Re-evaluation of weld root fatigue strength for load-carrying fillet welded joints using the notch stress concept

Re-evaluation


Introduction
There is a growing body of literature that recognizes the importance of the fatigue behavior of the load-carrying fillet-welded joints (LCFWJs) [1]- [3], playing a significant role in steel constructions.Fatigue strength of hidden slit notches is an essential feature of LCFWJs principally evaluated by nominal stress [4] and notch stress method [5], [6].
The past decade has seen considerable literature that has grown up around the theme of welded structures evaluation, which has been typically carried out using nominal stress   , computed from sectional forces or moments in the critical cross-section of the structural component [7].
Developments in the nominal stress approach have led to design codes and design guidelines such as the IIW recommendations [8], [9], and European recommendations and codes [8], [9], provided the S-N curves to compare   with endurable nominal stress amplitudes.Present design codes are limited to axially loaded LCFWJs and do not report the consideration of bending stresses.However, Ahola et al. [10] have proposed an analytical method for computing nominal weld stresses under out-of-plane bending.The evaluation of weld root failures using structural hot spot stress methods has not been officially included in design codes, although Fricke et al. [11] and Sørensen et al. [12] have given proposals to assess weld root fatigue strength using the structural weld stresses.
Since fatigue evaluation of welded joints by local methods was reported in 2006, modeling different types of welded joints, including notch and analyzing stress [5], have been attracted a lot of interest [16], [17] and led to feasible guidelines considering different situations such as IIW fatigue design recommendations by Hobbacher [18].A class of methods based on the critical distances categorized as TCD are also being proposed and investigated in welded components.In a study by Baumgartner et al. [13], [14] evaluating the steel welded joints, critical distances of 0.1 mm and 0.6 mm have been found considering the maximum principal stress hypothesis for the joints in the TIG-dressed and as-welded condition, respectively.Moreover, Karakas et al. [15] determined the critical distance of 0.06 mm for welded joints made by magnesium.The above-referred TCD studies consider weld toe and weld root conditions in their analysis.
To show the application of the notch stress method, Radaj et al. [5] provide a comprehensive background of the effective notch stress method, applying the Neuber's concept of the fictitious notch rounding [20].Besides, In a study conducted by Karakas [21], the application of Neuber's effective stress approach to determine fictitious notch radius based on microstructural length for magnesium welded joints has been demonstrated.Using the notch stress method, researchers have been able to evaluate weld toe and root failure for each type of welded joints [16].Generally, for steel with plate thickness  ≥ 5, the reference radius of 1 mm typically is implemented using finite element models coupled with FAT 225 curve [18] and maximum principal stresses, which is independent of the location of the failure.In 2010, Pedersen et al. [22] conducted a study to re-analyze more than 30 fatigue test series, employing the effective notch stress method, comprising different types of welded joints.They recommended a lower FAT class (FAT 200) for effective notch stress method, to obtain results as safe and conservative as the nominal stress approach.Previous research [19] has established that aluminum welded joints with plate thickness 5 mm ≤  ≤ 25 mm could be assessed independently from the weld geometries and alloy by the local stress concept.The study has proposed S-N curves considering different stress ratios for the fictitious radius of rref = 1.0 mm.There have existed different types of the notch stress approach, providing over the past years such as reference radius of 0.05 mm, combined with FAT class 630 to assess weld root failure in thin plates [17].However, there is some evidence [23], [24], proving results of weld toe evaluation that apply rref = 0.05 mm with its FAT class are non-conservative.Karakas et al. [25] and Sonsino et al. [26] were the pioneer of the evaluation of the welded joints made by the magnesium alloy AZ31 using the local stress concept (rref = 1.0 and 0.05 mm).They carried out research for the welded joints under different conditions to drive FAT classes based on IIW recommendation for comparison with welded steel and aluminum joints.Karakas [27] has also applied the reference radius and Smith-Watson-Topper approaches to examine the effects of mean stress on fatigue life of welded magnesium joints.Moreover, Sonsino et al. [28] have introduced rref = 0.3 mm and its FAT class (FAT 320) to evaluate plates with a thickness range of 5 mm < t <10 mm.Besides, Bruder et al. [23] have proposed lower endurable stresses for applying rref = 0.3 mm for evaluating weld toe failure.Furthermore, Baumgartner [29] has conducted a review of the notch stress method considering different radii, including rref = 1.0, 0.3, and 0.05 mm.Factors found to be influencing the fatigue strength have been explored in this investigation.In 2020, the application of the reference radius of 0.3 mm to assess the fatigue strength of welded magnesium alloy AZ31 joints has been examined [30].Data from this research show that rref = 0.3 mm for magnesium is not able to be a definite replacement for rref = 1.0 and 0.05 mm.
Although the notch stress method is reliable and flexible to utilize for all types of welded joints, the illuminative evaluation of LCFWJs, which considers root failure is quite limited so that a comprehensive assessment of LCFWJ failing from weld root focusing on the effect of yield strength and geometrical parameters have not been performed, despite its extensive application in connection types in shipbuilding or the fixed offshore constructions.Furthermore, the prior studies [31]- [33] have shown that the notch stress approach with FAT 225 might be nonconservative for evaluating the weld root fatigue strength.
The main focus of this paper is to re-analyze the fatigue strength of LCFWJs, failing from weld roots using notch stress and nominal stress approaches and make a comparison between them, considering yield strength of material and geometrical parameters such as the plate thickness, weld penetration depth, and weld throat thickness.To reach this goal, fatigue data is extracted from published literature, and finite element analyses are conducted to determine stress concentration factors to employ in the notch stress approach.Besides, different reference radii in the notch stress approach are investigated to provide a better understanding of its effect on FAT class proposals.Consequently, the reliability of the notch stress method to assess the fatigue strength of LCFWJ is ensured.

