## Abstract

To account for the effect of the stress concentration factor (SCF) at the weld toe on stress intensity factor range ΔK, a magnification factor M_{k} is used. The existing M_{k} solutions given in BS 7910 were developed for the fatigue assessment of flaws at the toes of fillet and butt welds and may not be suitable for assessing flaws at single-sided girth weld root toes, where fatigue cracking often initiates and the weld width is relatively small. Finite element (FE) modeling was performed to determine a 2D M_{k} solution for engineering critical assessment (ECA) of a flaw at the weld root bead toe. The weld root bead profile was uniquely characterized by five variables including weld root bead width, weld root bead height, hi-lo, weld root bead angle, and weld root bead radius. Following a parametric sensitivity study, defect size, weld root bead height, and hi-lo were identified as the governing parameters. A total of 6000 FE simulations was performed, covering different combinations of weld root bead height and hi-lo. A series of automation scripts were developed in the python programming language and the M_{k} solution for each type of defect model was developed and provided in a parametric equation. The accuracy of the 2D M_{k} solutions was confirmed by the experimental data. It was found that the methods and M_{k} solutions currently recommended in BS 7910 and DNVGL-RP-F108 are inappropriate for assessing a flaw at a girth weld root toe.

## 1 Introduction

For practical and economic reasons, pipelines and steel catenary risers are commonly welded from one side. For single-sided girth welds, the fatigue design codes define a higher class S–N curve for the weld cap (Class D according to BS 7608:2014) than for the weld root (Class E) to account for the effect of possible poor weld root profile on fatigue performance. Indeed, fatigue cracking is often found to initiate from weld toes on the root side, rather than on the weld cap side, during full-scale fatigue testing of girth welded pipes as shown in the example in Fig. 1 [1]. The weld shown in Fig. 1, in this case, was made with a copper backing shoe on the inside surface as is sometimes used in the production of girth welds. Compared with the profile at the weld cap, the weld root exhibits two distinctive features: a sharp surface discontinuity and a much smaller weld width. In the examples shown in Fig. 1, there was a step between the weld root and the neighboring pipe surface. The fatigue crack initiated from the corner where there was a high-stress concentration factor (SCF).

*da/dN*) can be described by the Paris power law which shows the relationship between crack growth rate and Δ

*K*which is a function of crack size and stress range. To account for the effect of the SCF at the weld toe on Δ

*K*, a magnification factor

*M*has been proposed [4]:

_{k}*M _{k}* quantifies the change in stress intensity factor as a result of the presence of the surface discontinuity at the weld toe.

*M*increases with increasing weld width

_{k}*w*and decreases sharply with increasing distance from the weld toe in the thickness direction and usually reaches unity at crack depths of typically 30% of wall thickness.

*M*solutions have been generated for cracks at the toes of welds e.g., Refs. [4–6]. They were derived from finite element (FE) models for butt and T-butt joints.

_{k}Although both the two-dimensional (2D) and three-dimensional (3D) *M _{k}* solutions are provided in BS 7910 [7] for engineering critical assessment (ECA), they are representative of the geometry for the weld cap and do not capture the typical geometry at the weld root seen in Fig. 1. Furthermore, the use of these

*M*solutions predicts failure at the weld cap, rather than at the weld root, due to under-estimation of the

_{k}*M*value at the latter. This is because of a smaller weld width

_{k}*w*and lower applied stresses on the inner surface under an applied bending moment. The importance of this issue to the industry is further highlighted by the frequent revisions in DNV-OS-F101 [8–10] and DNVGL-RP-F108 [11] with regard to ECA of a postulated flaw at the weld root:

The DNV-OS-F101:2010 version did not provide specific guidance on how to assess a flaw at the weld root toe.

The DNV-OS-F101:2012 version provided guidance on assessing flaws at both the weld cap and root. For the latter, it specified setting the weld width

*w*equal to that at the weld cap but excluding the misalignment-induced SCF.The DNV-OS-F101:2013 (current) version and DNVGL-RP-F108:2017 specify setting

*w*equal to the weld width on the weld root side and multiplying the nominal stress ranges by the misalignment-induced SCF. However, it is not clear whether this guidance is based on the numerical modeling.

In view of the situation described above, there is a clear need to develop an *M _{k}* solution specifically for assessing defects at the root of girth welds, which the present study aimed to do. Due to the complexity of 3D analyses, it was based on a 2D-FE model, which proves to be a conservative approach for flaws of a finite length and is correct in the case of a straight-fronted crack (

*a/*2

*c*= 0).

## 2 Brief Review of the *M*_{k} Solution in BS 7910

_{k}

The 2D *M _{k}* solution given in BS 7910 is based on the study by Maddox and Andrews [4] where a range of cruciform and butt weld geometries in plate was investigated, including different weld bead width to plate thickness ratio

*w/B*, weld overfill height (

*h*) to plate thickness ratio

*h/B*, and angle between joined plates

*α*. In all of their models, the top and the bottom parts were symmetrical around the mid-wall thickness. Figure 2(a) shows the butt-welded geometry. The

*h/B*and

*w/B*ratios analyzed were 0.1–0.2 and 0.5–1.0, respectively. As the plate thickness was 50 mm, the minimum

*h*and

*w*values investigated were 5 and 25 mm, respectively. Based on the results of the FE modeling, they concluded that

*M*depends strongly on the ratio of weld width to plate thickness

_{k}*l*/

*T*in Fig. 2(a), but not on the relative height of the weld overfill or welded attachment

*h/T*.

