Comparison between PSM and IBESS approaches for the fatigue life estimation of weldments

In the framework of the fatigue assessment of welded components, several methods are available in design standards such as the nominal stress, hot-spot stress, notch stress and linear elastic fracture mechanics approaches. The present paper aims at comparing two advanced local approaches for the fatigue strength assessment of different welded joints made of steel. The first one is IBESS which is based on short crack fracture mechanics. The second one is the PSM which instead involves the strain energy density approach. Both methods will be briefly presented, and the fatigue life prediction results discussed. The results obtained for the joint geometries considered in this work show advantages and drawbacks of the approaches, which are thoroughly analysed as well.


Abbreviations
Parameter in the crack propagation law c wi : Coefficients accounting for mean stress effect of ith loading mode (i = 1, 2 or 3) d: Average FE size of the adopted free mesh pattern d max : Maximum FE size for the PSM application d SED : FE size inside the structural volume R 0 adopted for the SED calculation da∕dN: Fatigue crack propagation rate 1 3 e i : Coefficients for calculating ΔW as function of 2 and (i = 1, 2 or 3) E: Young's modulus E ′ : = E For plane stress and E∕ 1 − 2 for plane strain conditions f (R): Mean stress correction according to IIW f wi : Coefficients for calculating eq,peak (i = 1, 2 or 3) h: Reinforcement height J: J-Integral (monotonic loading) k: Slope of finite life fatigue ( S − N ) curve in double logarithmic scale K i : NSIF parameters relevant to opening, sliding and tearing loading modes (i = 1, 2 or 3) K * FE , K * * FE , K * * * FE : Non-dimensional NSIF parameters based on the PSM l: Root face length L: Weld width m: Exponent in fatigue crack propagation law N: Number of loading cycles N A : Reference number of loading cycles equal to 2 × 10 6 N f : Number of cycles to failure p: Fitting exponent of the second term in the crack propagation law PS: Probability of survival R: Stress ratio ( min ∕ max ) R 0 : Size of structural volume in which the SED is averaged r, , z: Coordinates of cylindrical reference system s: Index to identify adjacent vertex nodes for the application of the PSM in 3D FE models S: Stress T: Plate thickness T : Scatter index U: Crack closure factor ( ΔK ef f ∕ΔK) U LC : Crack closure factor for long cracks U SC : Crack closure factor for short cracks z: Weld leg length in cruciform joints and longitudinal gussets 2 : Opening angle of a sharp notch Δa: Crack extension Δ ref : Reference strain range ΔJ: Cyclic J-integral ( ≠ J max − J min ) ΔK: Stress intensity factor range ( = K max − K min ) ΔK ef f : Effective K-factor range ΔK p : Plasticity-corrected ΔK ΔK th : Fatigue crack propagation threshold ΔK th,ef f : Intrinsic fatigue crack propagation threshold (no crack closure effect) ΔK th,LC : Fatigue crack propagation threshold in the long crack regime ΔK th,SC : Fatigue crack propagation threshold in the (physically) short crack regime ΔL r : Ligament yielding parameter for cyclic loading ΔW: Averaged SED range ΔW FEM : Averaged SED range calculated from the FE model Δ : Stress range (maximum value minus minimum value) Δ app : Applied stress range (refers to cross section without crack) Δ nom : Nominal stress range Δ eq,peak : Equivalent peak stress range according to the PSM Δ eq,peak,PSM : Equivalent peak stress range calculated applying the PSM Δ eq,peak,SED : Equivalent peak stress range calculated starting from the SED Δ eq,peak,PSM : Average equivalent peak stress range calculated applying the PSM Δ eq,peak,SED : Average equivalent peak stress range calculated starting from the SED Δ ij,peak : Peak stress range for opening or sliding or tearing local stresses Δ ij,peak : Average peak stress range for opening or sliding or tearing local stresses Δ ref : Reference stress range : Stress 0 : Reference yield stress 11

Introduction
The aim of the present paper is a comparison of two methods for the prediction of the fatigue strength of weldments. The peak stress method (PSM) has been developed at the University of Padova [1,2], and it is an engineering FE-oriented tool for the rapid estimation of notch stress intensity factors (NSIFs) thanks to the use of coarse 2D or 3D FE analyses. The PSM applies to weldments at the weld toe and weld root, whose geometry is idealised as sharp V-notches with null radius. Due to the sharp V-notches, external loads generate singular stress fields that can be used for the estimation of the time spent for initiation and propagation of a small crack in the vicinity of the V-notch tip. In principle, any fatigue model based on NSIFs can take advantage of the PSM and in the present investigation the PSM is complemented by the strain energy density (SED) approach for the fatigue strength assessment of welded joints. The IBESS method has been developed, among others, at BAM Berlin [3,4] for the determination of the fatigue life of welded joints based on short crack fracture mechanics. For the application to weldments with cracks originating at the weld toe, parametrized analytical solutions as a function of the local weld toe geometry are provided for input data such as the stress profile in wall thickness direction. Other special features are an option for establishing an initial crack size based on a crack arrest criterion and the treatment of multiple cracks at stress levels above the endurance limit.
Both methods and the nominal stress approach according to IIW are applied to 10 data sets of three different weldment types, two materials and as-welded as well as post-weld heat treated conditions. This will be exercised in Section 3. In the next sections, brief introductions are given to the nominal stress approach (Section 2.1), PSM (Section 2.2) and IBESS (Section 2.3).