Fatigue test data extraction
The experimental evidence of the constant amplitude fatigue tests extracted from the literature [3], [10], [12], [34], [31], [35]- [48] is shown in summary form in Table 1 and 2. Furthermore, all data are shown in Fig. 1 using nominal weld stresses.Since the notch stress method takes geometric parameters into consideration, only studies with detailed explanations of specimen geometry are chosen for the re-analysis.Only studies evaluating LCFWJs failing from weld root are considered when specimens are subjected to axial force, and welded joints are in the as-welded condition.The yield strength of materials varies from 250 to 1157 MPa, stress ratios are positive, and between 0-0.75, specimen thickness varies from 6 to 28 mm, flank angles are between 25° to 60° and, moreover, main and cross plates are identical in thickness for all specimens.According to the IIW recommendation [18], it is assumed that the knee point is located at 10 7 cycles, considering the constant amplitude.It is worth noting that this study concentrates on the finite life region of small-scale specimens, and stress ranges after the knee point are not considered.Figure 1 illustrates the extracted fatigue data related to LCFWJ in where the stress range in a weld throat (∆  ) and its relationship with fatigue life is provided.2.
The stress range ∆  is determined, considering the actual weld throat section, including the weld penetration depth based on the design codes [18], [50]- [52] in the following way: Where ∆  is the maximum nominal stress in the cross-section of the adjoined structural component, t is the plate thickness, and   is the effective weld throat thickness.Figure 2 shows the geometrical parameters of LCFWJ.

