The design curves are then described in terms of arbitrary letters [4,5,12,13] or, increasingly, by the stress range at 2 x 10 6 cycles [6,9,11,14] . A clear advantage of the latter is that it quantifies fatigue strength.
2.3. Influence of residual stress
Welding introduces residual stresses, which modify the mean stress experienced by a welded joint under fatigue loading. Long range residual stresses will also be introduced when sub-assemblies are connected together, due to imperfect fit-up. It must normally be assumed that tensile residual stresses up to yield strength magnitude will be present in a welded structure. [1-3]
As a result, its fatigue life will be independent of mean stress and depend only on the applied stress range, even if this is compressive. [1,3,19]
Consequently, all the widely used fatigue design codes accept this principle and base design on full stress range.
Many studies have addressed the benefit of stress relief, with a view to modifying the design approach. However, complete stress relief can rarely be achieved. The low design stresses for many welded joints mean that even low tensile residual stresses are sufficient to produce the same effect as high ones. Additionally, long-range residual stresses are unlikely to be relieved. These factors have persuaded most Code writers to ignore the potential benefit of stress relief.
Variations in the type and magnitude of residual stresses in welded laboratory specimens is a serious source of scatter in fatigue data and of potentially misleading results. Ideally, laboratory specimens should contain high tensile residual stresses.  Recognising this, but also that it may be too costly to perform all fatigue tests on large scale components, the technique of cycling small specimens down from a high tensile stress of S max ≅ yield has met with some success.  However, the statistical scale effect is still absent. Thus, whenever possible design data are related to results from large-scale specimens. [19,21]
2.4. Scale effect
One of the most controversial changes to design rules has been the introduction of a thickness effect design penalty. It is well established that the fatigue strength of a welded joint which cracks from the weld toe decreases with increase in plate thickness.  This is understood to be a consequence of the influence of the stress concentration due to the weld detail, such that weld toe cracks are influenced by the stress concentration to greater depth in thick sections than in thin ones. For plates thicker than some reference t ref (typically 12-25mm), many rules now include the design stress penalty (t ref/t) 0.25. However, it has been shown that this scale effect can be relaxed by also allowing for the overall geometry of the joint, notably the attachment size. [23,24] The new Eurocode on aluminium  recognises this. The IIW recommendations  go further and vary the exponent from 0.1 to 0.3, depending on the weld type, weld profile and mode of loading. Finally, there is some support for a corresponding 'thinness effect' bonus for thin plates. 
2.5. Effect of type and strength of material
One of the most important consequences of the dominance of fatigue crack growth in the lives of welded joints is the fact that fatigue strength does not increase with increase in material strength, rate of crack growth being largely insensitive to material tensile strength. This contrasts with unwelded material. Consequently, design curves for welded joints are independent of material tensile strength.
The principle was established over 30 years ago [25,26] and yet the quest for methods of utilising high strength steels to advantage in fatigue loaded welded structures still receives considerable attention.  The most promising finding is that post-weld improvement techniques which introduce a significant fatigue crack initiation period, such as weld toe dressing, provide greater benefit for high strength than low strength steels. [28,29] However, the dependence on steel strength is still rather weak ( Fig.3), reflecting the severe stress concentration represented by the welded joint. 
Some design codes provide guidance on the influence of elevated temperature, but up to the creep regime this is usually accommodated simply by allowing for the elastic modulus of the material at the temperature of interest.
3. Developments in the application of fatigue rules
3.1. Hot spot stress approach
Fatigue design rules are based mainly upon data generated from tests on either beams or small-scale plate specimens incorporating the weld detail of interest, subjected to uni-directional loading. Thus, the fatigue test results can be expressed in terms of the nominal applied stress in the region of the test detail and the same stress is usually specified for use with design S-N curves.
An alternative approach, which is gaining ground and indeed is already established for tubular connections,  is based on structural stresses.  More particularly, for weld details in which fatigue failure is from the weld toe, the structural stress at the weld toe, called the hot spot stress, is used. This includes the stress concentration effect of the welded joint, but excludes that due to the weld itself. It should be possible to express the fatigue strength of a whole range of welded joints in terms of the hot spot stress using fewer curves than at present. Indeed, the recent revision to the UK rules for offshore structures  has introduced this approach with a single design curve. The hot spot stresses for relevant details are specified in terms of stress concentration factors, to be applied to nominal stresses, and used with the curve for transverse butt welds. However, this overlooks the problem, which also affects current design rules, of defining nominal stresses. These are particularly difficult to extract from finite element analysis output. It is easier to determine structural stresses and hence these offer a more general design approach. Fatigue design in the new European pressure vessel design rules  will be based on structural stresses.
