HFJOINT: A high-fidelity numerical modeling tool for stress concentration factor analysis of welded tubular joints

Contents lists available at ScienceDirect Advances in Engineering Software journal homepage: www.elsevier.com/locate/advengsoft Research paper HFJOINT: A high-fidelity numerical modeling tool for stress concentration factor analysis of welded tubular joints Songhan Zhang a,∗, Wim De Waele a,b, Kris Hectors a,b aSoete Laboratory, Department of Electromechanical, Systems and Metal Engineering, Ghent University, Technologiepark-Zwijnaarde 46, 9052 Zwijnaarde, Belgium bFlandersMake@UGent–Corelab MIRO, 9000 Gent, Belgium A R T I C L E I N F O Keywords: Welded tubular joint Finite element analysis Structured mesh Load mapping Hot-spot stress A B S T R A C T Assessing the fatigue performance of welded tubular joints is crucial to the safety and durability of their host structures. Fast-computing beam-element models are insufficient to accurately capture local stress concentrations at the intersection region, leading to inaccurate lifetime predictions. In this work, a highfidelity numerical modeling tool, HFJOINT, is developed for stress concentration analysis of welded tubular joints, following a user-friendly process. The workflow begins with the creation of an elementary T/Y-joint using quadratic hexahedron elements, where the weld geometry is generated in accordance with the American Welding Society (AWS) standard. The chord and brace are divided into several regions allowing for an entirely structured mesh. The geometric transformations of multiple elementary joints enable creating more complex joints. After evaluating the stiffness matrix, the beam-element forces are converted to external tractions, and are transformed into solid-element nodal forces via Gaussian integral. The boundary conditions are defined from the geometric constraints formulated by the Lagrange’s equation of the second kind. Based on the nodal displacements, the postprocessing module evaluates the local stress at any point. Using linear extrapolation, the hot-spot structural stress and the stress concentration factors (SCFs) along the weld circumference are computed. The workflow has a computational complexity of (𝑁1.89). The mesh convergence shows that the relative changes are below 2% when refining the weld circumference from 64 to 96 segments. The tool is verified against the SCFs of benchmark T-, Kand Xjoint models, showing its potential for fatigue analysis of welded tubular joints in broad applications. 1. Introduction Tubular joints with welded connections are used in a variety of complex, large-scale structures such as offshore wind turbines [1] and bridges [2], because of the balance between strength, own weight and ease of fabrication. As shown in Fig. 1, the simplest case of a welded tubular joint consists of a single brace and chord connecting at a certain intersection angle. The cross section of the weld depends on the dihedral angle, which varies along the weld circumference, reaching its maximum at the crown and the minimum at the saddle. Stress concentration at the weld toe in tubular joints is a critical factor governing fatigue life. The weld toe, where the weld meets the base metal, introduces a geometric discontinuity that acts as a stress raiser, leading to localized high stresses. As a result, the lifetime of welded tubular structures is usually governed by the fatigue behavior of their joints [3]. Fatigue experiments on large tubular joints are extremely costly. High-fidelity numerical modeling techniques [4] provide more accurate, efficient and versatile approaches for the fatigue assessment ∗Corresponding author. E-mail address: [email protected] (S. Zhang). of tubular joints, but require expertise and are time intensive when a generalized modeling approach is considered. The numerical modeling of tubular joints has been an active field of research for several decades [5]. For complex structures consisting of a large number of tubular members, global models are conventionally developed using beam elements [6,7]. Such low-fidelity models are computationally efficient, but are insufficient to accurately describe the local stress fields