Conventional nominal stress approach
A widely used approach for the assessment of cyclically loaded weldments is the nominal stress approach based on FAT classes, whereby FAT is the acronym for "fatigue". FAT classes are predefined fatigue strength values in terms of stress range of an S-N curve at N = 2 × 10 6 loading cycles. Each class is lowered by certain and fix percentage. The principle is illustrated in Fig. 1. Documents such as the IIW guideline [5], the Eurocode [6] or the FKM guideline [7] contain FAT classes of structural details obtained on experimental basis. If the FAT class of a weld in a structure is known, the complete S-N curve can be constructed. The finite life branch is assumed to have a slope k of 3 for normal stress perpendicular to the weld seam and it passes through the point defined by the FAT class.
In case of IIW recommendations, the finite life branch of the S-N curve is terminated at 10 7 cycles where it passes into either the endurance limit or a very high cycle fatigue branch with a slope of, e.g. 22 [5]. Eurocode 3 instead defines the constant amplitude fatigue limit at N = 5 × 10 6 loading cycles, while for variable amplitude spectra, the fatigue strength has to be assessed basing on extended curves with a slope k = 5 up to N = 10 8 loading cycles.
Note that IIW, Eurocode and FKM use the same FAT diagrams, but they differ in the stress ratio R = min ∕ max which they use as their basis: R = 0.5 for IIW; R = 0 for Eurocode ;R = − 1 for FKM. The discrepancy is compensated by different mean stress corrections. No detailed discussion on this will be provided here since only the IIW approach will be applied (see, however [8]). IIW uses the following expressions: Equation 1a is valid for unwelded base material, wrought components with negligible residual stresses (lower than 0.2 Y ) and stress-relieved welded joints; Eq. 1b is reported to be valid for small-scale thin-walled welded components containing short welds. For complex two-or three-dimensional cases with global residual stresses and thick-walled components, it is suggested that f (R) = 1 is to be used [5].
In the general case, f (R) = 1 should not be exceeded. Note that the FAT classes are given in terms of the stress range Δ , and not the stress amplitude a (as it is frequently the case in conventional fatigue) and a lower percentile value of probability of survival PS = 97.7%.