To adopt the hot spot stress method more widely, hot spot stress S-N curves will be needed. It seems likely that, ultimately, it will be necessary to draw a distinction between at least three weld types, namely the toes of transverse butt and fillet welds and the ends of longitudinal fillet welds, rather than using a single design curve. 
3.2. Complex loading
There is also the need to improve current methods for considering welded joints under complex combined or multi-axial loading. At present, the main approach is to use an equivalent stress range (e.g. principal, von Mises) in conjunction with the S-N curve based on nominal normal stress range. Indeed, for failure from the toe of biaxially-loaded joints, the stress component parallel to the weld can be ignored as long as it is less than 75% of the stress acting normal to the weld. 
However, for combined loading which produces a change in principal stress direction during a cycle (non-proportional loading), Eurocode 3 requires a summation of the damage due to the different load types, using Miner's rule and appropriate S-N curves for the different stress types. 
It is this last situation which presents problems. For proportional loading, there is reasonable correlation of uni and multi-directional loading fatigue data using an equivalent stress. However, equivalent stresses can seriously underestimate the damaging effect of non-proportional loading, overestimating lives by more than ten times. 
This has been the subject of extensive research in recent years, focussing mainly on combined bending and torsion. 
The most promising methods seems to be those based on the critical plane approach, with the condition that the plane of fatigue cracking in a welded joint is likely to be dictated by the weld geometry (e.g. weld toe). 
In broad terms, an equivalent shear stress range Δτeq
is determined as a combination of the applied shear stress range on the critical plane Δτ and the maximum normal stress on the same plane σ max
(taken to be tensile yield in an as-welded joint to allow for residual stress) of the form
where k is a material constant (typically 0.2 to 0.3) that may vary with fatigue life. Some success of the method has been demonstrated, but further work is needed to develop the approach to a form more suitable for standardisation.
3.3. Variable amplitude fatigue loading
Miner's rule is universally adopted as the method for estimating fatigue lives under variable amplitude loading. In some cases it is assumed that Σni
= 1 at failure, where the stress history consists of n i
cycles of stresses S i
and N i
is the life at S i
from the design curve. In others, for conservatism, Σn i
<1 is required. The method became accepted for application to welded joints as a result of fatigue tests in the 1960s and early 1970s, mainly conducted under block programme loading.
Some rules take account of the damaging effect of stress ranges which are below the constant amplitude fatigue limit but which, under variable amplitude loading, gradually become damaging during the fatigue life as a crack develops. The method normally adopted is to extend the S-N curve beyond the constant amplitude fatigue limit at a reduced slope, for example 5 instead of 3 (see Fig.2).
Fatigue tests conducted over the past 20 years under random loading conditions have tended to throw doubt on the validity of Miner's rule and there are now many reported cases where Σn / N was <1 at failure, down to below 0.5.  This apparent contradiction with the early work is probably due to the fact that block programme loading tends to encourage crack growth retardation, which extends fatigue life.
The problem is associated with wide-band loading conditions, the damaging effect of small stress fluctuations being greater in a variable amplitude sequence than under constant amplitude loading. It is thought that this situation arises because, for a given stress fluctuation, crack closure may occur under constant but not under variable amplitude loading.  It is also thought that current methods are not taking due account of the damaging effect of stresses below the fatigue limit. 
Thus, it has been suggested that Miner's rule should be adequate provided the constant amplitude data are obtained under conditions where the crack tip will always be open, that is at high tensile mean stresses, and the slope change in the S-N curve is assumed to occur at a lower effective fatigue limit.  This has yet to be confirmed.
Alternative fatigue life prediction methods have been proposed, including that devised by Gurney.  Expressing the applied stress spectrum in terms of the number of cycles n i applied at proportions p i of the maximum stress range in the spectrum (S max) and referring to its total length Σn i as the block length, Gurney's rule states:
where m is the slope of the constant amplitude S-N curve.
However, Gurney's method is still not universal. It appears to be particularly suitable for relatively short block lengths. However, for long block lengths, when there is a tendency for Miner's rule to become over-conservative, Gurney's rule could be unsafe.
A final factor which could contribute to errors in life predictions is the method used to convert the stress history into recognisable cycles ('cycle counting'). The rainflow or reservoir methods are generally adopted, but little has been done to validate them for welded joints.
In spite of the superiority of Gurney's rule in some circumstances, it has not been adopted for design rules. The simple concept of Miner's rule, which is consistent with consideration of the fatigue process as the progressive growth of cracks,  remains the popular choice. Suitable adjustment of the cycle counting method could be a way to ensure that Miner's rule is always safe.
