S J Maddox and G R Razmjoo
Published in: Fatigue and Fracture of Engineering Materials and Structures vol.24, no.5. May 2001. pp.329-337
Current fatigue design methods for assessing welded steel structures under complex combined or multiaxial loading are known to be potentially unsafe. This has led to a number of research projects over the past 10 years. Some progress has been made in developing better methods, but they are not yet suitable for general design.
This paper presents an interim solution based on a review and analysis of relevant published data; all referring to fatigue failure from a fillet weld toe. These indicate that Eurocode 3/IIW S-N curve FAT80/3 (negative inverse slope of 3) is suitable for combined normal and shear stresses acting in phase, and possibly for out-of-phase (i.e. non-proportional loading) bending and shear if the shear stress is not due to torsion. However, shallower curve FAT80/5 is necessary for out-of-phase torsion and bending or tension. Both curves are used in conjunction with the nominal maximum principal stress range occurring during the loading cycle.
Keywords: Fatigue, multiaxial loading, design, welded joints, steel
||Applied bending moment
||Number of applied stress cycles
||Outside diameter of a tube
||Inside diameter of a tube
||Nominal normal stress
||Nominal normal stress in x direction
||Nominal normal stress in y direction
||Maximum nominal principal stress
||Equivalent nominal stress
||Nominal shear stress
||Nominal shear stress in x, y plane
Comprehensive international fatigue design rules for welded steel structures are now widely available [1-3] . However, one aspect which is known to be potentially unsafe is the method of assessing complex loading (combined or multi-axial) situations, particularly those in which the principal stress directions change during the fatigue load cycle (i.e. non-proportional loading) [4-6] . Such a situation may arise as a result of a moving load (e.g. vehicle on a bridge) or more than one load source (e.g. combined bending and torsion) acting out-of-phase. There are a number of possible design approaches  , but the one most commonly specified is to base design for both proportional and non-proportional loading on the maximum principal or equivalent stress range, and then refer to the design S-N curve obtained under uniaxial loading condition [6-8] . However, there are now extensive data [4-6] , including some obtained very recently under variable amplitude loading  , to show that these approaches can overestimate the fatigue lives of butt and fillet welds under non-proportional loading by more than an order of magnitude. This has prompted a number of research studies and has led to alternative methods of correlating the fatigue lives of welded joints obtained under simple uniaxial and complex combined or multi-axial conditions. One of the more promising is the critical plane approach, originally proposed by Findley but more recently shown to be suitable for considering welded joints subjected to non-proportional loading [4,11] . An alternative approach, which has also successfully correlated fatigue test results for welded joints obtained under uniaxial and non-proportional loading, is the effective equivalent stress hypothesis (EESH) developed by Sonsino et al [5,9] . A feature of both these approaches is that they use local notch stresses rather than nominal stresses. The determination of notch stresses requires detailed finite element (FE) modelling of the weld toe as a notch. In view of the range of weld details to be considered in design, and the inevitable variation in local geometry (weld toe angle and radius), this would be too time consuming for most design situations. It is also relevant to note that although the local approach utilising notch stresses is one possible method for designing welded joints  , it is not yet established in fatigue design rules for welded structures.
Therefore, an alternative, simpler approach for design engineers who are not familiar with local approaches will still be of value. This paper presents such an approach, developed on the basis of a review and analysis of available fatigue data for welded joints tests under complex loading conditions. The approach has been developed for use in conjunction with fatigue design rules that present S-N curves for weld details expressed in terms of nominal stresses [e.g 1-3] . Such design curves incorporate the full effect of tensile residual stresses in welded joints and therefore only stress ranges are used, independently of applied mean stress. For complex situations where nominal stresses are not well defined, hot-spot stresses, derived from the stress distribution approaching the weld but excluding its notch effect, would be used instead.
As well as the important case of non-proportional loading, the opportunity was taken to assess the design methods in the light of experimental data obtained under proportional loading conditions. Available data cover a number of fatigue failure modes, notably weld throat failure in load-carrying fillet welds [13,14] , fatigue cracking transverse to the length of a weld  and weld toe failure [4,5,13-22] . The present paper is confined to consideration of the last, which is the most common mode of service fatigue failure in welded structures. In all cases, the fatigue test results were considered in terms of the endurance to failure. This was complete rupture in the case of the tube specimens, which soon followed the achievement of a through-wall crack. The failure criterion in the case of beams was effectively the same, although tests were usually terminated when a through section crack was so large that the beam deflection was excessive. Endurances obtained from fatigue design rules for welded structures correspond to the same failure criteria.
