Validation of SCF Formulas for Joint Mislaignments in Fatigue Design Rules of the Coast Guard and Navy

1 Validation of SCF Formulas for Joint Mislaignments in Fatigue Design Rules of the Coast Guard and Navy Tara Larkin1,2, * , Pingsha Dong1 1 University of Michigan, Ann Arbor, MI, USA 2 United States Coast Guard Academy, New London, CT, USA Abstract. This study presents new SCF solutions for axially misaligned butt welds between two plates, derived using the “ASME B&PV Code” mesh-insensitive structural stress method. Two-dimensional finite element models are analyzed across varying boundary conditions and geometries. A direct analytical approach is also taken and results in closed-form solutions corresponding to various plate thickness and length combinations. Additionally, the solutions are validated by those derived using the energy method by Xing and Dong (2018). Results prove that current International Institute of Welding (IIW) formulas, along with other industry standards, contain empirical terms that lack physical basis, have limited applicability to realistic boundary conditions, and generally produce un-conservative estimates of stress concentrations—leading to overprediction of fatigue life by up to seven times for some misalignment conditions. The demonstrated inaccuracy of IIW formulas for simple misalignments, where analytical solutions are feasible, raises concern for more complex joints. The SCF solutions presented in this study are more accurate and broadly applicable for misaligned butt and cruciform fillet welds on naval vessels, enable identification of the critical weld toe for targeted structural health monitoring, and can be extended to more complex joints. The methods and solutions presented are also used to analyze the impact of tapered transitions in welds of varying thickness. Keywords: stress concentration factor, misalignment, taper 1 Introduction Fatigue design of welded joints has been studied for decades, yet premature fatigue failures continue to plague naval vessels, including recent U.S. Coast Guard and Navy acquisitions. Notably, the initial National Security Cutter design for the Coast Guard lacked sufficient structural fatigue considerations, leading to extensive structural enhancement work on the first two hulls and significant redesign for subsequent hulls. To begin addressing these issues, the Coast Guard launched the Fatigue Life Assessment Program (FLAP) [1] to evaluate fatigue design procedures, particularly those outlined in the Navy’s Fatigue Design Guidance for Surface Ships [2]. Industry standards, including the Navy’s guide, outline three primary methods to calculate fatigue life: nominal stress, hot spot stress, and the effective notch stress[2], [3], [4], [5]. Each method defines a stress to calculate, applies a resulting stress concentration factor for a welded connection not explicitly modeled, and classifies the joint into an appropriate SN curve. The nominal stress method, despite its simplicity and widespread acceptance, is challenging in application. Nominal stress is often ill-defined in complex structures and the categorization of joints is ambiguous as joint type, geometry, and loading mode may not perfectly align with existing curves. The hot spot stress and effective notch stress methods employ more detailed finite element models and require fewer SN curve categories. However, these methods have been shown to be sensitive to element type and size [6]. The International Institute of Welding (IIW) guidance even states a “high level of expertise is required on the part of the FEA analyst [5].” While the Navy recognizes all three methods, the nominal stress approach is required unless specific experimental criteria and approvals have been met [2]. Embedded within the fatigue design process is a significant amount of uncertainty. FLAP predominately focused on the wave loading and structural response aspects, which are critical inputs for determining the stress expected in a welded joint over the lifetime of the design. Wave sensors and