The PSM
The peak stress method (PSM) is a finite element (FE)-based method to rapidly estimate notch stress intensity factors (NSIFs) K 1 , K 2 and K 3 for the three loading modes at a sharp V-shaped notch, like for example in the case of a tube-toflange welded joint reported in Fig. 2a. By means of a linearelastic FE analysis with a relatively coarse mesh, the method correlates the peak stresses evaluated at the V-notch tip with NSIFs, by means of non-dimensional coefficients according to the following relationships: In the previous equations, the parameter d is the average element size of the adopted FE free mesh which typically the FE analyst has to input before running the free mesh generation algorithm of the employed FE software; i are stress singularity degrees which depend on the opening angle 2 of the V-notch [9,10]; , =0,peak , r , =0,peak and z, =0,peak are the peak stresses defined in the local cylindrical coordinate system and evaluated at the node located at the V-notch tip; as an example, , =0,peak means the opening stress acting in the direction normal to the notch bisector (see Fig. 2b). K 1 , K 2 and K 3 are NSIFs related to opening, sliding and tearing loading modes respectively, originally defined by Gross and Mendelson [11] with the following equations: relative to the cylindrical coordinate system depicted in Fig. 2b. The linear elastic local stress components , r and z are calculated close to the notch tip ( r → 0) and along is typically a burdensome procedure because it requires very fine FE mesh patterns to calculate accurately the local stress field at the V-notch tip [12]. Conversely, the PSM requires significantly coarser FE mesh patterns and therefore provides a tool for the rapid estimation of NSIFs.
The non-dimensional coefficients K * FE , K * * FE and K * * * FE of Eqs. (2)(3)(4) have been originally calibrated for 2D finite elements [1,13,14]; afterwards, calibration has been extended to 3D FE models [15][16][17] and to commercial FE software packages other than Ansys [18]. A comprehensive review of the PSM has been reported recently [2] to which the reader is referred for the operating instructions to apply the PSM Fig. 2 a Assumption of the NSIF-based approach in fatigue design of welded joints. Example of a partial penetration tube-to-flange welded joint. The sharp V-notch opening angle 2 is typically 0° at the weld root and 135° at the weld toe. The circularly shaped structural volume of radius R 0 is centred at the weld toe or at the weld root according to the averaged SED approach. b Cylindrical reference system ( r, , z ) centred at the weld toe and local stress components within its range of applicability, which will be recalled later on.
The PSM being a method for an approximate, rapid, FE-oriented evaluation of the NSIFs, any fatigue strength criterion based on the NSIFs can in principle take advantage of the PSM. In particular, the PSM has been applied to the fatigue model based on the strain energy density (SED) averaged over a circular-shaped structural volume of radius R 0 , as shown in Fig. 2a. According to Lazzarin et al. [19][20][21], the averaged SED over a structural volume ΔW can be used as damage parameter for the fatigue strength assessment of weldments and calculated with the following expression based on the NSIFs: In the previous equation, E is the material modulus of elasticity and parameter e i depends on the opening angle 2 and Poisson's ratio . R 0 is the radius of the circularshaped structural volume which has been found equal to 0.28 mm by Livieri and Lazzarin [19] in the case of steel weldments with weld toe failure. To take into account the effect of misalignments on the weldment, Fischer et al. [22] reviewed the SED approach and concluded an increased R 0 at the weld toe, i.e. 0.32 mm.
It is possible to equal the expression of the averaged SED valid for a triaxial stress state, Eq. 8, with the expression of the SED for an equivalent uniaxial plane strain state: By substituting Eq. 8 in Eq. 9, the equivalent peak stress can be made explicit according to the following expression [2]: The coefficient c wi accounts for the mean stress effect; therefore, their values depend on the load ratio R i of the relevant loading mode (i = 1, 2 or 3) [2]: In Eq. 10, parameters f wi depend on the opening angle 2 and the Poisson's ratio , the average finite element size d adopted in the FE analyses and the material structural volume with size defined by R 0 . The expressions of f wi are as follows [2]: Eigenvalues i and parameters e i are reported in the literature for a range of opening angles [2]. However, since the opening angles of the weldments reported in the present investigations are different from the exemplary ones reported elsewhere [2], f wi has been calculated for each opening angle 2 using Eqs. (12)(13)(14).
The PSM has been originally formalised for V-notch opening angles 2 ≤ 135°, and coefficients K * FE , K * * FE and K * * * FE are valid within the error in NSIFs estimations defined in [2]. When 2 > 135°, Eqs. (2-4) of the PSM cannot be applied, strictly speaking. In such circumstances, the SED parameter can be calculated directly from a FE analysis ( ΔW FEM ) according to Eq. 15: where the strain energy density ΔW FEM,i is evaluated at the integration points of the ith finite element included in the structural volume (or area in 2D problems) having radius R 0 . Then, the equivalent peak stress can be evaluated by means of Eq. 9, where now ΔW must be interpreted as ΔW FEM according to the so-called direct approach.
In what follows, all weldments tested in the IBESS project and considered later on in the present paper have angles greater than 135°. More precisely, butt joints have 2 = 148° and 155°, cruciform joints have 2 = 143° and longitudinal gussets have 2 = 146°. Bearing this in mind, for butt and cruciform joints, the equivalent peak stress has been calculated from ΔW FEM by using the direct approach (Eq. 9 with ΔW = ΔW FEM ), while for the longitudinal gussets, it has been calculated using the PSM by means of Eq. 10. Whatever the notch opening angle, the equivalent peak stress obtained both from Eq. 9 with ΔW = ΔW FEM and from Eq. 10 has been systematically compared. The equivalent peak stress calculated by means of Eq. 9 with ΔW = ΔW FEM has been reported as Δ eq,peak,SED .
The master scatter band of the PSM has been originally calibrated on approximately 180 experimental data taken from the literature and relevant to welded steel joints with weld toe failures ( 2 ≅ 135°) [23]. The welded joints had very different geometries and size and were subjected to simple axial or bending loadings. Figure 3 reports the master scatter band of the PSM along with approximately 860 experimental data (in terms of Δ eq,peak ) relevant to weld toe as well as weld root failures, which enabled to validate systematically the PSM [2]. The scatter index T resulted equal to 1.90 referred to probabilities of survival of 2.3 to 97.7%. It is worth mentioning that the scatter index of 1.90 is consistent with T = 1.50 referred to probabilities of survival of 10 to 90%, which was found for single test series by Haibach [24]; this outcome supports the conclusion that the PSM fully accounts for the different geometries, sizes and loading mode, and the curves reported in Fig. 3 are master fatigue curves for different probabilities of survival.

The IBESS approach
The IBESS approach uses short crack fracture mechanics to determine the S-N curve and the endurance limit. It is characterized by four main features: (a) the elastic-plastic determination of the cyclic crack driving force, (b) the consideration of the gradual build-up of the crack closure phenomenon, (c) a crack arrest-based criterion for determining a fatigue strength relevant initial crack size for the fracture mechanics analysis and (d) multiple crack analyses for stress levels above the endurance limit. Only the most important aspects will be presented here. For a comprehensive overview of the methodology, the reader is referred to [3,4].

