FEA Stress Determination for Weld Fatigue Using Hot-Spot Stress Method: Benchmarking and Rail Application

Ensuring the structural integrity and durability of rail welded equipment is critical for the safety of both the rail infrastructure and the equipment itself. The prediction of weld fatigue life using the stress-life (S-N) method is one of the most widely used approach in ensuring the structural durability of rail equipment. The main controlling parameter in S-N based fatigue life prediction is the stress range cycles that the equipment and welds will see during the use. The weld toe is one of the primary sources of fatigue cracking due to the severity of the stress concentration it generates [1-4]. Several geometric features in welded parts act as a stress raiser. Such stress raising discontinuities can produce essentially global and local effects producing high local stresses at welds. The stress concentration effect at welded joints, resulting from geometric changes, misalignments, material inconsistencies, and defects from the welding process, makes welded connections more susceptible to early failure compared to base materials. There are four primary methods to estimate the fatigue life of welded components: 1) The nominal stress method; 2) The hot-spot stress or strain method; 3) The local notch stress or strain method; and 4) The fracture mechanics method. These methods differ in the stress and strain levels they use, depending on how stress concentrators are accounted for in the calculations. Incorporating the Finite Element Analysis (FEA) method into the fatigue design of steel structures has facilitated the creation of localized stress analysis techniques. These include the Hot-Spot Stress (HSS) method, the notch stress method, and crack growth analysis [5]. In rail steel structures, welded joints are typically evaluated for fatigue using the nominal stress and HSS approaches when using S-N methods. The selection of the appropriate stress range method depends on several factors, including the geometry of the weld joint, stress orientations, loading conditions, desired accuracy level, and primarily the availability of relevant S-N curve data for predicting fatigue damage. The nominal stress range method is traditionally simplified approach, it tends to provide a conservative estimate, it lacks precision in accounting for stress concentrations. Conversely, the HSS method is a more advanced and refined approach that offers a more detailed evaluation of stress concentration, making it suitable for complex geometries [1-3, 5-9]. The nominal stress at a specific location is the combined total of membrane stress and bending stress at that point. Nominal stress is typically calculated using analytically derived formulas from theory. It excludes the effects of local structural discontinuities, stress concentration due to weld toe, thickness changes, residual stresses, and misalignments. A hot-spot in a structure is a location where a fatigue crack is likely to begin, resulting from the combined effects of fluctuating stress, geometric discontinuities, and weld geometry. The HSS approach evaluates stress near the weld; the hot-spot stress (HSS) is the structural stress at the weld toe or weld end. It includes the stress concentration effects of gross structural discontinuities, misalignment, and weld geometry but excludes the nonlinear peak stress caused by the local notch at the weld toe, although the HSS is positioned at the weld toe [2, 3]. Figure 1 shows the variation in structural stress approaching the weld toe.

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Figure 1. Structural stress

A local notch does not alter the membrane and shell bending structural stress. The primary effect of a notch is the local nonlinear peak stress distribution. This nonlinear peak lies within a radius of approximately 0.3t-0.4t from the notch root [2, 3, 5-8, 10]. Although the weld profile requirements are specified, the exact weld geometry and the shape of the local notch at the weld toe varies considerably along the weld and among different welds. As a result, the nonlinear stress peak at weld toe due to local notch exhibit unpredictable values. The nominal stress approach and HSS approach both include the random high stress caused by notches in the test results, which appears as a spread in fatigue S-N curves. As a result, calculating nonlinear stress peaks is not required; instead, they should be excluded from the calculated or measured nominal or hot-spot stress when applying these two methods in fatigue assessments [2, 3, 5-8]. The HSS method is applicable to welded joints under the following conditions [2, 3]:

1. The fatigue stress predominantly acts perpendicular to the weld or at the ends of weld.

2. The method is limited to potential fatigue crack initiation at the weld toe or end on the surface.

The hot-spot stress method does not apply to fatigue cracks that start and propagate from the weld root. It requires separate fatigue assessment using nominal stress or other approaches, is not part of this study. The HSS method is intended for assessing fatigue at the weld toe, where external geometric stress concentrations dominate. The HSS method primarily addresses stress concentrations at the weld toe, where geometric factors elevate stress levels. For fatigue cracks initiating from the weld root, alternative methods that address the unique stress distribution and initiation mechanisms at the root are required.  Fatigue cracks originating from the weld root exhibit a different stress distribution that is less affected by external geometry, HSS method does not accommodate these specific stress characteristics at the weld root. The weld root's internal position complicates accurate stress measurement using surface-based methods like HSS. Fatigue cracks at the weld root often initiate due to factors such as lack of penetration, inclusions, or root defects, which differ from the geometric stress concentration effects at the weld toe. The HSS method does not consider these specific initiation mechanisms. Therefore, assumptions and calculations from the HSS method are applied only to potential crack initiation at the weld toe.