strain gauges were installed at critical design locations on active cutters to assess the accuracy of current methodologies. In addition, FLAP also highlighted other critical areas of uncertainty in the fatigue design process, such as construction imperfections [1]. * Correspondence to: [email protected] 16th International Symposium on Practical Design of Ships and Other Floating Structures PRADS 2025 Ann Arbor, MI, USA, October 19th – 23rd 2025 2 Misalignments are common imperfections in shipyard welds that reduce fatigue life by inducing a secondary bending moment. Some representative axial misalignments are shown in Figure 1, where t represents thickness, and e is the misalignment caused by the offset of the centerlines of two adjoining plates. The impact of the induced secondary bending moment is assessed through a stress concentration factor, 𝑘𝑚, shown in the general form in equation (1). The membrane stress, 𝜎𝑚, refers to the uniform through-thickness stress, and the bending stress, 𝜎𝑏 is the linearly varying stress gradient. For the geometry of a butt joint, this equation can be rearranged into equation (2), where 𝜆 is a factor provided in industry standards. Depending on the reference, some 𝑘𝑚 equations only consider the secondary bending moment effects ( 𝜎𝑏 𝜎𝑚), rather than the total effect represented by equation (1). For clarity and direct comparison, equations in that form that are discussed in this paper have been adjusted. 𝑘𝑚=𝜎𝑚+𝜎𝑏 𝜎𝑚 (1) 𝑘𝑚=1+𝜆𝑒 𝑡 (2) Figure 1. Misalignment in a (a) butt weld [7] (b) butt weld with plates of varying thickness, and (c) fillet weld[7] Industry’s trend toward lightweighting and structural optimization has increased the prevalence of misalignments, resulting from welding-induced distortions [8], making it imperative to critically assess associated design guidance. Higher strength, thinner plates are more susceptible to distortion, and require joints with thickness transitions, which inherently have misalignments. The Navy recently drafted a newer version of their Fatigue Design Guidance for Surface Ships, citing major structural failures across three different classes of ships and issuing a warning about the lightweighting trend [2],[8]. Despite this warning, the updated draft fails to provide any additional discussion on misalignments, and still simply suggests finite element analysis or the American Bureau of Shipping’s (ABS) 1996 reference with equation (2), where 𝜆=1.5. External design firms contracted by the Coast Guard have not been required to strictly adhere to Navy design standards, and have been permitted to identify and apply comparable industry standards. When shipyard misalignments were discovered in the production of a recent Coast Guard acquisition, IIW guidance was followed [10], which provides additional details compared to the Navy guidance. IIW stress concentration factor formulas for axially misaligned buttwelds are presented in equations (3) and (4), with their application explained in Table 1. 𝑘𝑚= 1+𝜆 𝑒∙𝑙1 𝑡(𝑙1+𝑙2) (3) 𝑘𝑚= 1+6𝑒 𝑡1∙𝑡1 𝑛 𝑡1 𝑛+𝑡2 𝑛 (4) In IIW guidance, the SN curves are titled by the fatigue assessment (FAT) category that the joint is classified by. Each FAT curve already assumes a certain amount of 𝑘𝑚 as an acceptable misalignment. This value is 1.30 for butt joints, when using the nominal stress method and its associated FAT curve, and only 1.05 for the hot spot and effective notch stress methods. Therefore, equations (3) and (4) only need to be applied when a misalignment exceeds what is assumed in the FAT curve. When this is the case, an effective 𝑘𝑚 should be calculated using equation (5). Additionally, IIW states that a minimum 𝑘𝑚,𝑒𝑓𝑓𝑒𝑐𝑡𝑖𝑣𝑒, dependent on joint type, must always be applied to numerically determined 𝑘𝑚 values that did not explicitly include a misalignment in the model. Taras et al [6] concluded this to be essential, and that current industry standards lacks clarity and sufficient guidance. 