curve 265–7, 268 compressive loading 93, 94 design curves and experimental curves 10 and fatigue strength 8–10 joint design classification 21, 23–4, 25, 26 Miner’s rule 113–17 predicting fatigue life using 3 possible S-N curves 258–61 variable amplitude test results 359, 360–412 scatter bands 21, 22 Schilling, C.G., et al, Rayleigh distribution used by 338–42 Second World War 19 semi-elliptical cracks 52–5 sequence effects 284–7 see also stress interaction service loading 193–243 comparing loading types 196–207, 242 loading spectra 325–55 tests under service loading spectra 228–42, 243 servo-controlled fatigue testing machines 118 Severn Crossing 230 single peak overload 288–98, 299, 303, 304 single underload 289, 301–2 slag inclusions 399 small stress range cycles 244–62 block testing of low stresses 247–51, 252 comparative tests on stress relieved joints 251–8 design for variable amplitude loading 281–3 predicting fatigue life 258–61 SP4 spectrum 238, 242 spectrum form comparing loading types 200–7 variable amplitude test results 358, 360–412 spectrum shape 147–92, 334, 335 fatigue tests under concave upwards spectra 148–50 fatigue tests under Rayleigh and Laplace loading spectra 150–71 influence of block length and spectrum shape combined 183–9, 190 influence of spectrum shape and clipping ratio combined 177–80 Nordic offshore test programme 239–42

Index tests under Weibull stress spectra 171–7 splices, beams with 379 stalactitic loading 289, 295, 301 Markov matrix 332, 333 standard deviation 21, 23, 305, 308–24 mean life minus 2 standard deviations 21, 22, 23, 24, 25, 26 Standard Load Spectrum 231–2 static loading 3, 78–9 statistical methods 109–11 analysis of constant amplitude test data 305–24 steels 15–16, 17 step change of constant Kmax 289, 295, 301, 332, 333 step change down 289, 295, 300, 303 step change of Smean 289, 295, 301, 302 step change up 289, 295, 300 step change of Smean 289, 295, 301, 302 stiffener-to-web joints 22 stiffeners 229, 230 beams with transverse stiffeners 412 straight line extrapolated S-N curve 244–6, 258–62 strain 3 strain-gauge measurements 100, 101–2 as sawtooth pattern 103, 104 strain hardening 286 stress concentration 79 correction factor for K allowing for 11–12, 275–8 stress cycles 6–7 stress intensity factor 11–16, 263–4 stress interaction 100–1, 125–46, 284–304 experimental evidence 287–302 high-low overload 289, 295, 300–1 low-high overload 289, 295, 300–1, 303 multiple peak overload 289, 295, 298–300, 303 multiple underloads 289, 301–2 single peak overload 288–98, 299, 303, 304 single underload 289, 301–2 step change with constant Kmax 289, 295, 301 step change down 289, 295, 300, 303 step change of Smean down 289, 295, 301, 302 step change of Smean up 289, 295, 301, 302 step change up 289, 295, 300 stress ranges 6, 8–10, 79 block testing of low stresses 250–1, 252 distribution of stress ranges 109–11 narrow band testing 210–15