2. Treatment of complex loading in design codes
2.1 Principal stress directions constant
Design codes which consider fatigue under complex, combined or multiaxial, stressing are usually based on the assumption that S-N curves obtained under uni-directional loading (i.e. usually axial or bending) are applicable when used in conjunction with an equivalent stress. The two most widely used equivalent stresses are the von Mises stress, σ e, and the maximum principal stress, σ 1, given by:
von Mises stress:
Maximum principal stress:
Traditionally, the former has been use for pressure vessel design  and the latter for structural applications [1,2] . With regard to welded joints, an advantage of the principal stress, which has known direction, is that account can be taken of the orientation of the weld, which can have a very significant effect on its fatigue strength. When using the scalar von Mises equivalent stress, it is normally necessary to assume that the weld is stressed in the most unfavourable direction.
Justification for the use of the principal stress in structural design codes is based mainly on fatigue tests on welded beams with web stiffeners. In such cases, it has been found that fatigue cracks initiate at the weld toe on the web in a region of combined bending and shear and propagate at right angles to the direction of maximum principal stress [16,19] .
2.2 Principal stress directions change
It is only relatively recently that a significant database from fatigue tests on welded joints under non-proportional loading, when the principal stress direction changes during a load cycle, has become available [4-6,9,13] . Consequently, current fatigue design methods are based more on judgement than experimental evidence. Thus, the approach described in Section 2.1 is also used in some codes. The requirement is to establish the maximum algebraic difference between principal stresses occurring during the whole load cycle. This may then be used directly [3,23-25] or to calculate the von Mises (or sometimes Tresca) equivalent stress [3,24] , depending on the design code. In some cases  there is a further restriction based on the directions of the principal stresses such that the stress range should be taken as the greatest algebraic difference between the principal stresses occurring on principal planes not more that 45° apart in any one load cycle.
An implicit assumption when applying the equivalent stress approach is that the same S-N curve is applicable for any type of stress (i.e. combined, multiaxial, normal or shear stresses). However, this is not the case in the IIW  or Eurocode  rules, which give different S-N curves for normal and shear stress conditions. Consequently, an alternative procedure is recommended. This entails effectively applying the Palmgren-Miner cumulative damage rule for the separate nominal stress components σ x and τxy using the appropriate design S-N curves. Thus, the following must be satisfied in Eurocode 3:
at the end of the fatigue life, where n is the number of applied stress cycles and N is the endurance given by the design curve for that stress. The IIW rules are more conservative in that the summation should not exceed 0.5. An implicit assumption in the approach is that the two stress types act independently, and both contribute fatigue damage in the same way that they would when acting alone.
3. Comparison of design methods and experimental data
The validity of current design methods was explored on the basis of relevant published fatigue data. Particular attention was paid to the substantial database now available, referred to earlier, from tests under both proportional and non-proportional combined bending or tension and torsion on steel tube to plate welded joints. A typical specimen design is shown in Fig.1. The detail is, of course, of practical relevance (e.g. shafts) as well as being a convenient one for studying combined normal and shear stressing. A difference between this case and the one which provided most of the data upon which the current design approaches are based, that is an I-beam with web stiffeners situated in the field of combined bending and shear, is that the shear stress arises from torsion. In view of this, and the fact that the recent data cover a wide range of shear/normal stress values, the validity of the design method for proportional loading was also considered.
Fig.1. Typical tube to plate specimen used to investigate fatigue under multiaxial loading
The design curves considered were those in the IIW fatigue design recommendations  , which for the relevant details are identical with those in Eurocode 3  . They are referred to using the terminology FAT number/number where the first number is the fatigue strength in MPa at 2 x 10 6 cycles and the second number is the negative inverse slope of the S-N curve (i.e. 3 or 5).