Nominal stress approach
In order to assess the weld root fatigue strength of LCFWJs, the nominal stress method is applied.Data points have been statistically evaluated by the standard fitting procedure to achieve the characteristic design and mean strength.First, the least square linear regression in the form of   =  − ∆ has been applied to obtain the survival probability at Ps = 50% as FAT mean.Where m is known as the slope of the curve in log-log with data points regarding   as a function of ∆.  and ∆ are corresponded to the response and independent variable respectively.Generally, a calculation of a design curve at Ps = 97.7% is made using  ℎ =   −  1 .v.A multiplication factor of  1 is available in IIW recommendation and Stdv is the standard deviation of  [18], [53].
Based on the statistical evaluation of the test results, the mean stress-life curve is obtained by: Where ∆  is the nominal stress range which can be obtained using Eq. ( 1), N is assigned to the permissible number of cycles related to ∆  , ∆  is the reference fatigue strength (FAT) at 2•10 6 cycles.K is also introduced as the slope parameter of the S-N curve [18].
It can be seen from the fatigue data in the nominal stress system in Fig. 3, the survival probability at Ps = 50%, and 97.7% computed to obtain the characteristic design and mean strength.What is noticeable in Fig. 3 is that the resulting design curve with K = 3.18 obtained from free regression analysis comes to an agreement with FAT 36 coupled with K = 3 recommended by IIW.Besides, the scatter ratio (Tσ) between the fatigue strengths for Ps = 90% and 10% is shown in the Fig. 3.
As the data in Fig. 3 shows, there exists agreement between extracted data and FAT 36, and only very few data points are located below the S-N curves.Stress ratio, various thicknesses, misalignment, as well as weld quality might be given rise to observable large scatter in the results.Although IIW recommendations of FAT values are given for R = 0.5, the recent literature [31], [45] on root side failure has revealed that the effect of stress ratio (R) on load-carrying fillet welded joints is not significant compared to those with weld toe failure.Data points obtained from [45] in Fig. 4 shows that the difference in the fatigue strength for R = 0 and R = 0.5 is very slight.Besides, the effect of R on welded joints made of ultra-high-strength steel (UHSS) investigated by Ahola et al. [31] demonstrates that UHSS welded joints under load-carrying conditions are not highly sensitive to stress ratio.

Notch stress approach
The notch stress method perceives the escalation of local stress at the rounded weld root or weld toe in accordance with the elasticity theory.Averaging linear-elastic stresses on a definite area takes into consideration the micro-structural support effect using a stress gradient on fatigue behavior.The inspiration for this concept is that the decrease in stress in a notch caused by averaging linear-elastic stresses over a certain depth might be obtained by fictitious notch radius ρ [18], [54].
The determination of fatigue-effective notch stress at the weld root, was conducted using finite element analysis employing two-dimensional plane strain element models.The maximum principal stress criterion was applied in the determination of notch stresses.However, the von Mises equivalent stress might be applied if smaller FAT classes utilized since in sharp notches, the equivalent stress generally diminishes [54].There exist two alternative shapes for modeling the notch root, namely the keyhole and U-shaped notches.Based on [16], comparing the Ushaped notch with the keyhole notch, the high-stress concentration decreases in the U-shaped notch, and a slight underestimation of the notch stress to evaluate fatigue strength in root face could be assumed.The keyhole notch-shape is, therefore, applied in this study.
Mesh of the root side is implemented, considering Hobbacher recommendation [18].It shows the accuracy of the model is improved by applying the mesh refinement in the region around the weld root.All weld details are uniformly modeled using the finite element method to draws a meaningful comparison between the reference radii.Figure 5 shows the specification of the mesh applied in the root side.The steps of the finite element modeling are as follows: • 2D finite element models (plane strain) • the double-symmetry is used by analyzing a quarter of the structure • weld root is rounded with a keyhole shape • quadratic displacement function • linear-elastic analysis • the mesh refinement at a weld root and weld toe Fig. 5. Mesh applied for the notch stress analysis.