441

residual stresses 80–3, 86–90 compressive loading 92–5 small stress range cycles see small stress range cycles see also S-N curves stress ratio 6, 20, 70–4 comparing loading types 197–207 tests under two and three level loading 134–46 variable amplitude test results 357–8, 360–412 wide band loading 121–2 stress relieved joints 20, 83, 89, 90–1, 261, 262 comparative tests with small stress range cycles 251–8 compressive loading 93–5 tests under two and three level loading 134–46 structural steels 15–16, 17, 20 stud shear connectors 22, 24, 46–7, 318 submerged arc welds 42, 45, 311–12 butt welds 22, 24 superimposed sine wave loading 122 surfacing, bridges 233 T butt welds 33–5 T joints, tubular 67, 377, 390 tack welds 37 temperature distribution and residual stresses 86 fluctuations 2 traffic distribution on bridges 233, 234, 236 tensile loading 79–83, 86–92, 95, 96 tensile residual stresses 76–7, 79 thickness, and crack growth retardation 298 thickness correction factor 64–5, 67–8 three level loading 125–46 fatigue tests using stress sequences with excursions of two sizes 134 influence of stress ratio and residual stresses 134–44 theoretical analysis 130–4 threshold stress intensity factor 13 toe ground joints 207–10, 214–15, 393, 394–5 top limit stress 138, 143, 145 traffic flow 231–2 transition plates 31, 32 transport loading 228–36 transverse butt welds 29–31, 38–40, 69, 311–12 automatic 28, 397 cruciform joints 43–6 influence of weld shape on fatigue strength 40–2, 43, 44, 45

442

Index

K butt welds 67, 69, 383–4 manual 29, 398 mean stress 70–1 narrow band testing 216–19, 220 on a permanent backing bar 42–3 with slag inclusions 399 transverse fillet welds 68–9 transverse load-carrying fillet welds 47–9, 55–61, 73, 322–3 in bending 388 toe ground joints 207–10, 214–15, 394–5 cruciform 22 lap joints 22, 24–5, 55–61, 376 variable amplitude test results 390 narrow band loading tests 207–16, 217, 220, 221 variable amplitude loading test results 371, 408 transverse non-load-carrying fillet welds 22, 47–9, 50–5, 72, 74, 316–17 bending 65, 66, 67 variable amplitude test results 387 narrow band loading tests 216, 217 offshore structures 239–41 variable amplitude test results block programme 403–6 random order 368–70 transverse residual stresses 77–8 trough to cross-girder joints 231, 234–6, 392 trough splice joints 231, 234–6 tubular joints 68 tubular T joints 67, 377, 390 tubular Y joints 391 two level loading 125–46 fatigue tests using stress sequences with excursions of two sizes 134 influence of stress ratio and residual stresses 134–44 theoretical analysis 130–4 two-parameter Weibull distribution 334–8 two-step loading tests 118, 125 UKOSRP II wide band spectrum S4 350 underloads multiple 289, 301–2 single 289, 301–2 United States (USA) 229 AASHTO bridge rules 244–5 highway bridges 151, 152 Rayleigh distribution uses for variable amplitude testing 338–42

upper bound of fatigue life 126–7 ‘upside down’ stress cycles block testing of low stresses 249, 252, 254 wide band testing 222, 224, 225 VAMP spectrum 238, 242, 351 variable amplitude loading 99, 100–24, 125, 147–8 design for see design for variable amplitude loading history of variable amplitude testing 117–24 block programme loading 118–19 early test results 122–4 other types of loading 119–22 level-crossing counting 103, 107–8 Markov matrices 332, 333 Miner’s rule 111–17 rainflow counting 103–6 reservoir counting 103, 106–7 statistical interpretation of count data 109–11 test results 356–412 W Class 22, 23, 24, 25, 26, 35, 61 WASH spectrum 344–5, 348, 349 wave loading 228–9, 236–42 Weibull loading spectra 171–7, 178, 191–2, 334–8 weld defects 11 weld root 11, 31, 38, 40, 284 weld shape/profile 40 influence on fatigue strength 40–2, 43, 44, 45 weld size 50–5, 56 weld throat failure 34, 56–61, 407 weld toe failure 10–11, 31, 33, 38, 39, 40, 57, 284 wheel loads, lateral distribution of 232–3, 234 Wheeler model of crack growth retardation 286 wide band loading 121–2, 164, 193–243, 274 comparing loading types 196–207, 242 Markov matrix 332, 333 offshore structures 238, 239–41 tests under 221–8 Willenborg model of crack growth retardation 286–7, 298 Wye Bridge 232–3 Y joints, tubular 391 yield stress 81–2 variable amplitude test results 360–412