3.4. Weld quality
One possible interpretation of the use of lower bound fatigue data for design is that account is taken of the very worst weld quality (as it affects the particular failure mode being assessed). For general design rules, which will be used by a wide range of industries of varying ability, this is a reasonable state of affairs, especially as even the worst quality laboratory welded specimen is likely to be of reasonably good quality. However, the increasing need for economy and competitiveness in fabrications highlights the need for more flexibility and, in particular, the ability for manufacturers to gain credit, in terms of increased fatigue design stresses, for improving weld quality. Guidance is available  to enable allowable imperfections to be specified, or unexpected flaws to be assessed, on a fitness-for-purpose basis. However, the concentration is on imperfections which might reduce fatigue life. What is required now is similar guidance for features which have the potential to guarantee fatigue lives above the design curves. Preliminary steps have been taken to link fatigue design and weld quality in aluminium structures. [9,11,44] Further work is needed to rationalise them further, and to produce similar guidance for steels. A price to pay may be more extensive inspection, probably of both butt and fillet welds.
4. Concluding remarks
The most recent fatigue design rules for welded structures have a common basis and offer comprehensive coverage of key factors affecting fatigue life. Improvements would include wider average of different welding processes and corrosive environments (including for unwelded material); development of the hot spot stress approach to keep pace with the increasing use of FEA; a design method based on the good progress made in research into fatigue under non-proportional loading; safer cumulative damage methods. The adoption of a single classification scheme, preferably the fatigue strength at 2 x 10 6
cycles, would also reduce confusion.
A serious source of error in the application of fatigue rules is estimation of the service stress history. Any progress that can be made in improving the designer's ability to specify fatigue loading, or the monitoring of structures to detect evidence of premature fatigue, will be just as important as improvements to design rules.
Publications of special interest have been marked *. These all refer generally to fatigue or fatigue design of welded structures. *
1. Gurney T R: Fatigue of welded structures', 2nd Edition, Cambridge University Press, Cambridge, 1979. *
2. Fisher J W: 'Fatigue and fracture in steel bridges', Wiley Interscience, 1984. *
3. Maddox S J: 'Fatigue strength of welded structures', Abington Publishing, Abington, Cambridge, 1991.
4. British Standards Institute: 'Steel, concrete and composite bridges: Part 10 - Code of Practice for Fatigue', BSI, London, 1980.
5. AASHTO: 'Standard specifications for highway bridges', LRFD, 1st Edition, 1994.
6. Eurocode 3: 'Design of steel structures', prENV 1993, European Committee for Standardisation, Brussels, 1992.
7. Offshore Installations: 'Guidance on design, construction and certification', UK Health and Safety Executive, 4th Edition, 1990, Amendment No.3, London, 1995.
8. American Petroleum Institute: 'Recommended practice for planning, designing and constructing fixed offshore platforms', RP2A, 20th Edition, API, Washington, 1993.
9. BS8118, Part 1: 'Structural use of aluminium: Code of practice for design', British Standards Institution, BSI, London, 1992.
10. European Convention for Constructional Steelwork: 'European recommendations for aluminium alloy structures - fatigue design', Publication No. 68, ECCS, Brussels, 1992.
11. Eurocode 9 (Draft): 'Design of aluminium alloy structures, Part 2: Structures susceptible to fatigue', prENV 1999, Part 2 CEN/TC/250/SC9/N89, European Committee for Standardisation, Brussels, June 1996.
12. American Welding Society: 'Structural Welding Code - Steel', ANSI/AWS D.1., AWS, Miami, 1996.
13. BS7608: 'Code of Practice: Fatigue design and assessment of steel structures', British Standard, BSI, London, 1993. *
14. Hobbacher A: 'Fatigue design of welded joints and components', IIW, Abington Publishing, Abington, Cambridge, 1996.
15. Haagensen P J: 'Fatigue of tubular joints and fatigue improvement methods', Progress in Structural Engineering and Materials, Sept.1997, 1
16. Gurney T R and Maddox S J: 'A re-analysis of fatigue data for welded joints in steel', Welding Research International, 1973 3
17. Brozzetti J, Chabroline B and Raoul J: 'Background document on fatigue design rules in Eurocode 3, Part 2: Bridges', CTICM Report No.10.003-7, Sept.1992.
18. Draft Unfired Pressure Vessel Standard, Part 3: 'Design', European Committee for Standardisation, Doc. No. CEN/TC54, Brussels, 1996. *
19. Fisher J W: 'Improved performance through large scale dynamic testing of structures', IIW Intl. Conf. on Performance of Dynamically Loaded Welded Structures, Welding Research Council, New York, 1997.