Finally, the test results were considered only in terms of nominal principal stresses, since these are usually used as equivalent stresses in structural design codes. In the context of tubes under combined bending or tension and torsion, giving normal stress σ x and shear stress τxy, σ y = 0 so that Eq.(2) becomes:
3.2 Results obtained under uni-directional loading
Several studies have considered tube to plate joints subjected to either torsion or direct stressing, as well as combined torsion and tension or bending. These offer the opportunity to investigate the need for different S-N curves to cover normal and shear (torsion) stresses. Figure 2 presents all the results obtained under either bending or torsion from specimens made using steel tubes of similar dimensions (wall thickness from 7-10mm) [1,4,9,13,14,22] . The results are presented in terms of the maximum principal stress range (twice the amplitude) at the weld toe, which is the same as the nominal bending stress ( σ x) or shear stress ( τxy), as appropriate. The stresses plotted are the nominal stress ranges in the tube (outer and inner diameter D and d respectively) at the weld toe, given by:
where M is the applied bending moment at the weld toe and T is the applied torque.
As will be seen, although there is relatively good agreement between the results, there is a definite tendency for those obtained in torsion to follow a shallower S-N curve than those obtained in bending. The mean curves fitted to the two sets of data are included in Fig.2. Also shown in Fig.2 are the design curves for the two loading modes, FAT100/5 for applied shear stresses and FAT80/3 for applied normal stresses. Both curves refer to fatigue failure from the weld toe. The shallower experimental S-N curve obtained in bending is to be expected in view of the low level of welding residual stress that would have been present in the specimens. This, the fact that most tests were conducted at a stress ratio of R = -1, and the relatively small wall thicknesses will also have contributed to the high fatigue strength obtained, as compared with the design curves. Thus, in general the test results are reasonably consistent with the design curves and they justify having different S-N curves to cover normal and shear stresses. This implies that the damage summation approach represented by Eq.(3) will be required to consider combined loading.
Fig.2. Fatigue test results for flange-tube welded joints failing from the weld toe, tested under bending or torsion
3.3 Results obtained under combined loading
Turning next to test results obtained under combined bending and torsion, the available data [4,5,22] for in-phase (proportional) loading are presented in Fig.3(a), while those obtained under out-of-phase (non-proportional) loading [4,5] are presented in Fig.3(b). All the results are presented in terms of the maximum range of principal stress experienced during a load cycle, calculated by considering the variation of both normal and shear stresses during each cycle. These stresses were calculated using Eq.(5) or (6), as before, while the principal stress was calculated using Eq.(4). In the case of proportional loading, when the applied normal and shear stresses were in-phase, the resulting maximum and minimum normal and shear stresses, and therefore the maximum principal stress range in the cycle, are obvious (see Fig.4(a)). However, in the case of non-proportional loading, when the normal and shear stresses are not in phase, it is necessary to determine the change in principal stress throughout the load cycle to establish the maximum range, as illustrated in Fig.4(b). It will be observed that for the same applied normal and shear stress ranges, the maximum principal stress range is smaller for non-proportional loading ( Fig.4(b)), suggesting that it should be less damaging than for proportional loading conditions. However, as noted earlier, the opposite proves to be the case.
Fig.3a). Fatigue test results for flange-tube welded joints failing from the weld toe, tested under combined in-phase bending and torsion
Fig.3b). Fatigue test results for flange-tube welded joints failing from the weld toe, tested under out-of-phase combined bending and torsion
Fig.4. Principal stress variation for: Fig.4a) in-phase; and
Fig.4b) 90° out-of-phase loading
The results in  were obtained under out-of-phase sinusoidal bending and torsion. This resulted in changes in the maximum principal stress direction between ±45°. Changes in principal stress direction, usually between ±20°, were also introduced in the tests reported in  , but this was due to a reversal in the applied torque with the bending stress held at its peak value. Nevertheless, it will be seen that there is reasonable correlation between the results, suggesting that this difference was not significant.
Referring first to proportional loading conditions, the current design method is to assume that the fatigue life under combined loading will be the same as that under normal stresses for the same principal stress range. Thus, in the context of combined bending and torsion of the tube to plate joint, the results should agree with those in Fig.2 obtained in bending only. The mean curve fitted to those results is included in Fig.3(a). As seen, it provides a reasonable representation of the combined loading data. It will also be noted that there is no apparent influence of the τ/σ ratio on the fatigue test results. Thus, this database supports the current design method for proportional loading for τ/σ values from 0.14 to 1.0. These results, and others, are considered further below in comparison with design S-N curves.