3 Table 1. IIW Misalignment Guidance 𝑘𝑚,𝑒𝑓𝑓𝑒𝑐𝑡𝑖𝑣𝑒 =𝑘𝑚,𝑐𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒𝑑 𝑓𝑟𝑜𝑚 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛𝑠 𝑘𝑚,𝑎𝑠𝑠𝑢𝑚𝑒𝑑 𝑖𝑛 𝐹𝐴𝑇 𝑐𝑢𝑟𝑣𝑒 (5) Many studies have questioned the validity of 𝑘𝑚 equations, yet industry standards remain rooted in formulations by Berge and Myhre from the 1970s [11], that were subsequently experimentally validated under specific loading conditions [12]. In 1999, Cui et al [13] used plate theory to investigate equation (2), noting 𝜆 to vary amongst literature, and concluding 𝜆 to depend on geometry and loading. The study specifically proved the ABS version of equation (2) with 𝜆=1.5, still referenced by the new draft of the Navy fatigue guidance, has no justification. In 2004, a Ship Structures Committee report [14], found existing guidance to be too general, and boundary conditions to be an important factor that needs to be addressed. With industry standards largely remaining unchanged following these early conclusions, more recent studies have reignited the issue and questioned the existing approaches. Taras et al [6] assessed the numerical methods against the nominal stress approach for several construction imperfections. Xing et al [15] and Zhou et al [12] presented new analytical solutions for misalignments in welded joints, validated with FEA and experimental data. Mancini [16] developed thin plate solutions with experimental validation, and Rautiainen [17] assessed numerical methods for use on complex joints. Amongst these studies, the following issues have been raised regarding fatigue design guidance for misaligned welds: 1. Limited Applicability - Boundary conditions have been found to significantly impact 𝑘𝑚, yet IIW contains vague classifications, such as “fully restrained”, “un-restrained”, and “remotely loaded”, without clear guidance for adjusting for alternate conditions [9]-[12]. 2. Empirical Limitations – IIW formulations are rooted in empirical formulations that lack physical meaning and were derived under specific conditions. Studies have proven formulations to be wrong in certain cases [15], and question the use of n=1.5 for different thickness combinations, that is “supported by tests.” 3. Inconsistent Results – The three-industry accepted fatigue design methods produce different stress results, direct users to different FAT curves, and ultimately lead to different fatigue life estimates. Even within each method, ambiguous guidance, subjective joint categorization, and mesh sensitivity yield varying results [6], [12], [14], [15], [17]. 4. Non-conservative – IIW approaches have been found to underestimate stress in many cases, leading to overestimated fatigue life, which is especially concerning considering the issues above and the multitude of other factors of uncertainty involved. [6], [12], [15], [16], [17] 5. Lack of critical weld location information in a joint – current 𝑘𝑚 formulations do not identify the critical toe, which is critical for structural health monitoring efforts, such as those initiated by FLAP, and effective design. There is lack of clarity on how a taper at a butt joint between two different thickness plates reduces 𝑘𝑚, as IIW simply credits tapered transitions with an alternate FAT categorization. Type of Misalignment Equation Notes Axial misalignment between flat plates (3) λ is dependent on restraint λ = 6 for unrestrained joints For remotely loaded joints assume 𝑙1 = 𝑙2 Axial misalignment between flat plates of differing thickness (4) 𝑡2≥𝑡1 Relates to remotely loaded unrestrained joints. The use of n=1.5 is supported by tests. 4 Of note, Lotsberg [18] asks similar questions, but concludes IIW formulas for axially misaligned butt welds to be sufficient. However, the study was conducted under specific geometry and loading. This paper utilizes the “ASME B&PV Code” mesh-insensitive structural stress method, referred here as the nodal force method (NFM) for two-dimensional analyses, to address these five issues. The NFM is used to analyze joints with various boundary conditions and geometries. Direct analytical solutions will also be obtained when possible, including specific forms derivable from those presented in [12], [15]. These solutions will be used for not only validating the relevant IIW formulas, but also shed light on the regime of validity and corresponding boundary conditions. The NFM is used to assess the effects of weld presence in misaligned joints, which has been ignored in analytically derived closed-formed solutions, in addition to examining the validity of the empirical n=1.5 term and studying tapered transitions in welded joints of varying thickness. 