The 3D *M _{k}* solution given in BS 7910 is based on the work by Bowness and Lee [6] where

*M*values for semi-elliptical cracks in T-butt welds Fig. 2(b) were determined using 3D FE models. A series of parameters were investigated including loading modes (membrane and bending), crack size, crack aspect ratio,

_{k}*a/c*(crack depth over half crack length), weld angle, weld width (2L in Fig. 2(b)), and weld toe radius

*ρ*. The minimum ratio of weld width over plate thickness

*w/B*investigated was 1.0 and the effect of attachment height was not investigated. It has been verified that the 3D

*M*solution provides a more accurate (less conservative) prediction of fatigue crack growth than the 2D solution [12]. However, the 3D solution was developed within certain geometrical limits: 0.05 ≤

_{k}*a/B*≤ 1, 0.1 ≤

*a/c*≤ 1, and 0.5 ≤

*w/B*≤ 2.75. For example, for

*B*= 20 mm, the crack depth must be ≥0.1 mm and the weld width

*w*must be ≥10 mm. Beyond these limits, the 3D

*M*solution should not be used, and the 2D

_{k}*M*solution is recommended instead.

_{k}## 3 Finite Element Modeling

### 3.1 Background and Software.

*a*, in a structure, the stress intensity factor for mode I is

*σ*is the applied stress and

*Y*is a correction factor depending on loading, geometry, crack size, and crack front shape. To account for the effect of the SCF at the weld toe,

*Y*also includes

*M*.

_{k}Parametric FE modeling was undertaken to generate and analyze over 6000 axisymmetric FE models of pipelines containing fully circumferential, inner surface defects at the toe of an idealized representation of a girth weld root bead. The parametric study was used to determine the influence of the weld root bead geometry on the stress intensity factors for a range of defect depths subject to a remote axial (membrane) stress.

All FE models were generated using abaqus/cae v6.14 and solved using the general static solver, abaqus/standard v6.14. Once the modeling approach had been verified on several test geometries, the model generation and analysis processes were automated by developing a python script that interacted with the abaqus environment.

### 3.2 Geometry.

The geometry of each model comprised an axisymmetric slice of two lengths of pipe joined by a girth weld. The outer diameter of the pipe was fixed at 406.4 mm, and the nominal wall thickness was *B* = 20 mm. The total axial length of pipeline modeled was four outer diameters or approximately 1600 mm. The weld itself was assumed to have a flush ground weld cap on the outer surface and only the idealized geometry of the weld root bead on the inner surface was considered.

The weld root bead profile was uniquely characterized by five variables (see Fig. 3):

Weld root bead width (WRBW)

*w*: The resulting parametric*M*factor solutions were derived in terms of non-dimensional geometric variables, and therefore, the WRBW was normalized by the wall thickness_{k}*B*= 20 mm. The non-dimensional weld root bead width is referenced with an overbar $w\xaf=w/B$.Weld root bead height (WRBH)

*h*: The non-dimensional version of the weld root bead height is indicated by an overbar $h\xaf=h/B$.Hi-lo: Hi-lo is defined as the relative distance between the inner surfaces of the pipe on each side of the weld Fig. 3(c). In this study, hi-lo was a measure of the axial misalignment induced by joining pipes with different wall thickness.

Weld root bead angle

*θ*as shown in Fig. 3(a).Weld root bead radius. In linear elasticity, a geometrically sharp, re-entrant corner feature such as a weld toe generates a stress singularity. A consequence of this is that as the FE mesh is refined in the vicinity of the stress raiser, the solution will not converge. To overcome this numerical problem, the FE models incorporated a 0.05 mm radius into the transition between the inner surface of the pipe and the weld root bead profile. The radius size of 0.05 mm was selected because it was smaller than the smallest defect depth considered (0.07 mm) and a preliminary sensitivity study indicated that it was a suitable choice for the determination of conservative

*M*factors._{k}

### 3.3 Models of Weld Root Bead Profile.

Three distinct models were analyzed to characterize different weld root bead profiles:

Type I model: the weld root bead profile is symmetric about the weld centerline. An illustration of a Type I model is shown in Fig. 3(a). All of the geometry cases analyzed during the parametric study are listed in Table 1.

Type II model: the geometry is essentially two pipes with the same outer diameter but different wall thickness (and the weld root flush with the inner surface of the pipe with the larger wall thickness). An illustration of a Type II model is shown in Fig. 3(b). The Type II model parametric study used the same values of parameters shown in Table 1 except that the values of weld root bead height were excluded.

Type III model: the geometry is that of two pipes with the same outer diameter and different wall thicknesses joined by a weld with the weld root bead extending beyond the inner surface of both pipes. An illustration of a Type III model is shown in Fig. 3(c).

As outlined in Table 1, only the defect depth, weld root bead width, and weld root bead height were varied for the final parametric study, while the weld root bead radius and the weld root bead angle were held constant. This was done because a preliminary sensitivity study showed that the angle and root radius had only a weak influence on the solution. This enabled the parametric study to focus on the variation of a smaller set of parameters.

Note that in Table 1, the values of defect depth are logarithmically spaced between the smallest defect depth 0.07 mm and the largest defect depth 3.0 mm. This distribution of discrete depth values was chosen because, based on the previous work of Maddox and Andrews [4], it was observed that the *M _{k}* factor varies with the logarithm of the defect depth (or more specifically, the logarithm of the normalized defect depth). Therefore, since interpolation and surface fitting are best suited for data that is equally spaced, it was decided to use 20 different defect depths for which the log-normalized values were equally spaced between the smallest and largest values.

For all models, the usual axisymmetric coordinate system was used with the axis of the cylindrical geometry aligned with the global Cartesian *y*-axis, and the radial direction corresponding to the global Cartesian *x*-axis. For convenience, the crack plane was located at the position *y* = 0.

The original work by Maddox and Andrews [4] considered 2D plane strain representations of butt welds and cruciform joints. A key aspect of their work was that, in the absence of any crack-like defects, the neutral axis was constant throughout the length of the geometry. As a consequence, no additional bending was generated by the application of a remote tensile force that was aligned with the neutral axis. The geometry considered in this work focuses on the effect of the weld root bead profile and assumes that the weld cap is ground flush. If the Type I model considered in this work was to have been analyzed using the same kinematic representation (i.e., plane strain or plane stress) as the Maddox and Andrews work, then a global bending moment would have been generated due to the shift in the neutral axis in the weld region, and the calculated *M _{k}* factor would have been influenced not just by the geometry of the weld root bead width and height, but also by the additional, misalignment-induced bending stress. In contrast to the Type I model, for the Type II and III models, a secondary bending stress is present due to the mismatched wall thicknesses either side of the weld region. Therefore, the new solutions described in this paper are specifically for cylindrical bodies with an inner surface defect. If the Type I solutions are to be used for plate-type structures, then an additional misalignment SCF should be included to account for any secondary bending that might occur.