Figure 2 shows the two types of hot-spot in a structure.  Type ‘a’ is located at the weld toe on a plate surface, and Type ‘b’ is located at weld toe on a plate edge [2, 3]. The HSS cannot be directly probed or measured from FEA or experimental results. It is determined using linear Surface Stress Extrapolation (SSE) with two reference points for type 'a' hot-spot and quadratic surface stress extrapolation with three reference points for type 'b' hot-spot [1-3]. The Type ‘a’ hot-spot is more commonly observed in welded structure, for which the SSE method involves linear extrapolation from stresses on the surface at distances of 0.4t and 1.0t from the weld toe. The corresponding hot-spot structural stress is given by Eq. (1) [1-3]. In type 'a' hot-spot, the stress distribution approaching the weld toe is a function of plate thickness, whereas it is not in type 'b' hot-spot [1-3]. Extrapolation methods for type 'b' hot-spot use stresses located at absolute distances from the weld toe rather than proportions of plate thickness, as given in Eq. (2). There is detail guidance given in IIW and BS7608 documents on stress extrapolation based on joint type, hot-spot type, and mesh size. This study is based on relatively finer mesh for type ‘a’ hot-spot that uses Eq. (1) suggested in IIW and BS7608 guidelines [1-4].

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Figure 2. Type of hot-spots [2-4]

$\sigma_{H S}=1.67 \sigma_{0 \cdot 4 t}-0.67 \sigma_{1.0 t}$     (1)

$\sigma_{H S}=3 \sigma_{4 m m}-3 \sigma_{8 m m}+\sigma_{12 m m}$     (2)

The early design guideline for incorporating structural stress was outlined in Eurocode-3 ENV 1993-1 [11]. Subsequently, the International Institute of Welding (IIW) developed the extensive guidelines and recommendations for applying structural hot-spot stress, detailing elements such as type, size, and reference points [2-4]. While the HSS method has long been used for fatigue assessment of tubular structures, its application to plate structures is relatively recent. The HSS method has recently been incorporated into standards BS7608-2015, the Guide to Fatigue Design and Assessment of Steel Products [1]. The use of FEA for determining design stresses in fatigue life calculations has largely been increasing, driven by advancements in numerical applications. This trend is particularly notable for local approaches such as notch stress, fracture mechanics, and hot-spot stress methods. In fact, FEA has facilitated the development of local fatigue assessment methods, such as the Notch Stress and Hot-spot Stress approaches. The main disadvantage of the HSS method is its sensitivity to mesh type, element size, weld modeling techniques and accuracy in probing readings on reference points on the Finite Element (FE) model [5-9, 12, 13]. Even when well-built finite element models are used, FEA results can be highly sensitive to finite element modeling techniques, as the stress reference points for the HSS method are in areas of high stress gradients and stress singularities. As a result, the calculated stresses can vary based on element type, size, and weld modeling techniques in FEA, which can lead to over or under estimation of fatigue life.

The variation in hot-spot stress results primarily from how each element type represents and computes stress distribution under loading and geometric complexities, as well as the varied integration points used during calculations in the solution process. Shell elements are effective in capturing mid-surface stresses but may underestimate stress concentrations at sharp corners or complex weld geometries. Tetrahedral elements are popular and widely used for accurately capturing geometric features but provide less accurate results than hexahedral elements due to fewer integration points, which can lead to less accurate stress predictions, especially in regions with high stress gradients. Achieving accurate results with tetrahedral elements often requires a very fine mesh. If the mesh is not sufficiently refined, tetrahedral elements can produce higher stress values due to inadequate resolution of the stress field. Hexahedral elements are better suited for capturing localized stress variations. Modeling the weld surface explicitly allows the simulation to account for the additional stiffness and stress concentration effects caused by the weld geometry, providing a closer representation and more accurate results compared to simulations without weld surface modeling.

The primary goal of this work was to gain more clarity on applying the hot-spot stress method in assessing the weld fatigue in rail equipment, particularly focusing on how FEA stress determination and stress extraction for the HSS method works on FEA models with different mesh types and modeling techniques for welds. The identified welds were evaluated and compared for fatigue stress ranges using IIW and BS7608 guidelines for the hot-spot stress approach. The effect of various mesh types and finite element (FE) modeling on peak and structural stress was studied and presented. Subsequently, the hot-spot approach was applied to assess weld fatigue in rail equipment, and the results were compared with those obtained using the nominal stress approach. The study examined the lap joint and T-joint fillet welds, categorized as F2 with the nominal stress approach and as category D with the hot-spot stress approach according to BS-7608 guidelines. Appropriate fatigue strength S-N curves for welds in class D and F2 were used from BS7608 in fatigue damage factor calculation. The fatigue requirements for track-induced loads on rail equipment are described in various regulatory codes [1, 11, 14-18]. The regulations to apply vary depending on where the equipment will be operated and the specific requirements of the machine. This study aims to compare stress ranges by utilizing benchmark models subjected to consistent loads for deriving weld stress. It employs various mesh types and FE modeling techniques to assess hot-spot stress. The track induced vertical travel fatigue load applied to the rail vehicle model was referenced from EN12663 [15]. Additionally, BS7608 [1] was used to determine weld joint classifications and to derive fatigue estimates from the S-N curves for both nominal and hot-spot stress methods.