2 SCF Analysis To represent the range of boundary conditions possible for a welded joint on a naval vessel, several cases were studied, listed in Table 2. Cases 0 and 1 are statically determinate cases. Cases 1 through 3 match those studied by Xing [15], enabling validation of the direct analytical approach offered in this study with Xing’s energy method solutions. When comparing against IIW guidance from Table 1, it is assumed that IIW’s “unrestrained” and “fully restrained” are most like cases 1 and 2, respectively. Models for butt welded joints were developed based on Figure 1 (a) and (b) using thicknesses and misalignments provided in [10], with 𝑙1=10𝑡𝑚𝑎𝑥. To study the impact of weld placement, 𝑙2 varied. All FEA results were calculated using plane stress elements in ABAQUS. Table 2. Boundary Condition Cases Case Left Boundary Condition Right Boundary Condition 0 Edge fixed with no rotation Free 1 Pinned in x and y Pinned in y 2 Edge fixed with no rotation Pinned in y with no rotation 3 Edge fixed with no rotation Pinned in y 2.1 Nodal Force Method in ASME B&PV Code The “ASME B&PV Code” mesh-insensitive structural stress method presented in [19], addresses the ambiguities associated with stress identification and SN curve categorization in current industry standard fatigue design methods . This robust approach, adopted by ASME in 2007, is explained comprehensively in [20]. Due to the simple geometry of the misalignments in this study, the 2D structural stress definition, the NFM, is utilized. Unlike the hot spot stress method, which relies on a linear extrapolation of surface stresses to weld toe, the NFM imposes equilibrium equations by using the nodal forces acting on either side of the hypothetical crack plane to yield mesh in-sensitive results, that correlate well with experimental data [12], [15], [19], [20]. Depicted in Figure 2, a free-body cut at the weld toe exposes elements (shaded in blue) and their nodal forces (NFORC in ABAQUS). A summation of these nodal forces in the x-direction, 𝐹𝑥𝑖, and nodal moments yields the internal force, N, and moment, M, as written in equations (6) and (7), where i =1, n and n is the number of nodes exposed by the cut. For the example of three through-thickness elements, there is a total of six exposed 𝐹𝑥𝑖 to sum: one each at the top and bottom nodes, and two at each internal node. Mesh convergence tests conducted throughout the study confirmed the mesh-insensitivity of this method. Figure 2. An example of the 2D NFM, with three through-thickness elements. There is a total of six 𝐹𝑥𝑖 to be summed along the red line, representing a free-body cut at the hypothetical crack line at the weld toe [21]. 5 𝑁= 𝜎𝑚𝑡=∑𝐹𝑥𝑖 𝑛 𝑖=1 (6) 𝑀= 𝜎𝑏𝑡2 6=∑𝐹𝑥𝑖 (𝑦𝑖−𝑡 2) 𝑛 𝑖=1 (7) Additionally, this method enables calculation of an equivalent structural stress range parameter, using equation (8). The traction structural stress range, Δ𝜎𝑆, is the sum of the ranges of bending and membrane stresses, m is 3.6, and 𝐼(𝑟)1 𝑚 in Eq. (9) is a dimensionless life integral by using a two-stage growth model [22]. Using these definitions, a Master SN Curve is developed, that consolidates various SN curves into a single, unified curve, shown in Figure 3. This approach removes the subjectivity associated with FAT categories, offering consistent and reliable results across different weld types and joint configurations. Δ𝑆𝑆=Δ𝜎𝑆 𝑡2−𝑚 2𝑚 𝐼(𝑟)1 𝑚 (8) 𝐼(𝑟)1 𝑚= 0.0011𝑟6+0.0767𝑟5−0.0988𝑟4+0.0946𝑟3+0.0221𝑟2 +0.014𝑟+1.2223 (9) Figure 3. Master SN Curve with scatter bands representing 2 and 3 standard deviations from mean. 2.2 Analytical SCF Solutions Direct analytical solutions were derived to validate the NFM results. Figure 4 represents the four cases that were listed in Table 2, and their resulting moment distributions. The moment equation is listed for the left side of the model, when treating the applied force, P, as a moment, M, applied at the weld, as in equation 10. For the statically indeterminate cases, cases 2 and 3, the reaction forces and moments were solved for by imposing compatibility conditions in addition to the equations of static equilibrium. 