near the welds that govern the fatigue life due to the weld geometry and thin-wall nature. To address the limitation of beam models, shell-element models [8,9] have been introduced, where the geometric features of the tubular walls and their intersection area are presented. However, these models introduce stress singularities at the sharp edges, leading to highly overestimated fatigue stresses. For this reason, the surface stress extrapolation hot-spot structural (HSS) stress approach was proposed to evaluate the stress at the weld toe from the extrapolation of nearby local nodal stresses. To make the trade off between a beam model (computationally efficient) and a shell model https://doi.org/10.1016/j.advengsoft.2025.104046 Received 17 June 2025; Received in revised form 5 August 2025; Accepted 7 October 2025 Advances in Engineering Software 211 (2025) 104046 Available online 15 October 2025 0965-9978/© 2025 Elsevier Ltd. This is an open access article under the CC BY license ( http://creativecommons.org/licenses/by/4.0/ ). S. Zhang et al. Nomenclature 𝜎 The normal stress on a specified 3D plane after coordinate transform of stress tensor 𝝈Stress tensor in the vector form 𝜂The 2nd natural coordinate for Lagrange’s or transfinite interpolation   𝐫Local coordinates of an interpolation point in the weld cross section 𝜙rRodrigues’ rotation angle of the 3D rotation about a specified axis 𝜙wThe slope angle of the weld upper surface 𝛹Dihedral angle between the chord and brace walls 𝜓rEuler angle for the 3D rotation about global Z axis 𝜌Polar angle of the weld circumference 𝜎Normal stress (having the subscripts xx, yy and zz) 𝜎HS Hot spot stress at the weld toe 𝜎nm Nominal stress evaluated from a beam member 𝜏Shear stress (having the subscripts xy, yz and xz) 𝐀eBoolean matrix mapping the element DOFs into the global DOFs list 𝐁Element strain matrix 𝐃Constitutive matrix of the material 𝐅Global nodal force vector 𝐟Surface traction field resulting from beam element load 𝐉Jacobian matrix from natural coordinates to the physical coordinates 𝐊Global stiffness matrix 𝐊eElement stiffness matrix 𝐑Coordinate transformation matrix 𝐔Global nodal displacement vector 𝐔eElement nodal displacement vector 𝐔FThe displacement vector of the DOFs to be solve 𝐔im The global imposed displacement vector 𝐔IThe displacement vector of the imposed DOFs 𝜃brace intersection angle 𝜃rEuler angle for the 3D rotation about global Y axis 𝜑rEuler angle for the 3D rotation about global X axis  𝐞𝑗The unit direction vectors for the local coordinate system of the weld cross section (𝑗= 1,2,3)  𝐧Normal unit vector of a reference plane  𝐫0The global coordinates of a given point at the top surface of the weld.  𝐫[⋯]Global coordinates of the points in a transfinite mesh, where the subscripts T/B/L/R denotes top/bottom/left/right edges, respectively, and the subscripts 1/2/3/4 denotes the corner nodes  𝐫cb Global coordinates of the points at the inner surface of the chord  𝐫ct Global coordinates of the points at the outer surface of the chord  𝐫cGlobal coordinates of the node at the reference plane of the brace  𝐫ht Global coordinates of the points at the interface between B-HS and B-TR regions  𝐫I1Intersection line between chord and brace (inner side)  𝐫I2Intersection line between chord and brace (outer side)  𝐫IIntersection line between two cylinders (outer side)  𝐫mi Global coordinate of a point symmetric about a specified reference plane  𝐫mv Global coordinate of a point after 3D rotation  𝐫mv Global coordinate of a point after translational moving  𝐫tGlobal coordinates of the points at the top of the B-UN region  𝐫vA vector containing the coordinate of the weld vertices  𝐫w0 The global coordinates of the inner intersection point (I1) of the weld cross section  𝐫𝑖𝑗 Coordinates of the weld interpolation points in the local coordinate system 𝜉The 1st natural coordinate for Lagrange’s or transfinite interpolation 𝛯𝑗A geometry sequence for a transition mesh where the element sizes gradually change. 