Exclusions have been made where duplicate of data series was identified. Specifically, the study by Yousefi et al. [21] has been omitted, as it presents data originally obtained from Witt [9]. A similar case applies to Amstutz [22], which, despite not explicitly citing Witt [9], presents an identical data series. This conclusion was reached through direct comparison of reported values, highlighting the lack of proper citation in Amstutz et al. [22].

Additionally, the data study from Bufalari [23, 24] represents an extension of the earlier work by Lieshout [25]. Four data points have been removed in the transition from Lieshout [25] to Bufalari [24] due to revised interpretations based on further experimental insights. This refinement was confirmed through direct communication with the authors involved in the study. Consequently, the data study by Bufalari [24] is considered the reference for these experiments.

A further limitation arises in the study by Archer [15], which, while included in the database for reference, has been excluded from the final analysis. The primary reason for this exclusion is the use of strain gauge measurements at an unspecified location, making it impossible to replicate the stress data ideally in finite element modelling, without making significant assumptions.

The study by Garcia and Baptista [26, 27] is the only one in the database that reported a significant increase, by a factor of eleven, in fatigue life for NP cases, rather than a reduction. This unexpected result has started further investigation into this publication. As this study examines a biaxial loading case, where two perpendicular tensile forces are applied instead of a pure shear force in cases with e.g., torsion, it has only one comparable study in the database, conducted by Takahashi et al. [28, 29]. Their findings, however, showed an average fatigue life reduction of 28% under NP loading conditions as expected. It should be noted that NP stress states have, in some cases, been documented to increase fatigue life rather than reduce it. This phenomenon has been observed in brittle materials, such as cast iron [11]. However, since welding is rarely performed on brittle materials, NP loading in welded steel connections is generally expected to reduce fatigue life. Given that the study by Baptista and Garcia focuses on ductile materials, the significant increase in fatigue life observed in their results was unexpected. Due to the unusually high fatigue life reported in this study, it has been determined that this case should be considered an outlier. Consequently, this study has been excluded from the database until further investigations can be conducted.

Given the focus of this review on structural stress approaches, only fatigue failures occurring at the weld toe are considered. As a result, studies where failures occurred at the weld root or other locations have been excluded, further reducing the database to the final 9 publications including Siljander [17], Sonsino [30], Razmjoo [31], Witt & Zenner [32], Bäckström [33], Witt [9], Takahashi [29, 28], Lieshout & Bufalari [25, 24] and Winther [34]. This additional constraint further highlights the limited availability of experimental NP fatigue data for welded steel components.

For a dataset to be considered fully complete, it must include the necessary baseline data, consisting of the investigated NP case, the corresponding proportional case, and pure uniaxial data. The uniaxial data typically consists of two series to capture both pure normal stress and shear stress. Consequently, a complete dataset study should ideally consist of four experimental series to provide a robust basis for analysis.

Typically, individual data series are presented in the form of a log-log S-N curve, derived using Basquin’s power-law approximation [35]. The S-N curve provides a critical link between applied stress and fatigue life and is essential for the evaluation of NP fatigue criteria discussed in 1.8. Therefore, it is imperative to assess whether all required information is available and whether each data series contains a sufficient number of repetitions to reliably define an S-N curve.

According to the DNV design codes [2], a minimum of 15 specimens and at least three load levels are required to establish a statistically valid S-N curve. Additionally, these specimens should be tested at unique load levels rather than being repetitions at the same load, ensuring a distributed dataset for curve fitting. In the current collected data, only 6 out of 59 individual datasets contain 15 or more repetitions, and when considering unique load levels within a ± 10 MPa range, no series exceeds 11 distinct load levels. The average repetition count is 9, while the average number of unique load levels is 6. Given the limited availability of data, values in the orange range have been deemed acceptable within these constraints.

Upon reviewing the available datasets, several limitations were identified, primarily due to missing data or an insufficient number of data points. A full summary of the datasets is presented in the Appendix A.