### 3.4 Finite Element Mesh.

Each model was meshed entirely with eight-node, biquadratic, axisymmetric quadrilateral elements with reduced integration (type CAX8R in abaqus). The geometry of the part was partitioned in a way that facilitated a highly refined mesh in the vicinity of the weld root bead geometry and crack while permitting a coarser far-field mesh where the stress gradients were significantly reduced. The crack was modeled as a sharp crack, and the crack tip was meshed using a conventional radial spider web mesh composed of concentric rings of elements surrounding the crack tip. In each model, up to ten concentric rings of elements were generated around the crack tip, and each ring contained between 24 and 48 elements. The innermost ring of elements was composed of collapsed wedge elements with the mid-side nodes shifted to the quarter-point position and single-node degeneracy specified to accurately resolve the $1/r$ linear elastic strain singularity. The mesh gradually transitioned from highly refined to a coarser global mesh. Depending on the local weld root bead profile geometry, the global part typically contained six to eight elements through the wall thickness. A typical FE model used in this study is shown in Fig. 4.

### 3.5 Materials Properties.

Since the work was focused mainly on the calculation of stress intensity factors, only linear elastic material properties were considered. Both the lengths of pipe and weld were assumed to composed of the same homogeneous, isotropic, linear elastic material. The Young’s modulus was assumed to be 205 GPa and the Poisson’s ratio to be 0.3.

### 3.6 Loads and Boundary Conditions.

The objective of each simulation was to evaluate the stress intensity factor for a specific inner surface defect subject to a remote membrane stress. A remote 1 MPa tensile membrane stress was applied by prescribing a suitable concentrated force at a reference node located on the pipe axis at the end of the pipe geometry in the negative *y*-direction. This reference node was kinematically coupled to the end of the pipe to distribute the concentrated load among the FE mesh nodes. The opposite end of the pipe was restrained in the *y*-direction. All degrees-of-freedom except for the displacements in the axial direction (U2) were restrained at the reference node. All simulations were analyzed under the small strain assumption.

### 3.7 Post-processing.

From each simulation, the stress intensity factor, *K _{I}^{FEA}* was calculated by abaqus using an interaction integral method. The stress intensity factor was evaluated along ten contours surrounding the crack tip and path independence was verified. The value obtained from the outermost contour was then used to calculate the stress intensity magnification factor

*M*.

_{k}*M*factor obtained from FEA (

_{k}*M*) was calculated as

_{k}^{FEA}*M*factor in this study because the FEA-based

_{k}*M*factor solutions based on the curved shell geometry stress intensity factor solution in BS 7910 often exhibit non-monotonic behavior. This causes difficulty when defining a polynomial-based parameterization of the calculated solutions that is otherwise removed when one chooses the flat plate solution as the normalizing term. The BS 7910 K solution corresponds to the stress intensity factor for a flat plate with a long surface flaw having the same wall thickness (20 mm), defect depth, and applied remote stress (1 MPa membrane stress) as the FE model. In this way, the

_{k}*M*factor can be used to directly recover the FEA results, since upon re-arranging Eq. (3):

_{k}^{FEA}To be conservative, if *M _{k}*

^{FEA}was calculated to be less than 1.0 then the value was amended and assigned the value 1.0. The interpretation of a case where

*M*= 1.0 is that the stress intensity factor is no longer affected by the weld toe SCF and equals the value predicted using the handbook solution from BS 7910; therefore, the interpretation of a case where

_{k}^{FEA}*M*< 1.0 is that, for the given defect depth, the weld toe SCF no longer affects the stress intensity factor, and the stress intensity factor given by the BS 7910 flat plate solution is higher than the FEA axisymmetric stress intensity factor solution. This occurred for only a few cases where the defect depth was in excess of 10% of the wall thickness and is therefore considered insignificant.

_{k}^{FEA}### 3.8 Finite Element Model Automation.

Due to the need to understand the variation of stress intensity factors for a wide range of geometry parameters and defect depths, a series of automation scripts were developed in the python programming language, since abaqus allows for python scripts to interact with both the abaqus/cae and abaqus/viewer environments. There were two primary scripts:

*createM*This script was used to create an axisymmetric model of a weld root bead defect given pre-defined geometry variables. The script reads in all of the geometry variables from a structured text file and then calls the abaqus/cae environment using the “noGUI” option so that the script can be run in the background without the need for the user to interact in any interface. The script created the model, partitioned the geometry, assigned material properties, established the analysis steps, meshed the geometry, and then wrote the input file. The meshing was defined through a series of detailed parametric rules that took into account the geometry of the weld root bead profile and the defect height. A feature of this was that for very shallow defects, the meshing was exceptionally dense due to the need to accurately resolve the interaction of the weld root bead profile on the crack tip stress field._{k}Model.py:*submitAll.py:*This script was the “master” script for the parametric study. In this script, the user was allowed to specify the range of values for each parameter (WRBW, WRBH, defect depths, etc.). The script then entered a nested for-loop so that for each combination of user-specified parameters, the createM_{k}model.py script was called. Once the geometry-specific input file had been created, the script automatically submitted the job for analysis, waited for the job to complete, and then post-processed the output database (i.e., calculated the stress intensity factor and*M*factor)._{k}

Depending on the availability of computational resources, the submitAll.py script could run approximately 4000 analyses in 35 h or about one job every 30 s. Thus, the development of automation scripts for this project significantly improved the efficiency of the modeling activities, enabling a large increase in the generation of results against the small amount of time required to write the software.