Keyhole Shape Quadratic shape function Approx. 100 Elements over 360 °
Based on the above technical recommendations, the 2D finite element models using the doublesymmetry are applied, and only a quarter of the structure is analyzed to obtain the stress concentration factor (SCF) of LCFWJ.The fatigue effective Kf can be obtained from the SCF Kt using a fictitious notch radius of rref such that Kf = Kt (rref), on the assumption that there are no misalignments and also SCF due to bending loads, does not exist [5].The extracted data points in the nominal stress system ∆  can be converted into the notch stress system ∆  using Eq. ( 3).The detailed of the procedure is available in [16], [18], [55].
The specimen geometry and corresponding finite element model, from the study of Frank [37], (N-type), is shown as an example of the procedure of obtaining fatigue notch factors Kf.A unit stress is applied to the specimen.Figure 6 illustrates stress result that can be translated as SCF Kt or fatigue notch factors Kf.Notch stresses are computed for each specimen, and compared with endurable stresses, provided as S-N curves in the IIW recommendation [18].Using the above-described procedure, the assessment of the data points based on the notch stress approach applied reference radius of 1 mm, is shown in Fig. 7.For evaluating the fatigue data, FAT 225 for principal stresses provided in the IIW Recommendations, is utilized.
Nonetheless, as reported by Sonsino [17], von Mises stresses with FAT 200 might be regarded for the evaluation of the notch stress approach with rref = 1 mm.The graph reveals the design S-N curve is not conservative at all.What is interesting about the data in this graph is that about 90% of the uncovered data points assign to those plate thicknesses are between 8 and 10 mm.

Applicability of notch stress system using reference radius of 1.00 mm
Since the geometrical parameters such as plate thickness and weld thickness are explicitly taken into account through modeling in the notch stress approach, the macro effects on fatigue strength can be reflected, in the other words, SCF reflects the influence of the geometrical parameters on fatigue strength.This interpretation contrasts with the nominal stress method in which the cumulative effects of the parameters on the fatigue strength are presented.

Influence of plate thickness
In the present section, the effects of geometrical parameters, such as the plate thickness, weld penetration depth, and weld throat thickness, on weld root fatigue strength are examined in detail.Figure 8 provides a comparison between the results of the nominal and notch stress systems evaluating the thickness effect on fatigue strength.The data points with penetration depth equal to 0 are used to ensure comparability.There is proof of the existence of the thickness effect in the notch stress method, as shown in Fig. 8 (b), so that fatigue strength goes up along with the plate thickness, which might be related to the increased SCF of the thicker plate.

Influence of weld throat thickness
To determine the difference between nominal and notch stress methods to examine the weld throat thickness effect on fatigue strength, data points based on weld throat thickness (a), which are normalized by main plate thickness (a/t), are classified, see Fig. 9.As the correlation between weld throat thickness and fatigue strength is shown in Fig. 9, growth in weld throat thickness gives lower fatigue strength in the notch stress approach (Fig. 9 (b)).This is in agreement with the sensitivity study performed in [56] which proves higher FAT classes for load carrying joints with lower nominal weld throat thickness.On the other hand, because there does not exist a clear trend in the data points converted to the nominal stress approach, it is difficult to put an accurate interpretation on the results in Fig. 9 (a).

Influence of penetration depth
The relationship between weld penetration depth and fatigue strength is shown in Fig. 10  However, the notch stress method with the proposed FAT 225 leads to more non-conservative estimations especially for low penetrations.This stems from the fact that these joints showing higher initial crack sizes in fracture mechanics point of view, simultaneously causing higher notch stresses due to the decrease in the ligament size.