𝜁The 3rd natural coordinate for Lagrange’s or transfinite interpolation 𝐴Area of the cross section 𝑏cts Width of the chord transition region 𝐹ax Axial force applied on a beam member 𝐹ips In-plane shear force applied on a beam member 𝐹ops Out-of-plane shear force applied on a beam member 𝐼Moment of inertia of the cross section 𝐿Total length of the chord 𝐿bu The axial length of the B-UN region 𝐿hThe distance from the joint center to the reference plane of the brace 𝑀ipb In-plane bending moment applied on a beam member 𝑀opb Out-of-plane bending moment applied on a beam member 𝑁𝑗The bilinear interpolation functions for the mesh grid of weld cross section (𝑗= 1,2,3,4) 𝑛bhs Number of elements along the axis of the brace hot-spot region 𝑛eTotal number of elements in a finite element mesh 𝑛nTotal number of nodes in a finite element mesh 𝑞The factor describing the steepness of the changing element sizes in a transition mesh. 𝑅Outer radius of the chord cross section 𝑟Outer radius of the brace cross section 𝑅wThe gap in height between the chord and brace 𝑇bTorsional moment applied on a beam member 𝑡wThe thickness of the weld cross section 𝑤Weight factors for the Gaussian numerical integral Advances in Engineering Software 211 (2025) 104046 2 S. Zhang et al. 𝑥𝑗The distance from the stress extrapolation point to the weld toe (𝑗= 1,2) 𝑦cDistance from a specified point to the neutral surface (accurate), the stress concentration factor (SCF) [10], characterizing the ratio between the shell HSS stress and the beam nominal stress, is used. In engineering design, the SCFs are determined from empirical parametric equations (e.g., the Efthymiou equation [11]) that are fitted to simulation data and validated with limited experiment data. Therefore, these parametric equations have inherent limitations. Due to the involved exponential functions, the accuracy of the formulae is significantly reduced when a part of the parameters are outside their validity ranges [12]. With increasing computation power, high-fidelity solid-element modeling offers a robust solution for exactly taken into account the complex geometries of tubular joints [13–15]. This allows for the consideration of the detailed shape of the weld cross-section, for example based on the American Welding Society (AWS) standard [16]. Such high-fidelity models typically use 8-node linear [17] or 20-node quadratic hexahedron [13] elements which are available in commercial finite element software packages like Abaqus [18,19], Ansys [20,21] and Comsol [22]. In most cases, a linear-element mesh is generated, namely the raw mesh [23], and is then upgraded to a 20-node quadratic-element mesh, where the added edge nodes are conventionally assumed at the middle positions. High-fidelity modeling in fatigue assessments of welded tubular joints presents several challenges that remain to be addressed: (i) Mesh quality control. Due to the complex geometry, nonstructured meshes are usually considered for the sake of convenience and adaptivity. This may, however, result in distorted elements which affect the precision of computation. A fine, highquality mesh near the weld is required in order to capture the high stress gradient and related stress concentration [24]. Conversely, the global mesh size should ideally be relatively coarse in order to limit the computational cost. This further increases the difficulty in mesh quality control. An alternative approach is to divide the joint into a number of simpler pieces, ensuring that a tailored structured mesh is possible for each of the pieces [23]. The meshed pieces are then merged to form a meshed representation of the entire joint. However, a complex transition scheme remains needed to connect the fine mesh zone near the weld and the far-field zone having a coarser mesh, especially for the compatibility along the thickness direction. (ii) Trade off between accuracy and computational efficiency. The 8-node linear hexahedron and 20-node serendipity hexahedron (where the cross terms are omitted from its shape functions) elements provided in most softwares are insufficient to present the internal stress gradient without a very refined mesh. The complete quadratic hexahedron element has higher accuracy in describing large stress gradients, having potential in handling stress-concentration problems [25], but a larger number of DOFs have to be included. (iii) Complicated