## 4 Development of New *M*_{k} Factor Solutions

_{k}

### 4.1 Summary of the Results: Type I Model

#### 4.1.1 Effect of Weld Root Bead Angle.

The weld root bead angle varied from 70 deg to 110 deg in 10 deg increments and was found to have a negligible influence on the stress intensity factor (and hence on the calculated *M _{k}* factor). This was observed when changing crack sizes but fixing the WRBW and WRBH or when changing the WRBH but fixing the crack size and WRBW. Although the weld root bead angle does have an influence on the stress intensity factor, it is very small ≤3%. Consequently, it was determined that it was suitable to assume the weld root bead angle was constant at 90 deg for the remainder of the parametric study.

#### 4.1.2 Effects of WRBW and WRBH.

The existing 2D *M _{k}* factor solution in BS 7910:2013 is suitable for cruciform and butt weld geometries and is a function of the flaw depth and normalized attachment length

*L/B*(corresponding to

*w/B*for this study). For attachment lengths with normalized values less than 2 (i.e., attachments less than two wall thicknesses), the

*M*factor is an explicit function of the normalized attachment length. However, for L/B in excess of 2, the

_{k}*M*factor solution saturates and becomes independent of the attachment length.

_{k}The FEA results corresponding to different crack sizes and weld root bead heights are shown in Fig. 5 where the weld width is fixed at 5.0 mm (so that *w/B* = 0.25). The existing BS 7910 *M _{k}* factor solution is also included for comparison. In this figure, the

*x*-axis corresponds to the log-normalized defect depth log

_{10}(

*a/B*). When the WRBH is small (

*h*= 0.5 mm, a typical value at the weld root), the

*M*factor obtained from the FEA is significantly lower than the existing 2D BS 7910 solution at small crack sizes. As the height of the weld root bead or the crack size increases, the

_{k}*M*factors obtained from the FEA tend to converge monotonically toward the existing 2D BS 7910 solution. There is a small range of defect depths 0.02 <

_{k}*a/B*< 0.05 in which the existing

*M*factor solution is marginally lower for certain weld root bead heights. The results shown in Fig. 5 at a fixed crack size provide a consistent picture of the results from the FE parametric study: for fixed values of WRBW, the FEA results produce

_{k}*M*factors that are smaller than the existing solution and converge toward the existing solution as the WRBH increases. This demonstrates the benefits of having a new

_{k}*M*factor solution that accurately captures the effect of attachment height on

_{k}*M*.

_{k}An alternative slice of the results is shown in Fig. 6. In this figure, for a fixed defect depth (0.23 mm), the *M _{k}* factors are shown as a function of the WRBW for several different values of WRBH. Since the existing 2D

*M*factor solution does not depend on the WRBH, only one curve is shown. For very small WRBWs (

_{k}*w/B*= 0.15), the existing and new solutions agree and appear to be independent of the WRBH. However, as the weld width increases, there is an increasing spread between the new FE results and the existing solution. The same trend observed in Fig. 5 is reproduced in Fig. 6: for each distinct WRBW, the

*M*factor increases with increasing WRBH toward the BS 7910 2D solution. As noted previously, the existing 2D BS 7910 solution saturates (reaches a constant value) at a normalized weld root bead width of 2.0. This is not shown in Fig. 6, but it can be seen that the FE results tend to saturate more quickly, with almost all the curves shown becoming flat by a normalized WRBW of 1.25.

_{k}In Fig. 7, the *M _{k}* factors are plotted against the normalized WRBH for a fixed defect depth (0.23 mm) for various different WRBWs (3.0, 5.0, and 10.0 mm). In this figure, the lines corresponding to the BS 7910 solution are flat, because the existing 2D solution does not depend on the attachment height as a variable. The results from the FEA exhibit similar behavior to that of the dependence of

*M*on the attachment width: the

_{k}*M*factor increases with increasing WRBH until it reaches a maximum value. After the maximum value is obtained (typically around

_{k}*h/B*= 0.125), the

*M*factor reaches a constant value but tends to decrease slightly from the maximum before saturating.

_{k}In Fig. 8, a contour of the results from the FEA parametric study is shown as a surface, with the height of the surface representing the magnitude of the *M _{k}* factor, for the defect depth of 0.23 mm (

*a/B*= 0.011). This figure highlights the combined trends for

*w/B*and

*h/B*: for combinations of

*w/B*> 1.25 and

*h/B*> 0.15, which is the representative of a typical weld cap, the solution surface becomes flat (red regions). Thus, if the space of points in the parametric study had included larger weld root bead widths and heights, then a saturation value would have been obtained where the

*M*factor no longer depended on the weld root bead width and height. However, the values for

_{k}*w/B*and

*h/B*for which the saturation values would occur are beyond those expected for typical pipeline girth weld roots and were not included in the present study.

#### 4.1.3 Development of the Parametric Solutions for Type I Model.

*M*factor solution for a Type I model that depended not only on the defect depth and the normalized WRBW but also on the normalized WRBH. Based on Fig. 5, the non-dimensional variable α = log

_{k}_{10}(

*a/B*) was defined, where

*a*is the crack depth and

*B*is the wall thickness (20.0 mm). It was assumed that the best representation of the

*M*factors was an equation of the form:

_{k}*w*and

*h*analyzed during the parametric FEA study, the cubic polynomial coefficients

*c*

_{1}–

*c*

_{4}appearing in Eq. (5) were determined using least squares regression. The dependence of

*c*on (

_{i}*w/B*) $w\xaf$ and $h\xaf$ was represented in the form:

In Eq. (6), the variables $w\xaf$ and $h\xaf$ are log-normalized using a logarithm base 10, and a natural logarithm (log base e) is taken of the entire polynomial sum. For each cubic polynomial coefficient *c _{i}* (

*i*= 1, 2, 3, 4), least squares regression was used to determine the optimal

*d*coefficients with the resulting coefficients shown in Table 2. The results indicate that the cubic equation provides very accurate results.

_{jk}#### 4.1.4 Verification of the New Parametric M_{k} Factor Solutions.