𝑀= 𝑃∙𝑒 (10) Formulations for the stress concentrations at each toe, (𝑘𝑚)𝑖, are provided in Tables 3 and 4, where an i value of 1 or 2 represents the corresponding toe. These were developed using equation (1) and by setting x = li in the respective moment equations. Solutions are presented alongside IIW and modified versions of Xing’s[15], that enable direct comparison. Xing’s original expressions had an alternative numbering format, only considered the secondary bending effect, and used the membrane stress of the right plate, rather than the respective side. While Xing did not directly solve for Case 0, his approach yields identical results to the direct analytical solution. For 6 case 1, Xing had a typo, where he incorrectly listed 𝑡1 in the numerator instead of 𝑡𝑖. Once corrected, the analytical solutions for case 1 are also equivalent. Comparison of the solutions for the statically indeterminate cases is more difficult due to the introduction of higher order terms. Simplification of formulations for same thickness cases proves equivalency of the direct analytical method and Xing’s energy method, and evaluation of results with model geometry offers additional validation of the two methods. Therefore, Table 4 and subsequent discussion of analytical solutions references are singular. Figure 4. Representative model and direct analytical moment distribution for cases 0 through 3, when treating the misalignment and applied axial force as a moment applied at the center 7 Table 3. Solution Comparison – Statically Determinate Cases Case 0 Case 1 IIW 𝑘𝑚= 1+6𝑒 𝑡𝑙1 (𝑙1+𝑙2) or, with differing thickness: 𝑘𝑚= 1+6𝑒 𝑡1𝑡1 𝑛 𝑡1 𝑛+𝑡2 𝑛 where n=1.5 Direct Analytical (𝑘𝑚)1= 1+6𝑒 𝑡1 (𝑘𝑚)2= 1 Critical Toe = left toe (𝑘𝑚)𝑖= 1+6 𝑒∙𝑙𝑖 𝑡𝑖(𝑙1+𝑙2) Critical Toe = Larger 𝑙 𝑡 Xing (𝑘𝑚)1= 1+6𝑒 𝑡1 (𝑘𝑚)2= 1 (𝑘𝑚)𝑖= 1+6 𝑒∙𝑙𝑖 𝑡𝑖 (𝑙1+𝑙2) Table 4. Solution Comparison – Statically Indeterminate Cases Case Case 2 Case 3 IIW 𝑘𝑚= 1+3𝑒 𝑡𝑙1 (𝑙1+𝑙2) or, with differing thickness: 𝑘𝑚= 1+3𝑒 𝑡1𝑡1 𝑛 𝑡1 𝑛+𝑡2 𝑛 where n=1.5 Analytical [15] (𝑘𝑚)𝑖= 1+ 6𝑒𝑡𝑖2𝑙1𝑙2(4𝑙𝑖3𝑡1 3𝑡2 3 𝑡𝑖3+3𝑙𝑖𝑙1𝑙2𝑡1 3𝑡2 3 𝑡𝑖3+𝑙1 3𝑙2 3𝑡𝑖3 𝑙𝑖3) 𝑙𝑖(𝑙2 4𝑡1 6+4𝑙2 3𝑙1𝑡1 3𝑡2 3+6𝑙2 2𝑙1 2𝑡1 3𝑡2 3+4𝑙1 3𝑙2𝑡1 3𝑡2 3+𝑙2 4𝑡1 6) (𝑘𝑚)1= 1+ 3𝑒(2𝑙2 3𝑡1 4+3𝑙2𝑙1 2𝑡2 4+2𝑙1 3𝑡2 4) (𝑙2 3𝑡1 4+3𝑙1𝑙2 2𝑡2 4+3𝑙2𝑙1 2𝑡2 4+𝑙1 3𝑡1 4)𝑡1 (𝑘𝑚)2= 1+ 9𝑒𝑡2 3𝑙1(2𝑙2+𝑙1)𝑙2 (𝑙2 3𝑡1 4+3𝑙1𝑙2 2𝑡2 4+3𝑙2𝑙1 2𝑡2 4+𝑙1 3𝑡1 4) 2.3 Comparison Results from the statically determinate cases, cases 0 and 1, provide important insights into the IIW formulations. Figures 5 and 6 compare FEA results against the analytical solution and current IIW equations for varying misalignment, 𝑒 𝑡𝑚𝑖𝑛, and varying weld position, 𝑙1 𝑙1+𝑙2. These results indicate: 1. Validation of analytical solution with NFM. 2. Boundary conditions have a significant impact. Ignoring their effect and blindly applying IIW formulas can underpredict 𝑘𝑚. Case 0 gets significantly underpredicted by the IIW formulation where λ=6, and there is no clear guidance on how to treat this specific boundary conditions. For the worst case analyzed, where 𝑙1=0.25𝑙𝑡𝑜𝑡, the IIW life prediction is 7 times the FEA and analytical life prediction. 3. Statically determinate cases do not need the thickness correction term. The inclusion of this empirically derived thickness term, in the IIW formulas, results in an unconservative prediction of 𝑘𝑚 for both statically determinate cases. This indicates limited testing conditions in the formulation of the n=1.5 term. Despite IIW formula being derived like case 1, for weld positioning where 𝑙1<𝑙2, the IIW equation seems incorrect. The term should be 𝑙𝑖 𝑙1+𝑙2, instead of 𝑙1 𝑙1+𝑙2. Figure 6 shows that the IIW formulation only considers toe 1. When toe 2 is the critical toe, IIW is unconservative. For the worst case analyzed, a case 1 joint with 𝑙1=0.25𝑙𝑡𝑜𝑡, the IIW life prediction is 4 times the FEA and analytical life prediction. 