Elastic-plastic determination of the cyclic crack driving force
An elastic-plastic crack driving force is necessary because mechanically short cracks are considered. "Mechanically short" means that the crack size is in the order of the plastic zone ahead of its tip. In IBESS, a "plasticity-corrected" cyclic stress intensity factor ΔK p is determined which is a formally to ΔK transferred cyclic J-integral ΔJ.
ΔJ is obtained by The yielding parameter ΔL r is determined as with Δ app being the applied stress range (cross section without crack) and 0 a reference yield stress for which analytical expressions for weldments are provided in IBESS. The f L r function follows the highest analytical options of R6 [25] and the British standard BS7910 [26]: however, modified for cyclic loading. An important point is that different to ΔK = K max − K min the cyclic J integral ΔJ ≠ J max − J min . Instead, the Δ has to be applied to

Consideration of the gradual build-up of the crack closure phenomenon
Crack closure means the premature contact of the crack faces during unloading in the loading cycle. The phenomenon is caused by a number of different mechanisms. In plasticityinduced closure, the plastic zone ahead of the crack remains into the crack wake when the crack grows. This causes geometrical mismatch of the crack faces. Roughness-induced closure is caused by the asperity of the crack faces and enhanced by crack kinking or branching. Oxid-induced closure occurs when a material is prone to corrosion. At low mean stresses or R ratios, the oxide layer is partially rubbed off by friction when the crack is closed. The material corrodes again this way creating a corrosion debris which grows to a thickness much larger than the original layer. There exist further crack closure mechanisms which shall not be discussed here, but can be examined elsewhere [27]. The crack closure phenomenon is usually characterized by a crack closure parameter U = ΔK ef f ∕ΔK where ΔK ef f characterizes the part of ΔK over which the crack is open. The crack closure phenomenon requires a certain extension of the crack wake. This is the reason why it is not existent at the beginning when the crack initiates, e.g. at a material defect: U = 1 . During the stage of physically short crack propagation, the crack closure effects gradually build up and U decreases until the crack becomes a long one, where eventually U becomes independent of the crack length as illustrated in Fig. 4.
In IBESS, the curve of Fig. 4 is determined by the following expression: In the background is the assumption that the function U SC (a) is mirrored in the cyclic R-curve (Fig. 5).
In Eq. 20, U LC is the crack size independent long crack closure parameter determined by the NASGRO approach [28], U SC is its crack size dependent short crack equivalent, ΔK th,ef f is the intrinsic (closure free) component of the fatigue crack propagation threshold and ΔK th,SC (a) is the cyclic R-curve, i.e. the fatigue crack propagation threshold which increases with crack extension. With respect to the determination of the cyclic R curve, the reader is referred to [3,4] and [29].

Crack arrest-based determination of the initial crack size
This is based on a cyclic R-curve analysis, the principle of which is illustrated in Fig. 6. It combines three pieces of information. First, the cyclic R-curve is introduced (red line). Its origin is fixed by the initial crack depth a i at the abscissa and the intrinsic threshold ΔK th,ef f at the ordinate. Finally, cyclic crack driving force curves ΔK p − a are plotted for different applied loads Δ 1 to Δ 3 . That curve which tangentially touches the cyclic R-curve (in the example Δ 2 ) marks the transition from crack arrest to continuous crack propagation, in other words, the endurance limit. If the initial crack size a i is known, the scheme can be applied to the determination of the endurance limit of components. Vice versa, the principle can be used for the determination of a i . To that purpose, the crack driving force curve ΔK p − a is determined for a tensile plate with a semi-circular crack of varying size loaded at the endurance limit. The cyclic R-curve is still fixed by ΔK th,ef f at the ordinate but can be shifted at the abscissa. If this is done such that the crack driving force curve and the R-curve touch tangentially, a i can be taken from the diagram. It is the size of the crack which later on would just arrest at the stress level of the plain (or material) endurance limit. Actually, this provides a lower bound to a i which is, however, realistic for modern, high-quality welds without large defects such as lack of fusion or similar.

Multiple crack analyses
At stress levels above the endurance limit, multiple crack propagation is to be expected as the result of the varying local geometry along the weld toe which results in a number of hot spots for crack initiation. In IBESS, the weld toe is subdivided into equidistant partitions. Based on statistically processed scan results of the weld toe, each section is assigned a set of geometry parameters as well as an initial crack (Fig. 7). The transition from arrest to continuous growth is given by that crack driving force curve (referring to the load Δ 2 in the example) which tangentially touches the cyclic R-curve [4] A fatigue crack propagation analysis is carried out simultaneously for all the partitions. Depending on the stress level, it turns out that the cracks grow at different rates and that some of them even arrest. When the surface points of two adjacent cracks touch, crack coalescence is assumed.
If the principle is applied to the finite life branch of the S-N curve, the calculations are terminated when a failure criterion (e.g. fracture or a certain crack depth) is met. At each stress level, a number of specimens is simulated. This scheme is repeated at different stress levels. This scatter can be statistically processed for defining S-N curves for different percentile values of probability (Fig. 8).
The determination of the endurance limit (also illustrated in Fig. 8) follows a slightly different philosophy. The endurance limit has to be pre-defined, e.g. for no failure up to 10 7 loading cycles. Note that any other value can be chosen, e.g. N = 2 × 10 6 , which corresponds to the definition of the FAT class. It has been demonstrated that IBESS, in principle, can be applied to the determination of FAT classes when limit values of the weld geometry are used as the input information [30]. Generally, simulations are performed with a number of specimens at stepwise increased stress levels. The target information of the analysis is the number of failure events per trial at each stress level. Below a certain load only run outs can be stated, above another value 100% failure. In between, the ratio of failures to the total number of tests can be statistically processed. The result is a probability function of the endurance limit.