The parametric solutions have been verified against all of the simulation data. For each pair of *w* and *h* values, the simulation data for each defect depth were compared with the predicted *M _{k}* factor values for the same defect depths obtained by using Eqs. (5) and (6). To measure how well the simulation results were replicated by the parametric model, both the

*R*

^{2}score and the root-mean-square errors were calculated. The results indicate that the cubic equation provides very accurate results:

For all pairs of

*w*and*h*that were analyzed, the minimum R2 score was 0.992 and the mean*R*^{2}score was 0.997.The maximum root-mean-square error was 2% and the mean root-mean-square error was 0.03%.

The regression was further verified by choosing random width and height values. For example, consider the WRBW value *w* = 7.63 mm and the WRBH value *h* = 1.21 mm. The FE automation code was re-run to determine the *M _{k}* factors for Type I defect depths ranging from 0.07 to 3.00 mm for these specific values of

*w*and

*h*. The newly calculated values from FEA were compared with the predictions from the new parametric solution, and the results are very good. Considering all of the defect heights analyzed, the

maximum relative error from the parametric equations is 0.29% and

average relative error from the parametric equations is 0.12%.

Thus, the proposed new parametric *M _{k}* factor solutions can be considered valid for Type I models over the domain range considered:

wall thickness to the outer diameter (

*B/D*) equal to 0.049;normalized WRBWs

*w/B*between 0.15 and 1.0;normalized WRBHs

*h/B*between 0.0125 and 0.25;normalized defect depths

*a/B*between 0.0035 and 0.15.

### 4.2 Summary of Results: Type II Model

#### 4.2.1 Development of the Parametric Solutions.

The procedure that was applied for Type I models was applied for Type II, except that in the latter model, the WRBW is not an explicit parameter.

*M*factors for two Type II defects (one with

_{k}*h/B*= 0.025 and one with

*h/B*= 0.1) are shown in Fig. 9. As before, it can be seen that a cubic polynomial provides the optimal fit with an

*R*

^{2}score of 1.0, and therefore, the formulation of the

*M*factor equation is still the same as the Type I equation, except that the dependence on weld root bead width is removed:

_{k}A least squares regression, similar to that detailed for Type I defects, was undertaken, and the resulting coefficients *d _{jk}* at each

*c*are determined, and the resulting coefficients are shown in Table 3.

_{i}#### 4.2.2 Verification of the New Parametric M_{k} Factor Solutions.

The accuracy of the new parametric *M _{k}* factor solution for Type II defects has been verified in two ways. First, the result of representing the cubic polynomial coefficients from Eq. (7) by the quadratic form in Eq. (8) was examined. The results showed that, except for some very small variation in the cubic coefficients at low

*h/B*values, the quadratic equation provides a very good approximation to the calculated

*c*values.

_{i}Second, the accuracy of combining Eqs. (7) and (8) to represent Type II defect *M _{k}* factors was assessed. In Fig. 10, for four different normalized weld root bead height values, the

*M*factor solutions are shown. In this figure, the open symbols connected by solid lines correspond to the solutions calculated from the FE models; the solid symbols connected by dashed lines correspond to the solutions calculated using Eqs. (7) and (8). The agreement between the two curves is very good and is representative of the accuracy of the parametric solution for all other values of normalized WRBH not shown in the figure. Specifically, the largest absolute relative error of the approximations shown is 2.3%, corresponding to a very small normalized defect depth for the

_{k}*h/B*= 0.05 curve. The average relative error is less than 1% for all of the curves shown.

### 4.3 Consideration for Type III Model.

Having developed solutions for both Type I and II defects through least squares regression and detailed parametric studies, Type III defects were then considered. A series of Type III models were developed and the resulting *M _{k}* factor solutions were analyzed.

For a model with defect depth *a*, weld root bead height *h*, and weld root bead width *w*, the following variables are defined:

let λ be a variable representing the hi-lo 0 ≤

*λ*≤*h*;let

*M*_{k}^{I}(*a,h,w*) be the Type I*M*factor for the variables_{k}*a*,*w,*and*h*;let

*M*_{k}^{II}(*a,h*) be the Type II*M*factor for the variables_{k}*a*and*h*.

Verification of Eq. (9) is shown in Fig. 11 where simulations for a series of defects with WRBH equal to 1.0 mm and WRBW 5.0 mm were carried out. Three sets of simulations were performed: one each for Type I, II, and III (hi-lo equal to 0.5 mm) defects. The resulting *M _{k}* factors were calculated and are shown as solid curves in Fig. 11. Additionally, the convex combination of Type I and II defects is shown as a green dashed curve. The convex combination replicates the simulation results to within 1–2% accuracy. The same exercise was repeated but with the Type III defects having hi-lo equal to 0.25 mm. Again, the convex combination equation replicates the simulation results to a high level of accuracy.

Therefore, having developed detailed parametric equations for Types I and II defects, it has been demonstrated that the *M _{k}* factors associated with Type III defects can be produced by using a convex combination of the corresponding Types I and II defect solutions.

## 5 Assessment of Accuracy of the *M*_{k} Solution Derived

_{k}

### 5.1 Introduction.

In a Joint Industry Project (JIP) undertaken by TWI, fatigue performance of full-scale girth welded pipes was investigated under both constant amplitude (CA) and variable amplitude (VA) loading [13,14]. Comprehensive post-test examinations were carried out for all full-scale girth welds tested, including measurements of wall thickness, weld root hi-lo, and WRBH at each gauge location. Therefore, the opportunity was taken to use the results from that JIP to assess the accuracy of the FE *M _{k}* solutions developed in this study, in terms of both fatigue crack growth and S–N curve approaches.

The pipes used in the JIP had an outside diameter of 406 mm and a wall thickness of 19.1 mm and were made of seamless steel pipe to API 5L-X70 specification. The girth welds were made from the outside in the 5G position by pulsed gas metal arc welding/gas metal arc welding (PGMAW/GMAW) onto copper backing. The weld cap at each start/stop position was ground flush with the pipe surface. All welds were shown to be acceptable to a typical steel catenary riser specification.