4. Weld positioning impacts 𝑘𝑚. IIW contains no guidance on the proper use of the “remotely loaded” assumption. The “remotely loaded” is unconservative any time 𝑙1≠𝑙2, . 8 Figure 5. Km for varying misalignment, 𝑒 𝑡𝑚𝑖𝑛, based on IIW versus analytical and FEA solutions for statically determinate cases. FEA results validate analytical solutions and show underprediction by IIW. Figure 6. Stress concentration factors for varying weld position 𝑙1 𝑙1+𝑙2based on IIW formula, IIW “remote loading” assumption, versus derived solutions and FEA results for statically determinate cases. Figures 7 and 8 include results from all 4 cases and compare IIW and analytical solutions against NFM results. IIW does not have a solution for case 0. Cases 2 and 3 are assumed to be the “fully restrained” IIW condition. The results from these additional cases indicate: 1. Good correlation between NFM and analytical solution for all cases. 2. Good correlation of IIW equation only for the same thickness case 1 condition. IIW underpredicted critical toe for all other scenarios tested. 3. The IIW empirical different thickness term also underpredicts statically indeterminate cases, and is not validated by other prediction methods. Further investigation into this term is provided in section 3. 4. The non-conservative nature of IIW formulations is exacerbated by a combination of errors in geometries with different thickness combinations and 𝑙1<𝑙2. Case 3 yielded IIW life predictions 4.3 times the predictions from other methods. 9 Figure 7. Stress concentration factors at each weld toe for statically determinate cases with different thickness combination of 𝑡1=5𝑚𝑚,𝑡2=9𝑚𝑚 for (a) same length and (b) 𝑙1=0.5𝑙2 Figure 8. Stress concentration factors at each weld toe for statically indeterminate cases with different thickness combination of 𝑡1=5𝑚𝑚,𝑡2=9𝑚𝑚 for (a) same length and (b) 𝑙1=0.5𝑙2 2.4 Cruciform Joints A cruciform joint with four fillet welds, shown in Figure 9, was tested with the boundary conditions of cases 0 through 3 applied to the ends of the horizontal members. Results for thickness and length combinations matching those studied on the butt welds, produced identical results to Figures 7 and 8, indicating the analytical solutions and NFM method are valid for fillet weld cruciform joints, as well. Figure 9. Model of cruciform joint with fillet welds. NFM (a) (b) NFM NFM NFM NFM NFM NFM NFM NFM (a) (b) NFM NFM NFM NFM NFM NFM NFM NFM 16 manufacturing, it should not be given so much FAT categorization “credit” for reducing stress concentration. IIW currently significantly overestimates the reduction of 𝑘𝑚 due to tapering transitions of thickness in butt welds. Future extensions of this work include further analysis of the taper to produce closed form solutions. Validation will be done with available published experimental data, such as from a paper by Tariq et al, presented at the 2025 IIW Annual Assembly. This methodology will also be extended to more complex joints and ultimately attempt to develop more robust fatigue design procedures for the Coast Guard and Navy. References [1] K. Stambaugh, I. Drummen, C. Cleary, R. Sheinberg, and M. Kaminski, “Structural Fatigue Life Assessment and Sustainment Implications for a New Class of US Coast Guard Cutters”. [2] M. W. Sieve, D. P. Kihl, and B. M. Ayyub, “Fatigue Design Guidance for Surface Ships:,” Defense Technical Information Center, Fort Belvoir, VA, Nov. 2000. doi: 10.21236/ADA389305. [3] British Standards Institution, Ed., Guide to methods for assessing the acceptability of flaws in metallic structures: British standard BS 7910:2019, Fourth edition. in BSI standards publication, no. 7910. 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