The database
The methods are evaluated on the basis of experimental S-N curves which have been provided in the IBESS project. From the total of 31 curves, a number of 10 curves have been selected for the present study. This representative selection is summarized in Table 1. The variations included: Three weldment types (butt welds, cruciform joints and longitudinal gussets).
-Different weld geometries generated by MAG and TIG welding.
With respect to the materials, heat-affected zone (HAZ) tensile properties were required since the fatigue cracks initiated at the weld toes and propagated most of their life through the HAZ. A comprehensive characterization of the materials is found in [3,4] and shall not be repeated here. Only the hardness profiles across Fig. 7 Partitioning of the weld toe into equidistant sections in IBESS [4] the weld seams and the associated cyclic stress-strain curves are shown in Figs. 9 and 10. The HAZ micro-structure was martensitic-bainitic for both materials.

Application of IBESS to the selected weldments
The aim of the present paragraph is to give to the reader an overview of the input parameters of IBESS; all the details can be found in [4]. The global geometry of the joints and the local weld toe geometrical features are necessary for the calculation of the local stress profiles by means of analytical formulations implemented in IBESS (for butt and cruciform joints). These data are provided for each case study in Figs. 12 to 15. For longitudinal gussets, no analytical solutions for the stress profile are available; thus, the in-depth stress profile has to be input from the user (from a FE analysis for example).
A further geometrical input needed in a fracture mechanics-based analysis is the initial crack shape and size. In IBESS, the initial cracks have been considered semi-circular and the size has been derived from that of micro-notches found at the weld toe. It has been found that the micronotches were statistically distributed according to a normal or lognormal distribution. In case of the steel S355NL, the average depth was 62 µm with a standard deviation of 17 µm. In case of the steel S960QL instead, the average depth was 29 μm and the standard deviation 7 µm.
Material data such as the cyclic stress-strain curve, the cyclic R-curve and the crack propagation curve parameters   must be provided. The fatigue crack propagation within IBESS is described by the following equation: The value of the parameters in Eq. (21) for each material adopted in the IBESS simulations (S355NL and S960QL) is listed in Table 2.

Application of the PSM to full penetration butt and cruciform joints
Full penetration butt and cruciform joints have been analysed considering 2D models; in particular, only a quarter model of each geometry has been modelled due to the double symmetry. The 2D geometry has been meshed using 4-node plane elements (PLANE 182) using Ansys® 18.1 software, which enables the user to set the element technology according to four different options by selecting a proper parameter called "K-option 1": (0) full integration method, (1) reduced integration, (2) enhanced strain formulation and (3) simplified enhanced strain formulation. From Ansys® Mechanical APDL element library, the K-option 1 has been set to 3 (simplified enhanced strain formulation) to be consistent with the original 2D calibration of the PSM [1,18].
The FE mesh has been generated using the free mesh algorithm of Ansys® by setting the so-called global element size parameter d, i.e. the average finite element size. Since the critical point is the weld toe, only mode I stresses have to be assessed there; the minimum FE mesh density ratio a ref ∕d must be equal to or greater than 3, a ref being half the main plate thickness, in order to guarantee that K * FE = 1.38 ± 3% (see Eq. 2) [2] Referring to Table 3, the plate thickness of the doublesided full penetration butt joints is T = 10 mm; thus, the maximum FE size is d = (10/2)/3 = 1.67 mm [2]. Since butt joints analysed have a reinforcement height of h = 1.57 mm and h = 2.02 mm and an opening angle at the weld toe of 148° and 155°, an element size d = 1 mm has been chosen instead of d = 1.67 mm in order to have at least 3 elements along the weld flank [31]. Regarding the full penetration cruciform joints, the plate thickness T is 10 mm according to Table 4. The maximum element size for the application of the PSM is again d = (10/2)/3 = 1.67 mm; therefore, a global element size d = 1.5 mm has been chosen in this case. Another recommendation for the appropriate application of the PSM with 2D models is that 2 elements must share the node located at the singularity point in the case of 2 > 90° [2]. Such condition has been largely verified to be automatically fulfilled by the free-mesh generation algorithm available in Ansys®, and the mesh reported in Table 3 and Table 4 provides confirmation that this condition was met. Table 3 and Table 4 summarize the PSM parameters adopted for the calculations performed on full-penetration butt and cruciform joints. The last row of each table shows the percentage difference between the Δ eq,peak obtained applying directly Eq. 10 ( Δ eq,peak,PSM ) and obtained calculating the SED from FEM (Eq. 9 with ΔW = ΔW FEM ). The difference is higher, the higher is the deviation of the opening angle beyond 2 = 135°, as highlighted previously in Section 2.2. Table 3 and 4 highlight that the deviation between Δ eq,peak,PSM and Δ eq,peak,SED is in the range 15-22%, because Eq. 10 has been used outside its range of applicability. Therefore, Δ eq,peak,SED has been used in these cases.
The SED in the case of 2D models has been evaluated by means of Ansys® numerical code. Four-node plane elements (PLANE 182) have been adopted with a free mesh having a global size of 0.28 mm as reported in Table 3 and according to recommendations reported in the literature [32].