Full-scale girth welded pipes were fatigue tested under either CA or VA loading in resonance rigs in the air at ambient temperature. Details of these full-scale fatigue tests have been reported [1,14]. In summary, the VA loading spectra were divided into six sub-blocks and, in each sub-block, the stress range increased from the minimum value to a peak value and then decreased to the minimum value. The mean stress for both CA and VA loading was constant—150 MPa. The stress ratio varied between 0.25 and 0.74 for CA loading depending on stress range magnitude. The stress ratio in a VA loading spectrum varied between 0.25 at the maximum stress range of 180 MPa and ∼0.7 at the minimum stress range (e.g., 0.71 at a minimum stress range of 50 MPa). Almost all cracking was found to occur at the weld root toes. The failure locations were often characterized by relatively poor weld root bead profiles, with the average WRBH at the worst location along the circumference of the girth welds being ∼0.5 mm.

### 5.2 Estimating Fatigue Crack Growth.

For those specimens tested under VA loading, beach marks were produced on the fracture surfaces. They provide a good opportunity to assess the accuracy of the *M _{k}* solution developed in this project by comparing the crack growth measured on the fracture surfaces with those calculated using the fracture mechanics-based approach, based on the integration of the fatigue crack growth law as detailed in BS 7910. Within each block, there were six beach marks, each corresponding to the peak stress in the loading spectrum.

As *M _{k}* has a strong effect on fatigue crack growth only near the surface, beach marks corresponding to short fatigue cracks are required to examine the accuracy of the

*M*solution. Although beach marks were seen on the fracture surfaces of many specimens, beach marks from relatively short cracks were obtained from only one specimen as shown in Fig. 12. The pipe (No. 17) was tested under a spectrum with a minimum stress range of 50 MPa and a block length of 603,330 cycles. The weld failed after 31.3 blocks (i.e., at 18,884,229 cycles). The first visible beach mark corresponded to a crack depth of 1.6 mm and length of 13.5 mm. After three blocks of the VA loading spectrum, the crack had grown to a depth of 4.65 mm. The weld root bead of this girth weld corresponded to the Type II defect model, with the WRBH = hi-lo = 0.85 mm and

_{k}*w*= 3 mm.

*da/dN*can be expressed by the following Paris power law equation:

*C*and

*m*are material constants and ΔK is the stress intensity factor range. The mean crack growth law given in BS 7910 for steels in the air with stress ratio

*R*≥ 0.5 was used. Use of this curve with high-stress ratio is considered appropriate since numerical modeling of the girth welds predicted high-tensile residual stresses present at the weld root [15]. This is in two stages with

*m*= 5.10 and

*C*= 4.8 × 10

^{−18}initially and then

*m*= 2.88 with

*C*= 5.86 × 10

^{−13}after Δ

*K*= 196 N/mm

^{3/2}.

For the semi-elliptical crack, crack growth was calculated in both the *a* (depth) and *c* (surface) directions. For such calculations using the 2D *M _{k}* solution developed in this project, the solution is applied directly for calculating Δ

*K*at the deepest point in the crack. Thus,

*M*decreases as “

_{ka}*a*” increases. However, in the absence of a solution for

*M*, at the ends of the crack, a constant value is assumed for assessing crack growth in the

_{kc}*c*-direction. As suggested in BS 7910 [7], the

*M*value at a crack depth of 0.15 mm was used for the intersection of surface flaws with the weld toe, i.e.,

_{k}*M*is equal to

_{kc}*M*

_{k}_{a}at 0.15 mm. This is a conservative approach [16].

Figure 13 compares the prediction and the experimental data for the fatigue crack extensions with an increasing number of blocks. It can be seen that the prediction agreed well with the experimental data, suggesting that the *M _{k}* solution developed in this project was appropriate for characterizing the effect of the weld root bead profile on fatigue crack growth rate (FCGR) in this case.

Although more analyses from fatigue crack growth beach marks would be desirable to assess the accuracy of the *M _{k}* solution developed, it was difficult to obtain beach marks from short fatigue cracks (<1 mm). As a result, further assessments using an S–N curve approach has been undertaken as described below.

### 5.3 Estimating the Constant Amplitude S–N Curve.

Six pipes, with a total of eleven girth welds, were tested to establish the CA S–N curve. The test results are plotted in Fig. 14 in terms of nominal stress range. Regression analysis produced the mean S–N curve ΔS^{3.47}N = 4.53 × 10^{13}, with a standard deviation of log(N) of 0.148. On the basis that some failures were obtained at endurances beyond 10^{7} cycles including one at 4.4 × 10^{7} cycles, a two-stage S–N curve was proposed, with a slope transition at 5 × 10^{7} [1,17]. By assuming a slope of *m* + 2 for the curve after 5 × 10^{7} cycles, the parameter A for the curve after 5 × 10^{7} cycles was determined to be 1.22 × 10^{17}. The two-stage mean, lower and upper bound (mean − 2SD and mean + 2SD) S–N curves are shown in Fig. 14.

Welds inevitably contain flaws, notably the sharp non-metallic intrusions at the weld toes from which fatigue cracks have been observed to propagate [3]. In that work, a close examination of fillet welds in steel identified weld toe flaw sizes ranging from 0.15 to 0.4 mm. Based on post-test examination of macro-sections of the girth welds tested in the JIP, the quality was generally better than that found by Signes et al., with weld toe flaw depths of ∼0.07 mm on average. Two examples of the weld root toe flaws are shown in Fig. 15, with that in Fig. 15(a) showing initiation of a fatigue crack at the weld root bead toe.

Therefore, in the following fracture mechanics analyses, the initial flaw size was assumed to be 0.07 mm. The initial flaw was assumed to have a semi-elliptical shape and a shallow crack aspect ratio with *a/c* = 0.14 (2*c* = 1 mm). Again the *M _{kc}* was assumed to be constant and equal to the

*M*in-depth direction (

_{k}*M*) at

_{ka}*a*= 0.15 mm.