A notable limitation is observed in the dataset from Sonsino [30], where the original studies involving flange-to-tube and tube-to-tube welded specimens did not include all corresponding uniaxial S–N curves. In the case of the flange-to-tube specimens, it was possible to identify similar geometries in other studies [36, 37], which were used as substitutes. However, for the tube-to-tube specimens, no equivalent experimental data for pure torsion was found. As a result, only design code- or recommendation-based S–N curves could be used, preventing a fully experimental comparison.

A similar issue exists in the dataset by Bufalari [24]. This study includes a pure torsion S-N curve only for the case of R = -1, whereas the combined loading tests were performed at R = 0. Despite the mismatch in load ratio, the available torsion S–N curve was used as a reference due to the absence of alternative experimental data.

The dataset by Siljander contains only two repetitions and load levels for the pure torsion tests, which limits its statistical reliability. Bäckström's dataset includes a maximum of three repetitions per load case, making it statistically weak as well. Additionally, in the study by Winther et al., only a single load level was tested under combined loading, making it impossible to establish a full S–N curve for these cases. However, since the evaluation primarily relies on pure uniaxial data, this limitation does not significantly impact the overall applicability of the dataset.

While the absence of pure uniaxial data presents a significant limitation when evaluating directly against experimental results, this issue is mitigated when comparing results to design code S-N curves, such as the focus in this study, where the necessary reference curves are available through the design codes. These limitations do not render the dataset unusable, but they indicate that many series do not fully align with best practices for experimental validation. Consequently, while the dataset remains valuable for analysis, its usability depends on the intended evaluation method. When using design code S-N curves, missing data can be compensated for, but for strictly experimental analyses, certain data studies may lack the completeness required for reliable fatigue assessments.

Ultimately, despite the small collection of data studies, it represents the most comprehensive and reliable resource available for evaluating NP fatigue behaviour in welded joints for structural stress approaches. The findings and conclusions drawn from this study must, therefore, be understood within the constraints imposed by the current experimental evidence.

In the fatigue assessment of welded components, residual stresses introduced by the welding process play a critical role. However, in the conventional S-N approach, these residual stresses are not explicitly modelled. Instead, they are inherently accounted for in the S-N curves themselves, which are typically derived from full-scale welded specimens tested in the as-welded condition. These curves therefore include the influence of tensile residual stresses, making them suitable for design without further adjustment.

In the present study, the fatigue database includes both as-welded and stress-relieved specimens. Since residual stresses affect crack initiation and early crack growth, their absence in stress-relieved specimens can lead to artificially improved fatigue life. Consequently, a correction factor has been applied to the stress-relieved data [2], given by:

To ensure a practical and focused investigation, only established design codes have been considered that explicitly address NP loading in a structural stress manner. This includes: IIW [1], DNV [2], ASME [3], FKM [4] and BSI [5]. Although Eurocode 3 [38] is widely used and shares structural similarities with other recommendations, it does not specifically address NP stress state and has therefore not been included in this study.

Across the reviewed design codes and recommendations, three stress evaluation concepts are consistently referenced: the nominal stress approach, the hot-spot stress method, and the equivalent structural stress (ESS) method. The nominal stress approach is suitable only for simple geometries and will be used in this study as a baseline reference. For more complex geometries, the design codes generally recommend the hot-spot method except for ASME, which instead refers exclusively to the ESS method.

To ensure broad applicability and alignment with international practice, this study adopts all three methods: nominal stress as a benchmark, and both hot-spot and ESS as practical structural stress evaluation methods for complex welded joints. This selection ensures coverage of the full range of methods recommended by the major fatigue design standards which handles NP stress state.

To estimate fatigue life from stress for any fatigue method, the logarithmic relationship by Basquin [35] is used in this study. This expression relates the stress range to the number of cycles to failure through the slope (m) of the S-N curve and the logarithmic fatigue resistance coefficient (log C):

Here, represents the equivalent stress range corresponding to a specific number of cycles to failure. It can also be interpreted as the fatigue resistance at that particular number of cycles, and denotes the number of cycles to failure. Please note that the term ‘equivalent stress’ is used to describe stress states that combine both normal and shear components. Therefore, it is not limited to von Mises or maximum principal stress but serves as a general term for combined stress expression.