3.1 Results: model 1 and model 2

The benchmark models were solved with and without weld surface modeling, utilizing tetrahedral solid mesh, hexahedral dominant solid mesh, and shell mesh on midsurfaces, incorporating both 1st and 2nd-order elements for each element type. Due to the length of the document, all stress plots and graphs for each version are not included here; however, all results are summarized in Tables 1-8. Some stress plots are displayed in Figure 6 for model 1 versions with T-joint fillet welds and in Figure 7 for model 2 versions with lap joint fillet welds. Three hot spots were identified, as shown in Figures 6 and 7, for the HSS study. The stress readings were probed at 0.4t and 1.0t thickness distance away for each solved model. Split lines were created at and next to welds on the geometry surfaces to precisely probe stress readings at distances of 0.4t and 1.0t. An alternative method is to extract stress along paths created at the identified critical welds in the direction perpendicular to stress, which was also done to study the stress variation approaching the weld toe for comparison. The hot-spot stress is calculated from the probed values at reference points using BS7608 and IIW guidelines, with Eq. (1) discussed in the introduction section. The hex dominant mesh indicated consistent results across all models, falling somewhere between the results obtained with tetrahedral and shell elements.  The ratios are calculated in each table to indicate the relative difference between results, with a reference value of hex dominant mesh of higher order in each respective table.

The results in Tables 1-3 and the bar charts in Figures 8-10 correspond to spot 1 in model 1, whereas the results in Tables 4-6 and the bar charts in Figures 11-13 correspond to spot 2 in model 1. The results in Table 7 and Figure 14 correspond to spot 3 in model 2. Table 1 compares the HSS within different element types and their order, maintaining consistent weld surface modeling across all models in this table. Each table presents two comparisons: one within the same element type regarding element order, and the other across different element types. The results indicate that the shell element modeling predicts higher HSS values compared to both hexahedral and tetrahedral element modeling. It is well known that linear order tetrahedral elements do not produce accurate results; they significantly underestimate the results. However, both hexahedral and shell mesh models predict very close results regardless of their element order. Across all three element types with higher order, the predicted HSS values shows the variations up to 4% within each group. Table 2 lists similar results as in Table 1 without weld surface modeling in FE model weld connections. It indicates comparable findings within group results, with predicted HSS values showing variations up to 7% within each group. Additional findings from the inter-group comparison between Tables 1 and 2, for models with and without weld surface modeling, are separately presented in Table 3. It is observed that without weld surface modeling, higher HSS values are predicted in each element type. Within element types with weld surface modeling, there is less variation; the results show variations within 4%, compared to 7% variations in models without weld surface. Based on the outcomes of the first set of results, model 2 was examined with higher-order elements both with and without weld surface modeling for lap joint fillet weld. The findings from model 2 are similar to model 1, except for one variation: where the shell models predicted lower HSS values than other elements, the tetrahedral elements predicted the highest values in the group. The percentage variation within the group is within 4%.

The stress extracted at weld toes for the identified welds shown in Figures 15 and 16 for spot 1 and 2 respectively.  The elements with linear order, especially the tetrahedral, exhibit an erratic stress pattern approaching the weld toe from the toe to 0.4t distance. The peak stress value and stress variation are significant from model to model and within each element type; however, the stress values and trends are very similar after 0.4t distance in all models. The graphs shown are not to scale; they are presented to observe stress variation behavior approaching the weld toe within 0.4t and 1.0t thickness distance away. The models with shell mesh show a significant drop in stress at and right next to the weld toe intersection modeling which improves when weld surfaces are modeled and with a finer mesh. In FEA of welded structures using shell elements, it is common to observe a drop in stress at and near the weld toe intersection. Shell elements are typically used to model the mid-surface of the structure. When welds are not explicitly modeled, the weld geometry is simplified, which can lead to inaccuracies in stress distribution, particularly at stress concentration points like the weld toe. This can be improved by refining the mesh and modeling the weld surfaces. The finer mesh enhances resolution and captures stress gradients more effectively due to a greater number of nodes and elements, while the weld modeling accounts for the additional stiffness of weld geometry, accurately capturing stress concentration effects. The results showed that this inaccuracy is within a 0.4t thickness distance for the relatively finer mesh size used. The calculated HSS is not significantly affected by this inaccuracy, as the first readings were probed at a 0.4t distance.