The weld root was assumed to have the Type II profile. A WRBH of 0.5 mm, which represents the average size at the worst location along the circumference of all the girth welds, was assumed in the calculations. In addition, two more predictions were made by adopting the lower and upper bound values for the WRBH 0.25 and 0.75 mm. The predicted S–N curves are shown in Fig. 14 for comparison with the experimental mean, lower and upper bound S–N curves. The predicted mean S–N curve agreed well with the experimental mean S–N curve at fatigue lives ≤5 × 10^{6} cycles. At fatigue lives greater than 5 × 10^{6} cycles, the predicted lives were greater. This is not surprising since there were few failures in the long life regime and the data exhibited large scatter. Despite the failure at 4.4 × 10^{7} cycles, several welds were unfailed (marked with an arrow in Fig. 14) at fatigue lives far beyond the mean S-N curve.

It can also be seen that the scatter between the lower and upper bound S–N curves for WRBH 0.25 and 0.75 mm is much smaller than the experimental scatter. This suggests that, except for the WRBH, other parameters could also contribute to the experimental data scatter, for example, variations in initial flaw size, residual stresses, misalignment, etc. It should also be noted that crack initiation life was not considered in the analysis. For a blunt flaw like that in Fig. 15(b), fatigue crack initiation life could play an important part in the total fatigue life, especially in the long life regime.

## 6 Examination of the Guidance in Standards

As described in Sec. 1, DNV GL F101 provides guidance on ECA of flaws at the girth weld root toe. The most recent version [10] requires an input of the WRBW and an inclusion of the misalignment-induced SCF. BS 7910 does not provide specific guidance on ECA of girth weld roots. The 2D solution requires an input for only the weld width *w*. It is understood that, when assessing the weld root, the appropriate *w* should represent the WRBW. Therefore, the method is the same as that given in the DNV F101, 2013. Therefore, an opportunity was taken to examine the guidance by comparing the predicted fatigue lives, based on the approaches suggested in the standards, with both the experimental data and the predicted lives based on the *M _{k}* solution developed in the present study.

In all fatigue crack growth life calculations, the input parameters were the same as those used in Sec. 5.3, i.e., an initial surface-breaking flaw 0.07 mm deep and 1.0 mm long, the two-stage mean FCGR law from BS 7910, WRBW = 3 mm, WRBH = 0.5 mm, and weld root profile corresponding to the Type II defect model. Figure 16 shows four S–N curves for comparison:

The experimental mean S–N curve determined from the JIP (the mean curve shown in Fig. 14 up to 10

^{7}cycles).The S–N curve predicted by using the FE

*M*solution developed in the present study. As described before, the predicted S–N curve agreed well with the experimental one._{k}The S–N curve predicted in accordance with the guidance given in DNV F101:2012, i.e.,

*w*= weld cap width (13 mm in the present case) and excluding misalignment-induced SCF. It can be seen that it significantly under-estimated the actual fatigue life, by about 60%. This was because WRBW has a strong effect on*M*solution._{k}The S–N curve predicted in accordance with the DNV F101:2013, i.e.,

*w*= weld root bead width (3 mm in the present case) and inclusion of the misalignment-induced SCF (calculated assuming hi-lo = 0.5 mm so the center-line offset is 0.25 mm). The predicted S–N curve agreed well with the experimental one but over-estimated the fatigue life in the long life regime (>10^{6}cycles). The predicted life was also longer than that calculated based on the present*M*solution._{k}

The above results suggested that, if there is a hi-lo at the weld root (Types II and III defect models) and the WRBW is relatively small, the use of the guidance given in DNV-OS-F101:2013 could underestimate the actual *M _{k}* effect as can be seen in Fig. 17. In this figure, all DNV-OS-F101:2013

*M*curves were based on the BS 7910 2D

_{k}*M*multiplied by the misalignment-induced SCF to allow for a direct comparison with the present FE

_{k}*M*solutions which already include the SCF in the model. It will be seen from Fig. 17 that, when the WRBW is relatively small, at 3 mm, DNV-OS-F101:2013 under-estimates the

_{k}*M*as developed in this work, for most crack sizes of interest, especially at the higher hi-lo (1 mm). WRBW has a strong effect on the BS 7910 2D

_{k}*M*solution. When

_{k}*w*was increased to 5 mm in the BS 7910 2D

*M*solution, the

_{k}*M*curve calculated based on the DNV-OS-F101:2013 guidance is still largely below the present

_{k}*M*solution for a hi-lo of 1 mm. However, a further increase of WRBW to 10 mm sees the

_{k}*M*curve significantly above the latter.

_{k}In the above analysis, only one weld root profile (Type II), with WRBH = hi-lo = 0.5 mm, was evaluated. In the following, evaluations were also carried out for other types of weld root models for different WRBH and at two different stress levels. As before, the initial flaw size was assumed to be 0.07 by 1 mm, the two-stage mean FCGR curve given in BS 7910 for steels in the air with *R* ≥ 0.5 was used. The weld root bead width *w* was 3 mm. The results are summarized in Table 4. The following can be seen from the table:

For Type I flaws, the fatigue lives calculated based on the current DNV guidance agree well with those calculated by using the

*M*solution developed in this project for high WRBH (1 mm in the present example). For smaller WRBH, the current guidance under-estimates the fatigue life—a difference of about 22% at a WRBH of 0.25 mm._{k}For Type II flaws, the predicted fatigue crack growth lives, based on the guidance given in DNV-OS-F101:2013, are greater than those calculated based on the present

*M*solution—the difference increased with increasing WRBH (or hi-lo) and decreasing stress level._{k}A similar trend was also observed for the Type III flaw models—the difference increased with increasing hi-lo and decreasing stress level. For example, for a weld root with WRBH = 1 mm, hi-lo = 0.75 mm, the fatigue life predicted, based on DNV-OS-F101:2013 at a stress range 120 MPa, is 1.7 times that calculated based on the new FEA-based