In design codes and fatigue classifications e.g., FAT or detail categories, this relation is often expressed in the following form:

From this formulation, either the number of cycles to failure or the equivalent stress range can be isolated:

Typically, the reference number of cycles is taken as [1], depending on the chosen design code. This formulation will be used to evaluate NP correction methods that yield an equivalent stress range as their output.

In cases where the NP correction method does not directly yield an equivalent stress range, such as interaction equation-based formulations, an iterative approach is required to determine the number of cycles to failure. These methods describe fatigue life in terms of the cyclic degree of utilization, defined as the ratio between the applied stress range and the fatigue resistance, i.e., the reference fatigue resistance range obtained from the uniaxial S-N curve for a given cycle count N. Examples include methods from IIW and FKM, which will be presented in the following sections. A general interaction formulation is shown below:

To determine the number of cycles to failure, the reference fatigue resistance can be substituted with the expression with the equivalent fatigue resistance at a specific number of cycles to failure . The same methodology applies for the shear fatigue resistance. The interaction equation then becomes a non-linear expression in terms of the number of cycles to failure :

Here, and are the slopes of the normal and shear uniaxial S-N curve, and are the exponents of the normal shear utilization term and A is usually equal 1 but it depends on the methods, which will be presented for the individual methods respectively. As no closed-form solution exists for this type of equation, an iterative numerical method is employed to solve for the fatigue life. The iterative solution process is based on the methodology outlined by Fass et al. [40].

It is important to recognise that the S-N curve relationships used to relate stress states to fatigue life are typically based on uniaxial fatigue data. Specifically, data derived from pure normal stress loading. Such data are widely available in design codes and recommendations. However, pure shear fatigue data are considerably less frequently available, with most design codes providing only two reference cases [1, 5]. This applies to the previously mentioned standards, with exception of ASME [3], which instead refers to a master S-N curve that does not differentiate between shear and normal stresses. In cases involving pure shear, it is assumed that the same Master S-N relationship can be directly applied by shear stress as that of normal stress.

Applying the S-N curve approach in a multiaxial setting is challenging, as the S-N curve is based on uniaxial data. To reflect common engineering practice, the S-N curves in this study are interpreted in a way that allows direct fatigue life prediction for combined loading scenarios by converting them into an equivalent stress range. Hence, this does not apply to interaction equations, as the number of cycles is determined by an iterative approach, yielding no equivalent stress range. A common way to achieve this equivalent stress range is through models such as the maximum principal Stress or the von Mises approach, both of which reduce the multiaxial stress states to a scalar equivalent value. To provide a broader context and establish reference points for comparison, these two widely used methods have been included in this study as benchmarks as the NP loading effects are not explicitly considered.

As previously mentioned, only a selected group of design codes were identified to include methods for handling NP loading conditions. These are IIW [1], DNV [2], ASME [3], FKM [4], and BSI [5]. The aim of this section is to introduce each of these assessment approaches and highlighting their advantages and limitations. Additionally, the two commonly used multiaxial methods, maximum principal stress and von Mises, are included for benchmark purposes. A short overview of the selected methods is presented in Table 1.

It is noted that a majority of NP correction methods, including those in the IIW [1], DNV [2], and FKM guidelines [4], are based on this interaction equations likely inspired from the Gough-Pollard formulation [44]. This is particularly interesting considering that the original Gough-Pollard model is empirical and based solely on proportional loading data and deducted from fatigue tests of ductile non-welded materials.

As a sub-investigation, the interaction equation was explored to assess the influence of different exponent combinations for the normal and shear stress components, as originally proposed by Archer [15]. The primary objective was to evaluate whether specific exponent values could provide a better fit to experimental fatigue data. The results, shown in Fig. 4, include several interaction curves plotted using varying exponent values. The analysis was based on nominal stress data combined with experimentally derived S-N parameters, thereby inherently accounting for the notch effect at the weld toe, in order to be true to experimental results. The resulting data exhibit considerable scatter, which makes it difficult to identify a clearly superior exponent combination. This outcome is not unexpected given the natural variability in fatigue test data.