In the case of non-proportional loading, since different S-N curves were produced for bending and torsion, Eq.(3) should apply. To explore this, Fig.3(b) includes S-N curves calculated using Eq.(3) in conjunction with the experimental mean S-N curves for torsion and bending in Fig.2 for the range of τ/σ values covered by the test data. Most of the experimental results were obtained with τ/σ = 0.58, but other values ranged from 0.42 to 1.0. Thus, the curves shown in Fig.3(b) bound those for τ/σ between 0.42 and 1.0. Note that the lower curve for τ/σ = 0.42 is only slightly higher than that for pure bending ( τ/σ = 0).
Referring to Fig.3(b), it will be evident that Eq.(3) is certainly not applicable for out-of-phase loading, overestimating the actual fatigue lives by factors of more than 15. Thus, even use of the IIW summation value of 0.5 would still result in unsafe estimates of fatigue lives under non-proportional loading. A potential weakness in this approach is that the maximum variation in each of the two stress types during a load cycle is used to deduce the N values. Depending on the phase difference between them, these two values may not be the ones that combine to give the maximum principal stress range. This could be one reason for the failure of the design method.
The other design approach is to assume that the S-N curve for normal stressing is directly applicable to combined loading in terms of the maximum principal stress range, as in the case of proportional loading. Recalling that the lower curve in Fig.3(b) is effectively the same as the mean curve fitted to the results for bending in Fig.2, the present data do not support this approach. However, as in the case of the results obtained under proportional loading, they do appear to correlate regardless of τ/σ for the range covered. Thus, they could be represented by a single S-N curve. However, in contrast to the database for proportional loading, the slope of that curve would appear to be closer to that obtained for torsion loading than that obtained for bending. This is considered in more detail in the next section.
4. Assessment of data as basis for design
4.1 Combined loading with shear stresses from torsion
In order to consider a possible single design S-N curve for non-proportional loading, the database from tube to plate welded joints was considered further in comparison with design curves. At the same time, the database for proportional loading was compared with current design curves. In both cases, results obtained in tension and torsion [6,13] , in which both the normal shear stresses varied sinusoidally, as in  , are also considered. The data from Fig.3(a), obtained under proportional loading, are replotted in Fig.5, distinguished only in terms of whether the loads were applied in-phase or out-of-phase. Regression analysis of the in-phase data gave the 95% confidence intervals (mean ± two standard deviations of log N) shown. These may be compared with the two design curves for shear and normal stresses, both used here for comparison with data expressed in terms of the maximum principal stress range. As will be seen, the data are more consistent with the slope of the normal stress FAT80/3 curve than the shear stress FAT100/5 curve; even better agreement would be expected from welded joints containing high tensile residual stresses. The FAT80/3 also looks more reasonable when additional results obtained by Razmjoo 
under combined in-phase tension and torsion are introduced. Indeed, regression analysis of all the in-phase data, including those obtained under tension/torsion, gives a mean S-N curve with a slope very close to 3. Clearly, more data are required from a wider range of tube sizes and τ/σ values, but at present it seems that FAT80/3 is a suitable design curve for weld toe failure in flange-tube joints subjected to in-phase torsion and bending or tension. This implies that for in-phase loading conditions the normal stress component has the most influence on the fatigue life.
In contrast, the results obtained under out-of-phase loading, also included in Fig.5, imply that the shear stress component has more influence. Again the results exhibit relatively little scatter, but in this case they follow an S-N curve negative inverse slope close to 5, the slope of the current IIW/Eurocode 3 FAT 100/5 design curve for use in conjunction with the shear stress. Furthermore, the same is true for the results obtained under combined tension and torsion which, as in the case of in-phase loading, tend to lie on the lower bound to those obtained under bending and torsion. Regression analysis of all the data for joints subjected to non-proportional out-of-phase loading leads to a lower 95% confidence interval close to the FAT80/5 curve currently used to assess weld throat failure in welds loaded in shear. Thus, FAT80/5, used with the maximum principal stress range, may be a suitable choice for a design curve for weld toe failure in flange-tube welds subjected to non-proportional combined bending or tension and torsion. Clearly, further data are needed to substantiate this choice of design S-N curve for a wider range of shear to normal stress ratios and tube sizes. In this respect, the present data were all obtained from relatively thin tubes, between 7 and 10mm wall thickness, and lower S-N curves may be needed for thicker material. A further limitation is that all the data considered so far were obtained under bending or tension and torsion. Therefore, it is also necessary to consider the applicability of the tentative design curves to other types of combined loading, as discussed in the following section.