workflow. The SCF analysis of a welded tubular joint involves multiple steps including geometric modeling [26], mesh generation, load definition, static solver and surface stress extrapolated HSS stress determination. These refer to the interaction between multiple software tools and considerable manual interventions, which are challenging to handle, especially for non-expert users. It can be concluded that most modeling techniques are limited in accurately capturing the weld geometry, creating a high-quality mesh and performing a stress concentration analysis. To address these challenges, a structured mesh strategy based on 27-node completely quadratic hexahedron elements (Hexa27n), along with the solver for evaluating the stress field and SCFs, is proposed in the present work. It allows for the numerical modeling and fatigue assessment of welded tubular joints using a semi-automatic, user-friendly and controllable approach. In the proposed mesh strategy, a joint is always modeled from the integration of one or multiple T/Y-joints (which is called elementary joints in the following). Such elementary joint is divided into a number of interconnecting regions. Some regions are further divided into a number of parts allowing for structured mesh through either linear or transfinite interpolation in the 3D space. The highfidelity model of the elementary joint is then obtained by merging the structured Hexa27n meshes of all parts. This supports the modeling of more complex multi-brace, multi-planar tubular joints by creating, transforming and merging multiple elementary joints. By considering the boundary conditions from geometric constraints and element surface tractions (e.g., the distributed stress converted from beam-element forces), the stress field of the joint can be solved, allowing for the evaluation of hot-spot stresses and stress concentration factors along the full circumference of the weld. The proposed methodology has been integrated into a novel tool called ’high-fidelity numerical modeling of welded tubular joints (HFJOINT)’. The tool presents a user-friendly interface for (semi)-automatic generation of Hexa27n models of welded tubular joints, allowing engineers to accurately perform SCF calculations for a variety of actual joint geometries without being affected by the validity limitations of empirical formulae. The remainder of this paper is organized as follows. The highfidelity model generation of an elementary joint is demonstrated in Section 2. Based on the geometric transformations of elementary joints, the modeling of complex joints is illustrated in Section 3. The finite element formulation, involving the stiffness matrix, geometric constraints, the force vector and the evaluations of stress field and SCFs, is derived in Section 4. The SCFs evaluated from the developed HFJOINT are verified by means of case studies in Section 5, including a benchmark T-joint model for mesh refinement, computational cost and accuracy, as well as both Xand K-joints for comparison with experiment data. Finally, conclusions are made in Section 6 2. Structured mesh of an elementary joint In this section, the mesh generation of an elementary joint (i.e., a Tor Y-joint having a single pair of chord and brace) will be demonstrated. First of all, a subdivision with respect to the member, region and part levels, is applied on the elementary joint (Fig. 2) in order to cope with the geometric complexity. In HFJOINT convention, the elementary joint is partitioned into brace, weld and chord members. The brace is divided into the hotspot region (B-HS) having a refined mesh, the uniform region (B-UN) having a coarser mesh and the transition region (B-TR) where the mesh size is gradually varying. The chord is divided into the plug region (C-PL) at the center, the hot-spot region (C-HS) below the weld, the transition region (C-TR) having an edge varying from circle-like to quadrilateral-like, and the extend region (C-EXT). The C-PL region is further divided into the center part (C-PL-CT) and the transition part (CPL-TR). The C-EXT region is further divided into four parts towards the X-negative (C-EXT-Xn), X-positive (C-EXT-Xp), Z-negative (C-EXT-Zn), and