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Figure 6. Model 1 stress plots on various mesh types

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Figure 7. Model 2 stress plots on various mesh types

Table 1. HSS with weld surface modeling at model 1-spot 1

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Figure 8. HSS bar charts of Table 1 data

Table 2. HSS without weld surface modeling at model 1-spot 1

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Figure 9. HSS bar charts of Table 2 data

Table 3. HSS with and without weld surface at model 1-spot 1

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Figure 10. HSS bar charts of Table 3 data

Table 4. HSS with weld surface modeling at model 1-spot 2

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Figure 11. HSS bar charts of Table 4 data

Table 5. HSS without weld surface modeling at model 1-spot 2

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Figure 12. HSS bar charts of Table 5 data

Table 6. HSS with and without weld surface at model 1-spot 2

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Figure 13. HSS bar charts of Table 6 data

Table 7. HSS with and without weld surface at model 2-spot 3

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Figure 14. HSS bar charts of Table 7 data

Table 8. Hot-spot stress, nominal stress and fatigue damage factor compared on rail vehicle model

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Figure 15. FEA stress extracted at weld toe on various mesh types and weld surface modelling in model 1 at spot 1

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Figure 16. FEA stress extracted at weld toe on various mesh types and weld surface modeling in model 1 at spot 2

3.2 Results: Rail equipment model

The results shown in Figures 17 and 18 are the nominal and hotspot stress results on the rail vehicle model. Figure 17 illustrates both the analytical and FEA extracted stress data before and after adding welds to critical areas of the I-beam. Figure 18 presents an image of the analytical calculations performed to obtain the nominal stress. The identified welds were evaluated for nominal stress using analytical methods as well as by curve fitting on extracted FEA stress data. The assessment of hot-spot stress was carried out using higher-order quad-dominant shell, tetrahedral, and hex-dominant element meshes, with the comparison of stress and fatigue damage factors detailed in Table 8. Figures 10, 11, and 12, in conjunction with Table 8, provide insights into two specific aspects: 1) The comparison of fatigue damage factors between Nominal and hot-spot stress, and 2) The variation in hot-spot stress ranges across different mesh types and its impact on fatigue damage factors. The equipment and other structural mass were distributed on the top decking plate of the frame, with an estimated total weight of around 160,000 lbs. The structural mass of the equipment mounted to the mainframe was accounted by creating point mass elements at respective CG points and attaching those to mounting areas on the mainframe. The miscellaneous mass was accounted by creating distributed mass across the mainframe. The geometry model was almost symmetric, with loads and boundary conditions applied symmetrically as well. The left side was created with a shell mesh on midsurfaces, while the right side was created with a solid mesh. The corresponding spots on the left and right are studied and compared for fatigue damage. The FE model was analyzed for 1g load, and the results were scaled down for fatigue load. Analytical calculations were also conducted for 1g load, and the nominal stress was scaled down for fatigue stress range. This evaluation is for HCF; the stress values are well within elastic limits, so scaling of stress for different load magnitudes is possible.

Overlaying the FEA and analytical stress data with potential weld classes on the graph enables the determination of whether a particular type of weld is suitable for the level of stress in that area. The analytically calculated nominal stress is higher than the FEA predicted nominal stress. Historically, Track Maintenance Equipment (TME) rail vehicle assessments show that most welds on the structure fall under class F2 joint classification with the nominal stress approach and class D with the hot-spot stress approach. The nominal stress results indicate that any possibility of an F2-type weld in the highlighted area within the rectangle on the I-beam shown in Figure 17 would likely not meet the fatigue requirements. However, the other overlaid results and further assessment of the hot-spot approach, as shown in Figure 19, indicate that there is a possibility of meeting the fatigue stress criteria if specific welds are assessed more closely using the hot-spot stress method. The six graphs in Figure 19 show extracted stress perpendicular to the weld at location 4, as identified in section 2. Location 4 was assessed for two welds: one at the weld toe and the other at the end of the weld. The extracted data is used for hot-spot stress range calculations using Eq. (1) given in the introduction section. The results are summarized in Table 8, which suggests the importance of the hot-spot stress method when the nominal approach tends to give conservative results. The use of the nominal theory remains a significant time saver, as its conservative results can be obtained very quickly to deem large portions of the structure safe, and then the hotspot method can be used to focus on areas of concern. The stress range obtained in tetrahedral elements is predicting slightly higher hot-spot stress range than on shell and hex-dominant solid mesh; however, the effect on fatigue factor is notable.

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Figure 17. Nominal and hot-spot stress from analytical calculations and extracted FEA stress along I-beam

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Figure 18. Analytical nominal stress calculations along the length of beam for 1g load

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Figure 19. Stress extracted at welds for hot-spot stress and fatigue damage calculations