Application of the PSM to longitudinal gussets
A 3D, one-eighth of the whole model of the specimen has been defined using the symmetry conditions. The main dimensions of the longitudinal gusset model adopted for the application of the PSM are shown in Table 5. For 3D models, the PSM has been calibrated considering different element types such as 8-node brick, 4-node and 10-node tetra elements [2]. Hereinafter, only the results obtained applying the PSM with 10-node tetra elements (SOLID 187 of Ansys® library) will be shown. Differently from 2D FE models, in 3D models meshed with tetra elements, the variation of number of tetra vertexes sharing a node along the weld toe line must be taken into account [2,17]. Therefore, a simple engineering approach has been adopted, which consists in using a moving average of peak stresses evaluated at three adjacent vertex nodes, named s − 1, s and s + 1 [2].

3
Afterwards, the average peak stress referred to node n = s is calculated with the following equation: Table 3 Double-sided, full-penetration butt-welded joints geometries and 2D 4-node quadrilateral element FE model for the application of the PSM Having in hands the average peak stress range relative to node n , the equivalent peak stress range is calculated again by means of Eq. 10, simply by substituting the peak stress ranges Δ ij,peak with the average peak stress ranges Δ ij,peak (Eq. 22). A summary of all parameters adopted for the PSM application to longitudinal gussets is reported in Table 5.
Longitudinal gusset dimensions have been taken from [33], and the typical weld seam of longitudinal gussets tested is shown in Fig. 11. It is seen that the opening angle at the weld toe is equal to 2 = 146 • , which was determined by means of experimental measurements. It is worth mentioning that the upper weld toe of all longitudinal gussets has been hammer-peened to avoid crack initiation there.
As a result of the FE stress analysis, the percentage difference between the Δ eq,peak,PSM and the Δ eq,peak,SED resulted equal to − 6%. The small error obtained in this case justifies the use of Eq. 10, without the need for evaluating the SED from dedicated FE analyses.
The SED in the case of 3D models has been evaluated by means of Ansys® numerical code. Ten-node tetra elements (SOLID 187) have been adopted with a free mesh having a global size of 1.5 mm and a local size of 0.07 mm inside the control volumes as reported in Table 5. Indeed, the weld toe line has been divided into small control volumes along its length having both radius and height equal to 0.28 mm in order to analyse the distribution of the averaged SED parameter.

Applied loads and boundary conditions
For butt, cruciform and longitudinal joints, a tensile stress equal to 1 MPa has been applied to the main plate, as shown in Tables 3, 4 and 5, respectively. Symmetry boundary conditions have then been applied to appropriate edges of 2D (or surfaces for the 3D) models.

Postprocessing of FE analyses
Starting from the FAT class according to the PSM (PSM FAT = 156 MPa, relative to N A = 2 × 10 6 loading cycles and to a survival probability of 97.7%; see Fig. 3), the FAT class in terms of range of the applied nominal stress (PSM FAT nom ) can be calculated as follows: (22) Δ ij,peak,n=s = Δ ij,peak,s−1 + Δ ij,peak,s + Δ ij,peak,s+1
The postprocessing followed for the longitudinal gusset was slightly different, because in this case, both weld toe and weld root have been analysed. In particular, at the weld toe, only mode I contribution has been considered in Eq. 10, while at the weld root, all modes (i.e. opening, sliding and tearing) have been taken into account.
By using Eqs. 23a and 23b, the experimental data in terms of nominal stress range Δ nom vs number of cycles to failure N f can be compared with the PSM estimations.