Influence of material strength
This section contributes to examine the yield strength effect of base material on weld root fatigue strength.Applied data points, chosen for this observation, have the same plate thickness (t = 10 mm), and the zero-penetration depth to eliminate the plate thickness and weld penetration effects and thus ensure the comparability of the results.Figure 11 compares the results obtained from nominal and notch stress approaches.As shown in Fig. 11 (a) and (b), there is a clear trend of decreasing fatigue strength while yield strength increases.What stands out in Fig. 11 (b), related to the notch stress system, is the sensitivity to the changes in yield strength ≥ 600 MPa, which creates a gap separating its data points from other categories.
Surprisingly, data points with yield strength ≥ 600 MPa are not covered by FAT 225 in the notch stress method, see Fig. 11 (b).Expectedly, the higher notch sensitivity of materials with higher material yield and ultimate strength leads to decreasing fatigue strength for high-and ultra-high-strength steels [57].However, it should be noted that a limited number of test results were found for LCFWJs made of high-and ultra-high-strength steels, and further verification should be conducted.Moreover, further studies that address the notch sensitivity in accordance with the material strength should be undertaken.al. [28] for evaluating a thickness range of 5  ≤  ≤ 10 .The design S-N curve for principal stresses with FAT 320, corresponding to FAT 280 for the von Mises stresses provided to apply with the reference radius of 0.3 mm for both root and toe failure, see Fig. 12. Sonsino et al. [28] recommended using shallower slope (K = 5) when the utilized radius is rref = 0.3 mm.In the following diagram (Fig. 13), the data points assessment of LCFWJ, failing from the root with a thickness range of 5  ≤  ≤ 10  shows the recommended S-N curve is conservative for N < 10 6 .However, the S-N curve underestimate the weld root fatigue strength at high load levels.The scatter range diminishes and equals to 1:1.83.Nevertheless, the number of data points, applied in this approach is less than the previous one.On the other side, a statistical analysis of the data points is carried out and indicated that a lower FAT class (FAT 255) with K = 3 can be considered to cover all data points which leads to conservative results at low and high load levels with the same scatter rang obtained by FAT 320 coupled K = 5.It should be noted that the statistical analysis of the data indicates the negative inverse slope of the design curve (FAT 255) is 3.1, but for making a simplification, K = 3 is used.This proposal also keeps the consistency of the curve slope with FAT 255 proposed for rref = 1 mm.To have a general view of applying rref = 0.3 mm for evaluating the root failure in LCFWJs, Fig. 14 presents the results achieved from the evaluation of LCFWJ, considering all thickness range of 5 mm ≤  ≤ 28 mm.From the graph, it can be observed that FAT 240 coupled K = 3 might be covered all data points.However, compared with the data obtained from rref = 1.0 mm in Fig. 7, the scatter range increases from 1:2.27 to 1: 2.34.The results of Fig. 15 indicate that there has been a decrease in the scatter range from 1: 2.37 to 1:2.30 when K reduces from 3.6 to 3, whereas the reduction of K from 3.8 to 3, in Fig. 16, shows a 5 % increase in scatter value.

Overview of the reference radii investigated
In Fig. 17, there are the reference S-N curves with K = 3 driven for assessment of LCFWJ using different radii for thickness range between 6 mm and 28 mm considering the location of failure (root side).The literature [17], [28], focusing on the notch stress concept, has revealed that various parameters, such as loading mode (bending, axial, torsion), stress ratio, thickness, stiffness (flexibility) and stress concentration, are involved determining the slope of the S-N line, yet a complexity arises in the evaluation of the interaction of these parameters.As described in previous sections, the impact of thicknesses can be clarified for the reference radii defined.In order to evaluate the effect of changing reference radii, separate S-N curves have been implemented, including the IIW-design lines and the S-N curves gained from Ps = 50% and 97.7 % for FAT mean and the design curve, respectively.
It is apparent from Fig. 8 (b) that for the root failure of LCFWJs with a thickness range of  ≤ 10 , FAT 225 provided in IIW recommendation is nonconservative.FAT 168 with K = 3, thus, is proposed to obtain conservative results in the notch stress approach.
From the conservative results in Fig 13, it can be seen that FAT 255 with K = 3 might be considered to evaluate the fatigue strength of LCFWJ failing from the root to prevent from underestimating the fatigue strength at high load level in spite of the fact that FAT 320 with K = 5 is originally proposed by Sonsino et al. [28], for a reference radius of 0.3 mm to assess the fatigue strength of the welded joins with thin plate thickness (5  ≤  ≤ 10 ).
Another reported S-N curve is related to rref = 0.5 mm proposed for assessing the fatigue strength considering a thickness range between 6 mm and 28 mm.However, the scatter value obtained from FAT 220 coupled with rref = 0.5 compared with FAT 168 is about 2% higher.
Investigation in the notch stress method using rref = 0.5 mm also shows in contrast to earlier findings achieved from the notch stress method with rref = 0.3 mm, rref = 0.5 mm coupled with FAT 210 and K = 3 (Fig. 16), might be employed to evaluate the weld root fatigue strength of LCFWJs with plate thickness between 5 mm to 10 mm leading to a lower scatter range  The results, as mentioned in section 4.4, indicate that although the negative inverse slope (K) of the curves employed in the previous recommendations is varying between 3 and 5, those of the FAT classes recommended in the current study are kept equal to 3 without sacrificing the accuracy in fatigue life estimation, maintaining the consistency with the IIW recommendations.