20. Ohta A, Suzuki N and Maeda Y: 'Effect of residual stresses on fatigue of weldment', IIW Intl. Conf. on Performance of Dynamically Loaded Welded Structures, Welding Research Council, New York, 1997.
21. Jaccard R, Kosteas D and Ondra E: 'Background document to fatigue design curves for welded aluminium components', IIW Doc. XIII-1588-95, 1995.
22. Gurney T R: 'The influence of thickness on the fatigue behaviour of welded joints'. Proc. 2nd Intl. Conf. in Behaviour of Offshore Structures, BOSS'79, London, 1979.
23. Maddox S J: 'The effect of plate thickness on the fatigue strength of fillet welds', Abington Publishing, Abington, Cambridge, 1987.
24. Maddox S J: 'Scale effect in fatigue of fillet welded aluminium alloys', Proc. 6th Intl, Conf. on Aluminium Weldments, American Welding Society, Miami, FL, 1995, 77-94.
25. Signes F S, Baker R G, Harrison J D and Burdekin F M: 'Factors affecting the fatigue strength of welded high strength steels', Br. Weld. J, March 1967, 14
26. Watkinson F, Bodger P H and Harrison J D: 'The fatigue strength of welded joints in high-strength steels and methods for its improvement', Proc. Conf. on Fatigue of Welded Structures'. TWI, Abington, Cambridge, 1991.
27. Blom A (Ed): 'Welded high-strength steel structures', North European Engineering & Science Conference, NESCO, Stockholm, 1997.
28. Haagensen P J: 'Weld improvement methods - applications and implementations in design codes', Intl. Conf. on Fatigue of Welded Components and Structures, les Editions de Physiques, les Ulis, France.
29. Bigonnet A: 'Improving the fatigue strength of welded structures', Steel in Marine Structures, Elsevier, Amsterdam, 1987.
30. Maddox S J: 'Developments in fatigue design codes and fitness-for-service assessment methods', IIW Intl. Conf. on Performance of Dynamically Loaded Welded Structures, Welding Research Council, New York, 1997.
31. Razmjoo G R: 'Design guidance on fatigue of welded stainless steel joints', Proc. of Offshore Mechanics and Arctic Engineering Conference (OMAE'95), Vol.3 Materials Engineering, ASME, 1995, 163-171.
32. Maddox S J: 'Fatigue design of welded aluminium alloys structures', Proc. 2nd Intl, Conf. on Aluminium Weldments, Aluminium-Verlag, Dusseldorf, 1982.
33. Maddox S J and Razmjoo G R: 'Fatigue tests on large girth welded steel tubes', Offshore Mechanics and Arctic Engineering Conference (OMAE'98), Vol.3, Materials Engineering, ASME, 1998.
34. Muller F: 'Fatigue of weathering steels in as-received and welded material states within a weathering period of six years', IIW Doc. No. XIII-1479-93, 1993. *
35. Niemi E: 'Stress determination for fatigue analysis of welded components', IIW, Abington Publishing, Abington, Cambridge, 1995.
36. Dexter R J, Tarquinio J E and Fisher J W: 'Application of hot spot stress fatigue analysis to attachments on flexible plates', Offshore Mechanics and Arctic Engineering Conference (OMAE'94), Vol.3, Materials Engineering, ASME, 1994.
37. Sonsino C M: 'Multiaxial and random loading of welded structures', IIW Intl. Conf. on Performance of Dynamically Loaded Welded Structures, Welding Research Council, New York, 1997.
38. Brown M W and Miller K J (Ed): 'Biaxial and Multiaxial Fatigue', EGF Publication 3, Mechanical Engineering Publications Limited, London, 1989.
39. Marquis G B, Backstrom M and Siljander A: Multiaxial fatigue damage parameters for welded joints: design code and critical plane approaches', Proc. Conf. on Welded High Strength Steel Structures', NESCO, Stockholm 1997, 127-141.
40. Gurney T R and Maddox S J: 'An alternative to Miner's rule for cumulative damage calculations', IASBSE Workshop Remaining Fatigue Life of Steel Structures, IABSE Report No. 59, 1991, 189-198.
41. Marquis G: 'Long life spectrum fatigue of carbon and stainless steel welds', Fat. Fract. Engng. Mater. Struct, 1996 19
42. Niemi E: 'Random loading behaviour of welded components', Proc. IIW Conf. on Performance of Dynamically Loaded Welded Structures, Welding Research Council, New York, 1997.
43. PD6493:1991: 'Guidance on methods for assessing the acceptability of flaws in fusion welded structures', British Standard Published Document, BSI, London, 1991.
44. Ogle M H: 'Production weld quality standards for steel and aluminium structures' Welding in the World, 1991, 29