Fig.5. Fatigue test results for flange-tube welded joints failing from the weld toe, tested under combined bending or tension and torsion loading
4.2 Combined loading excluding torsion
Although applying torsion is a convenient way to induce shear stresses, they can arise from other types of loading. In this respect, as noted earlier, fatigue data are available for beams loaded in bending with weld details situated in regions of combined bending and shear. Most of these were obtained under in-phase loading [16-19] , but limited data are also available for beams subjected to out-of-phase loading  . In one case, the weld detail was located on the neutral axis of the beam so that it experienced only shear. The results for fatigue failure from the weld toe, expressed in terms of the maximum principal stress range, are shown with the proposed design curves in Fig.6. As will be seen, all the data lie above the FAT80/3 design curve, even those obtained under non-proportional loading. Clearly more data are needed for non-proportional loading to draw firm conclusions, but the impression given from the available data is that there is no need to adopt the shallower FAT80/5 design curve for combined loading when the shear stress does not arise from torsion. In this basis, the FAT80/5 design curve might be considered to be specifically applicable to weld toe failure in joints subjected to combined torsion and bending or torsion and tension. For other cases of weld toe failure under combined loading, FAT80/3 may be sufficient.
Fig.6. Fatigue test results for welded beams with fillet welded web attachments under combined bending and shear
A rationale for a distinction between the fatigue behaviour in torsion and other types of shear loading is that they lead to different fatigue cracking modes. In this respect, available information suggests that the Mode III shear failure mode only occurred for weld toe cracks under applied torsion. In contrast, the fatigue cracks in regions of combined bending and shear in beams loaded in bending were of the usual mode I type, propagating normal to the direction of principal stress. Such a fatigue fracture mode would be expected to result in an S-N curve closer in slope to the FAT80/3 curve than the FAT100/5 curve, as implied by Fig.6.
Clearly, further results are needed to confirm these indications, including more attention to the modes of fatigue failure obtained in fatigue tests performed under complex loading. Certainly, it seems reasonable that a change in failure mode from Mode I to Mode III will result in a different S-N curve. What is not clear is if the Mode III shear mode of fatigue cracking can be produced by loadings other than torsion. It is possible that for all other practical cases of fatigue loading on welded joints, when the weld toe is the most likely location for fatigue cracking, shear stresses are always accompanied by normal stresses (bending or direct) and that they encourage the Mode I failure. This warrants further study, as does the influence of applied shear stress in cases when fatigue cracks may occur at other locations, notably the weld root in load-carrying fillet welds.
Available fatigue test results for fillet welded steel tube to plate joints, failing from the weld toe under combined bending or tension and torsion loading, were used as the basis of improved nominal stress-based design rules. The following conclusions were drawn:
- The damage summation procedure specified by IIW and in Eurocode 3 for out-of-phase (non-proportional) loading is potentially un-conservative, by a factor of around 15 on life.
- FAT80/3 design curve proves to be suitable for in-phase (proportional) loading, but the FAT80/5 is needed for out-of-phase combined torsion and bending or tension. Both curves would be used in conjunction with the maximum principalstress range.
- There is no clear influence on fatigue life of the shear to normal stress ratio and the above design curves are suitable for values from 0.14 to 1.0.
- Extending the review to include data obtained from beams, the FAT80/3 curve proves to be suitable for both proportional and non-proportional combined loading when the shear stress component is not due to torsion.
- Possible implications are that the FAT80/5 curve is only required if the shear stress component is due to torsion, or to loading which induces Mode III type shear fatigue fracture.
This work formed part of a project undertaken by Fraunhofer Institut fur Betriebfestigkeit-LBF with funding from the European Coal and Steel Community (ECSC). The authors are grateful to Prof. C.M. Sonsino for valuable discussions.
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