Conclusion
In this investigation, the main goal was to assess the fatigue strength of LCFWJs, failing from the weld root using notch stress approach with different radii and compare with the nominal stress approach results.The second aim of this study was to investigate the effects of plate thickness, weld penetration depth, weld throat thickness, and yield strength on the fatigue strength of LCFWJs.To summarize the key findings, the following conclusions can be drawn: • The analysis of the data confirmed that, although the root failure in LCFWJs can be conservatively evaluated by FAT 36 recommended by IIW, by the use of the notch stress method with rref = 1 mm, the scatter range can be decreased by 10% and reliably estimate the weld root fatigue strength of LCFWJs employing FAT 168 with K = 3.
• When the whole range of thickness is considered (5 mm ≤ t ≤ 28 mm), either of rref = 0.3 mm with FAT 240 or rref = 0.5 mm with FAT 220 are applicable in fatigue analysis of LCFWJ.
• The results of this research demonstrate that to the weld root fatigue strength of LCFWJs considering the thickness range of 5  ≤  ≤ 10 , the notch stress method (FAT 210) with rref = 0.5 mm might be utilized, which brings lower scatter compared with the FAT 255 and rref = 0.3 mm.But the use of the latter is justified too, due to the slight scatter difference.

Fig. 1 .
Fig. 1.Fatigue data points extracted from the literature listed in Table2.

Fig. 3 .
Fig. 3. Extracted data points in the nominal stress system.

Fig. 6 .
Fig. 6.N-specimen: By applying a nominal stress of 1 MPa, principal Stress distribution and notch

Further analysis in this
study indicates that in contrast to earlier findings, a smaller FAT class (FAT 168) characterized by K = 3, Ps = 97.7% and R ≤ 0.5 might be suggested to obtain conservative results.Observed results from Figs. 3 and 7 show an 11% reduction in the scatter range has been achieved from the nominal stress concept (1 = 1: 2.54) to the notch stress approach (1 = 1: 2.27).Moreover, an S-N curve (FAT mean) calculated by Ps = 50%, R ≤ 0.5 and K = 3 is driven in where a stress amplitude at  = 2 × 10 6 cycles equals 290 MPa.At this point, it is worth mentioning that the negative inverse slope (K) of the curves obtained for FAT mean and the design curve equaled 3.08.But this study makes a simplification and employs K = 3.

Fig. 7 .
Fig. 7.The data points in the notch stress system with rref = 1.00 mm.

Fig. 9 .
Fig. 9.The influence of weld throat thickness on fatigue strength in nominal stress (a) and (a)    and (b) obtained from nominal and notch stress methods, respectively.The trends in both approaches indicate fatigue strength decrease through a reduction in weld penetration.

Fig. 10 .
Fig. 10.The effect of weld penetration on fatigue strength in nominal stress (a) and notch

Fig. 11 . 4 . 2 .
Fig. 11.The examination of yield strength material on weld root fatigue strength in nominal

Fig. 13 .
Fig. 13.The data points extracted from thickness between 5 mm t 10 in the notch stress system

Figure 15 is
also provided for comparison with the results of Fig.14, which is related to reference radius of 0.3 mm.Statistical analysis of Fig.15shows a slight reduction in scatter value compared with Fig.14from 1:2.34 to 1:2.30.

Fig. 15 .
Fig. 15.The data points in the notch stress system with rref = 0.5 mm.

Fig. 16 .
Fig. 16.The data points extracted from thickness between 5 mm t 10 in the notch stress system

Fig. 17 .
Fig. 17. References S-N lines extracted for the notch stress method for thickness range between 6 mm

Table 1
Overview of data collected.

Table 2
Data series of experimental fatigue tests.Kt is defined by means of the notch stress method.
* Information/data not available