*M*solution._{k}

## 7 Discussion

The present analysis indicated that it would be inappropriate to use the existing *M _{k}* solution in BS 7910 to assess a flaw at the weld root bead toe. This is because the existing

*M*solutions were based on models which did not characterize the features of the girth weld root toe. The girth weld root bead width and height are typically small with

_{k}*w/B*and

*h/B*ratios typically ∼0.15 and ∼0.03, respectively, while they were much greater, with

*w/B*≥ 0.5 and

*h/B*≥ 0.1, in the existing

*M*solutions. Furthermore, there is often a step between the weld root bead and the neighboring pipe surface which could introduce a high SCF at the sudden surface transition location. The misalignment-induced SCF was greater in the present non-symmetric model than in the symmetric model where the weld profiles in the weld cap and root were assumed to be identical [4]. As a result, the existing

_{k}*M*solutions tend to be conservative for Type I defects, but non-conservative for Types II and III defects.

_{k}The fatigue performance of girth welds in risers and flowlines strongly depends on the severity of the weld root bead profile. The present analysis indicated that the WRBH, WRBW, and hi-lo at the weld root are key parameters influencing the *M _{k}*. Therefore, to improve the fatigue performance of girth welds, measures should be taken to control these parameters, especially the hi-lo. Industry is focused on improvement of fatigue performance of risers and flowlines through control of these parameters [18]. Considerable work has been carried out in this area with respect to careful welding parameter control, surface inspection (including laser profilometry), volumetric inspection, control of out-of-roundness at the pipe mill, and by laser measuring and sorting during fabrication.

Although the *M _{k}* solution developed was based on the welds made with a copper backing shoe on the inside surface, it can be conservatively used for assessing defects at weld root with favorable profile, i.e., no steps between the weld root and the neighboring pipe inner surfaces.

The *M _{k}* solution developed in the present project was focused on application to girth welds in risers and pipelines where membrane loading is predominant. The

*M*solution under bending loading was not considered in the present study. Users should, therefore, exercise caution when applying the new FEA-based

_{k}*M*solution to other situations where the proportion of bending element is significant, e.g., assessing a flaw at the single-sided weld root in a plate or pipes with high wall thickness. However, comparison of the BS 7910 2D

_{k}*M*solutions between axial and bending loading modes, Fig. 18, suggests that:

_{k}the difference between the two is negligible for a typical WRBW, and

the use of the

*M*developed under axial loading for cases under bending loading is conservative._{k}

The *M _{k}* solution was developed from a model in which the weld cap was assumed to be ground flush with the pipes and perfectly aligned (no offset at the weld cap) and the SCF due to the hi-lo at the weld root was included in the solution. If there is a hi-lo at the weld cap, a misalignment-induced SCF due to this hi-lo needs to be considered in the use of the present

*M*solution for assessing a flaw at the weld root bead. If the hi-lo occurs on the same side of the weld as that on the weld root, the misalignment-induced SCF should be included in the assessment, using a method similar to that suggested in BS 7910 [7]. On the other hand, if it occurs on the opposite side of the weld to the hi-lo at the weld root, it would be conservative to ignore this misalignment-induced SCF.

_{k}If the DNV-OS-F101:2013 guidance is used to assess a flaw at the weld root toe, the assumed WRBW must be increased to avoid non-conservatism. The Types II and III defect models in Table 4 were re-analyzed by increasing the WRBW and the results are given in Table 5. It can be seen that it would be conservative to increase the weld width by 3 mm for the Type II defect and 2 mm for the Type III defect. This simple approach can be applied to weld root profiles with WRBH up to 1 mm. Whether this approach can be conservatively used for assessing flaws of all types at the girth weld root toe needs further analysis.

## 8 Conclusions

The FE

*M*solutions and the parametric equations for the Types I, II, and III defect models have been developed and established._{k}Weld toe angle at the weld root has a negligible effect on the

*M*._{k}The derived

*M*depends on the WRBH at_{k}*w/B*> 0.15, and on WRBW. For the Type I defect, the*M*increases with increasing_{k}*w/B*and*h/B*, and eventually reaches a plateau. In all cases analyzed, the BS 7910 2D solution provides an upper bound to these curves.For the Types II and III defects, the FE

*M*values are significantly greater than for the Type I defect and increase sharply with increasing hi-lo._{k}For the Type III defect, the

*M*solution can be obtained through the convex combination of the Types I and II defect_{k}*M*solutions._{k}The accuracy of the

*M*solution developed was confirmed by the experimental data in terms of both fatigue crack growth and S–N curve approaches._{k}The guidance given in DNV-OS-F101:2013 predicts fatigue lives comparable with those predicted using the present

*M*solution for relatively small hi-lo (0.25 mm). However, for girth weld roots with greater hi-lo, the guidance is non-conservative._{k}A simple modification to the current DNV-OS-F101:2013 guidance to avoid non-conservatism is proposed, i.e., increasing the WRBW by 3 mm and 2 mm for Types II and III defect models, respectively.

## Acknowledgment

This work was supported by the Industrial Members of TWI.

## Nomenclature

*B*=wall thickness

*M*=_{k}a magnification factor to account for the effect of the stress concentration factor at the weld toe on Δ

*K*- Hi-lo =
the offset between the inner pipe surfaces on each side of the weld

- Type I defect model =
the weld root bead profile is symmetric about the weld center line and the wall thicknesses of the pipes either side of the weld are the same (hi-lo = 0)

- Type II defect model =
the geometry is essentially two pipes with the same outer diameter but different wall thickness, and the weld root is flush with the inner surface of the pipe with larger wall thickness (WRBH = hi-lo)

- Type III defect model =
the geometry is that of two pipes with the same outer diameter and different wall thicknesses joined by a weld with the weld root bead extending beyond the inner surface of both pipes (WRBH ≠ hi-lo)

- WRBW or
*w*=weld root bead width

- WRBH or
*h*=weld root bead height