Case study A: butt weld of S355NL, as-welded condition, R = − 1
The experimental S-N data and the estimations of PSM and IBESS, together with the FAT class according to IIW, are given in Fig. 12a. In Fig. 12b, IBESS simulations are shown whereby the mean curves are shown in addition to the 97.7 percentile curves of IBESS and PSM. Moreover, the IIW FAT class curve referred to R = −1 by Eq. 1b (enhancement factor f (R) = 1.3 , with R = −1 ) relative to the 97.7% probability of survival is shown. Some considerations can be obtained from the figure. While a slope of k = 3 of the fatigue curves is pre-implemented in the PSM, the slope regarding IBESS is a result of the analysis. What shows up is a slope of approximately k = 3 in the upper part of the IBESS curve, but the curve becomes flatter at low values towards the endurance limit which was predefined for N = 10 7 in the present case. Instead of a sharp knee, IBESS predicts a smooth transition. Another eye-catching difference concerns the scatter band. In the PSM, the scatter index T = 1.90 is fixed, while in IBESS, the scatter is the result of probabilistic simulations in which the geometrical parameters are randomly extracted from their probability density functions. In the present case, 10 simulations have been performed at each stress level which consequently result in 10 (Δ , N ) points. The data can then be processed statistically. In comparison with the experiment, the scatter band appears to be too large in PSM and too small in IBESS. The PSM scatter band is larger than the IBESS one since the same PSM band is valid for different weldments given that the original scatter band has been obtained considering the results of T-joints and cruciform fillet-welded joints showing weld toe failures. The reason for the small scatter band in IBESS is the omission of the scatter in the da∕dN − ΔK data, i.e. the mean value of the parameter C of the crack propagation law has been (23b) PSM FAT nom = PSM FAT Δ eq,peak,SED ∕Δ nom →= 156 MPa Δ eq,peak,SED ∕Δ nom considered. The assessment of a realistic order of magnitude for this and the implementation in IBESS is subject of ongoing work. Note that PSM predicts a constant scatter in double-logarithmic scale along the finite life branch of the S-N curve while the scatter becomes larger towards the endurance limit in IBESS.
What is not shown in the figure is the simulation of the FAT class by IBESS. This has been done according to the method for determining the endurance limit described in Section 2.3.4 but with the N = 5 ⋅ 10 6 limit. The result is a IBESS FAT class of 193 MPa. The corresponding value of the PSM (named PSM FAT nom ) is 100 MPa, that of IIW but corrected for R = −1 is 104 MPa (all results calculated at 97.7% probability of survival and 2 × 10 6 cycles), as Fig. 12b shows.
The experiments of case A have been carried out on butt welds in the as-welded condition. Consequently, PSM takes into account welding residual stresses by means of the mean stress correction parameter c w = 1 (Eq. 11). In IBESS, residual stresses are neglected although the as-welded residual stresses were available as through-wall as well as surface profiles from measurements and finite element simulations [3,34]. However, it turned out (a) that they were negligible at the position of the weld toe and (b) experiments and finite element simulations demonstrated that existing residual stresses (tension as well as bending) would disappear due to cyclic loading at a stress ratio of R = −1 [3,34]. Therefore, the IBESS analysis without considering residual stresses makes sense for configurations such as the data set of case A.
At higher R ratios, cyclic loading has a similar effect. Finite element simulations even point to the build-up of compressive residual stresses. However, a discrepancy with experimental results is apparent from various reasons which will not be discussed here (but see [3,34]). As a consequence of this evidence, IBESS provides some guidance of how to treat residual stresses in fatigue crack propagation analyses but recommends their omission at the present state of the approach. This at least provides conservative assessments.

Further case studies: B to E
The results of the PSM and IBESS analyses of the four cases including the materials S355NL and S960QL in aswelded and PWHT state and at R ratios of − 1, 0 and 0.5 are shown along with the experimental data in Fig. 13. Only the mean curve of the finite life branch of the predicted S-N curve is shown. It shows up that, different to case A, PSM tends to overestimate the experimental fatigue strength in case B. Comparing Figs. 12a and 13a, experimental data do not show a noticeable difference, although in the former case, weldments were in the as-welded state, while in the latter, they were PWHT. According to Eq. 11, the PSM for the PWHT state and R = −1 adopts a mean stress correction parameter c w = 0.5. This has the effect to increase PSM curves towards higher stress ranges of a factor equal to 1∕ √ 0.5 = 1.41 according to Eq. 10. This justifies the higher PSM curve of the PWHT case B with respect to the as-welded case A. For other cases, the PSM agrees with experimental data.
IBESS provides better predictions with the exception of data set C where it showed up to be conservative. The authors suspect that R = 0 might be the reason. In Section 3.4, it was mentioned that there is some indication that the cyclic loading could lead to compressive residual stresses at the weld toe. It has been also mentioned that further work is required to be certain about this. Let us, for the moment, assume that there are really compressive stresses. What would be the consequence? The real stress ratio, which the crack tip experiences, would be lower than the nominal one for which the IBESS analyses were carried out. In other words, the loading conditions assumed in the analysis have been harsher than in reality. Therefore, the predicted S-N curve has been shifted to lower N values.
This immediately brings up the question why a similar trend is not observed for case D where the R ratio has been even 0.5. A possible explanation is the higher yield strength of the HAZ material of the S960QL steel (cases D and E) when compared with S355NL (cases A-C) (Figs. 12 and 13). Residual stress relaxation is caused by local plastic deformation. Whether and to what extent this will occur depends on the stress concentration at the weld toe and, of course, also on the yield strength of the material. The higher the latter is, the lower the probability of extended plasticity. Note that this explanation is still a speculation which needs further investigations.

Further case studies: cruciform joints of data sets F to H
In this section, one cruciform joint of S355NL (case F) and two of S960QL (cases G and H) are investigated. The former is loaded with R = −1 and the latter with R = 0 . According to Fig. 14, the PSM gives conservative results for data set F and G, while it seems to slightly overestimate the S-N curve for data set H. Except for case H, where results are in agreement with the PSM, IBESS tends to underestimate the experimental S-N curves, even for R = −1 . It is important to understand that the method uses two-dimensional influence or weight functions for the determination of the K factor which then is plasticity-corrected. In any case, the stress profile in wall thickness direction refers to the real component while the geometry function of the K factor solution is provided for a substitute geometry, in the present case a flat plate. Analytical solutions implemented in IBESS for such substitute 1 3