In conclusion, while the choice of exponent influences the shape of the interaction curve, the question of optimal exponent selection remains unresolved. Further research into this topic has been carried out by Bauer et al. [41] and proposed possible exponents. Overall, when considering welded fatigue data, the results tend to follow an interaction-type relationship, supporting its applicability for fatigue life estimation. However, the choice of a perfectly fitted interaction relationship cannot be concluded from this sub-investigation, highlighting the uncertainties that remain in this field.

Another important aspect is the practical implementation of each method. When ranking the criteria by ease of use, the BSI approach is clearly the most user-friendly. It requires only a classification of the load case and selection of an appropriate S-N curve. Next in terms of practicality are the equivalent stress range methods, such as those in DNV and ASME, which rely on clearly defined stress combinations and simple correction factors. At the more complex end of the spectrum are interaction equation-based methods, such as those from IIW and FKM, which require iterative numerical solutions to estimate fatigue life. This added complexity is only necessary when a direct fatigue life calculation is required, as it involves additional post-processing with the iterative solution. In contrast, if the goal is merely to verify the stress state, the method involves simply comparing the stress against a comparison value, which keeps the approach relatively straightforward.

These observations provide useful context for the comparative evaluation of NP correction methods in the following chapters, particularly regarding their balance between accuracy, theoretical consistency, and usability in engineering practice.

Figure 5 presents the plots for the benchmark methods: maximum principal stress and von Mises. It is evident that the data under proportional loading conditions lie on the conservative side, suggesting that the use of a 97.7% confidence interval contributes to a conservative fatigue life prediction.

In this context, conservativeness is represented by a prediction ratio greater than 1, while a ratio below 1 would indicate non-conservativeness. The prediction ratios for proportional loading are 6.33 and 10.00 for the maximum principal stress and von Mises methods, respectively. These values reflect the expected conservative behaviour inherent in design methods and serve as a baseline for evaluating other approaches. While the von Mises method is generally more conservative than the maximum principal stress method, both remain within the bounds of safe design practice.

Under non-proportional loading, a reduction in predicted fatigue life is observed. Specifically, the prediction ratio decreases by 47% for the maximum principal stress method (from 6.33 to 3.34) and by 43% for the von Mises method (from 10.00 to 5.56). This reduction supports the widely accepted understanding that non-proportional stress states in welded steel structures have a reducing effect on fatigue life. Highlight the need for a correction method to achieve the expected inherent conservatism.

After establishing the standard inherent conservativeness expressed by the prediction ratios, the same approach can be applied to the five design codes. The results are visually illustrated in the Fig. 6 and the bar plot in Fig. 7, further summarized in Table 2, and discussed in the following.

For reference, an ideally well-calibrated NP correction would bring the prediction ratio for NP loading (yellow bar) to the same level as the proportional case (blue bar), restoring the same level of inherent conservativeness while avoiding excessive overprediction.

The estimated range of inherent conservatism, based on the benchmark methods, lies between prediction ratios of 6.33 and 10.00. Within this context, the DNV and BSI methods fall below the range, ASME slightly exceeds it, while IIW and FKM lie within the expected bounds. Methods falling below this range are considered less conservative, but still demonstrate prediction ratios around 3, suggesting they remain on the safe side while providing estimates closer to experimental results. The IIW and FKM methods align well with the expected level of conservatism, making them more suitable for practical design purposes. ASME, although slightly more conservative than the range, still presents an acceptable safety margin for design purposes.

Regarding the NP stress state, all methods confirm a reduction in predicted fatigue life, consistent with the known effects of NP loading. However, the extent of this reduction varies across methods. When evaluating the NP correction approaches provided by the five design codes, none fully restores the original level of design conservatism observed under proportional loading.