Further case studies: longitudinal gussets of data sets I and J
In this section, two data sets of longitudinal gussets of S355NL are considered which are loaded at R = 0 (Fig. 15). One set of specimens has been tested in the as-welded condition; the other has been post-weld heat treated. The experimental data reveal a common observation on this type of geometries: that there is not much difference between these conditions for R ratio equal to 0 [35]. A difference compared to the butt welds and cruciform joints considered in the previous sections is that no reaction or long-range residual stresses did exist there. This is different for longitudinal gussets which are restrained in themselves. As a consequence, residual stresses of some 10% of the yield strength have been measured at the weld toe of the longitudinal gussets [3,34] which seem the crack to keep open. That the IBESS predictions are nonetheless conservative is probably the result of the difference in stiffness between the component and the substitute geometry for K factor determination such as discussed in Section 3.6. Therefore, although IBESS finally yields conservative predictions, the situation is unsatisfactory.
Longitudinal gussets have been hammer-peened in correspondence to the upper weld toe. This has been done to avoid the crack to start from the upper weld toe and to induce the failure from the lower weld toe. The PSM let to determine what is the most critical point regarding the initiation of the crack by comparing equivalent average peak stress ranges Δ eq,peak at the weld toe and weld root, respectively. As shown in Table 5, the equivalent peak stress range Δ eq,peak at the weld toe is 2.34 MPa while at the weld root is 1.32 MPa. Since the Δ eq,peak of the weld toe is bigger than at the weld root, the PSM predicts that the failure occurs at the weld toe. Indeed, this is confirmed by experimental results. By analysing Fig. 15, it emerges that the PSM is slightly conservative with respect to experimental data, providing however acceptable results.

Conclusions
This work aimed at comparing the IBESS and the PSM approaches for the fatigue life assessment of welded joints. The approaches have been tested against ten case studies comprising different geometries (butt welds, cruciform joints and longitudinal gussets), two steels with different strength (S355NL and S960QL), two residual stress conditions (as-welded and post-weld heat treated) and three stress ratios (− 1, 0 and 0.5). In all cases, the well-established IIW FAT nominal stress approach has been considered for comparison. Fig. 11 Longitudinal gusset weld seam [4] Fig. 12 Estimated S-N curves for butt welded joints (case A) according to PSM and IBESS and comparison with FAT class according to IIW [5]. a Experimental data, PSM and IBESS fatigue curves for 50% survival probability. b PSM and IBESS fatigue curves for 50% and 97.7% survival probability

Case A
The comparison has some significance because the approaches are based on different philosophies. While the PSM is an engineering FE-oriented technique which relies on coarse FE models for the rapid evaluation of the NSIFs at the weld toe and weld root, which are idealised as sharp V-notches, and on the SED approach for the fatigue modelling, the IBESS approach is based on fracture mechanics, considering short and long fatigue crack propagations. The underlying philosophies and working hypotheses result in the following advantages and drawbacks: • The PSM has been calibrated on a large experimental dataset of welded joints. Therefore, it considers a fixed slope of the S-N curve ( k = 3 ) and a fixed scatter index ( T = 1.90 ) independent of joint geometry, steel strength and testing conditions. On the contrary, in the IBESS approach, the slope and the scatter band are the results of rected stress intensity factors. Furthermore, IBESS does not apply to weld root failures. • In contrast, IBESS considers the large scatter of the local geometrical parameters along the weld toe and it can deal with multiple crack propagation by a geometrical partition of the weld seam. This is not the case for the PSM approach, in which the weldment is modelled by average geometrical values. • The IBESS model can predict the fatigue limit as nonpropagating condition of the simulated cracks (all the cracks simultaneously simulated in a multiple crack propagation analysis are arrested), whereas the PSM provides a technical definition of the fatigue limit, i.e. the stress level for a given number of cycles technically considered as runout (typically 2 × 10 6 or 10 7 ).
In light of the previous statements, the results can be interpreted as follows: • In case of butt welds, good predictions of the mean S-N curve are provided by IBESS, except for R = 0 , for which the PSM approach works better. The IBESS approach can predict fairly good also the fatigue limit, but it shows a too narrow scatter band because material parameters have been considered deterministic. • The PSM provided better predictions than IBESS for cruciform joints. The authors suspect that this is mainly due to the analytical solutions implemented in IBESS which provide conservative estimates of the stress intensity factors for this type of joint and therefore conservative fatigue life predictions. • This problem has been magnified for the simulations of the longitudinal gussets, where no specific analytical solutions were available. Furthermore, the in-depth residual stress profile was not considered in the simulations carried out with IBESS. This led to overconservative predictions, whereas the PSM was closer to the experimental results. • In general, the IIW FAT classes showed always conservative estimates.
The comparison with the experimental results pointed out areas of improvement for both approaches. The range of analytical solutions implemented in IBESS should be extended, and information about realistic in-depth profiles of residual stresses and scatter of the crack propagation data is needed to improve the fatigue life predictions. Concerning the PSM approach, the conditions of applicability in terms of geometrical parameters should be extended, in particular to weld toe angles 2α > 135°. Indeed, for the longitudinal gussets, the PSM was successfully used even though the weld toe angle was 2α = 146°. However, for the butt and cruciform joints, the PSM could not be applied outside its range of applicability and the FE determination of the SED has been necessary in order to obtain the equivalent peak stress with good accuracy. For further improving the PSM, an estimation of the fatigue limit of different welded joint geometries should be included.
Funding Open access funding provided by Università degli Studi di Padova within the CRUI-CARE Agreement.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http:// creat iveco mmons. org/ licen ses/ by/4. 0/.