Z-positive (C-EXT-Zp) directions. Based on this strategy, each destination (part/region) can always be modeled from a perfect structured mesh using hexahedron elements. Considering the accuracy, adaptability and computational efficiency, a complete Lagrange’s 27-node quadratic hexahedron (Hexa27n) element (Fig. 3), is developed and is employed by default in HFJOINT. The convention of the node numbering and the natural coordinates of the nodes are listed in Table A.5. The structured mesh for each region/part will be demonstrated next. Advances in Engineering Software 211 (2025) 104046 3 S. Zhang et al. Fig. 1. Geometric features of a representative welded tubular joint: (a) overview of the joint; (b) close-up view at the intersection between brace and chord walls. Fig. 2. Subdivision strategy of an elementary joint in terms of member, region and part levels. 2.1. Weld In order to properly locate the discretized weld cross sections, the global coordinate system (GCS) (X-Y-Z in Fig. 4(a)) is defined to describe the geometry of the entire joint. The center of the GCS is placed at the intersection point of the chord-brace axes. The X-axis is parallel with the central axis of the chord. The Y-axis is inside the plane of the chord-brace axes and is meantime perpendicular to the X-axis. The Z-axis (out-of-plane) can then be determined by the right-hand rule. In the GCS, the overall geometric shape of the weld is governed by the intersection lines I1 (between the inner surface of the brace and the outer surface of the chord) and I2 (between the outer surface of the brace and the outer surface of the chord). To describe the position of a weld cross-section, the polar angle 𝜌 is defined where 𝜌= 0 indicates the crown at the positive X-side (recall Fig. 1(a)) and 𝜌 increases with the rotation about the global Y-axis. The outer intersection line between two cylinders having the radius 𝑅 (chord) and 𝑟 (brace) can be described by the parametric function [27]:  𝐫I=(1 sin 𝜃{𝑟sin 𝜌−[𝑅−√𝑅2−𝑟2cos2𝜌]cos 𝜃},√𝑅2−𝑟2cos2𝜌, 𝑟 cos 𝜌)(1) where 𝜃 is the brace intersection angle which has been denoted in Fig. 1(a). At each polar angle, a local coordinate system is attached to the Advances in Engineering Software 211 (2025) 104046 4 S. Zhang et al. Fig. 3. The 27-node quadratic hexahedron element used in HFJOINT: (a) an example in the global coordinate system; (b) ordering of the nodes in the natural coordinate system. Fig. 4. Geometric profile of the weld: (a) position and orientation in the global coordinate system; (b) shape of the cross-section in the local coordinate system. weld cross section, see Fig. 4(b). The radial unit vector:  𝐞1=  𝐫I2− 𝐫I1 ‖ 𝐫I2− 𝐫I1‖(2) is pointing from I1 to I2, where both  𝐫I1 and  𝐫I2 are evaluated from Eq. (1) but the brace radius 𝑟 is taken 𝑟−𝑡 (where 𝑡 is the thickness of the brace) and 𝑟, respectively. The tangent unit vector:  𝐞2=1 ‖d 𝐫I1∕d𝜌‖ d 𝐫I1 d𝜌(3) is pointing in the tangent direction of the inner intersection line, where, d 𝐫I d𝜌=(1 sin 𝜃{𝑟cos 𝜌+𝑟2sin 𝜌cos 𝜌cos 𝜃 √𝑅2−𝑟2cos2𝜌},𝑟2sin 𝜌cos 𝜌 √𝑅2−𝑟2cos2𝜌 ,−𝑟sin 𝜌)(4) is explicitly derived from Eq. (1) using the chain rule. The upward unit vector:  𝐞3= 𝐞1× 𝐞2(5) is perpendicular to the previous two. The shape of the weld crosssection can then be described within the planar coordinate  𝐞1− 𝐞3, as shown with the red and green arrows in Fig. 4(b). In the local coordinate system of the weld cross-section, a set of geometric relationships governing the shape of the weld cross section are proposed in the AWS standard D1.1:2025 [16]. For complete joint penetration (CJP) welds, the detailed cross-section features depend on the dihedral angle 𝛹 (the spatial angle between  𝐞1 and the brace axis which can be found in Fig. 1(b)), as shown in Fig. 5. Within the LCS of the weld cross-section, the geometric dimensions shown in Fig. 5 allow to solve the coordinates of the vertices I2, A, B, C, D and E, based on the original point I1(0,0). Taking detail B (Fig. 5(b)) as an example, the coordinates can be calculated from:   𝐫I2=(𝑡w sin 𝜓,0),  𝐫A=(𝑅w tan 𝜓,0),  𝐫B=(𝑅w tan 𝜓, 𝑅w),   𝐫C=(0, 𝑅w),  𝐫E=(ℎw+𝑡w+𝐹w sin 𝜓,0)   𝐫D=(𝑡 sin 𝜓+cos 𝜓(𝑡tan 𝜙w∕ sin 𝜓+𝑅w) sin 𝜓− cos 𝜓tan 𝜙w , 𝑅w+sin 𝜓(𝑡tan 𝜙w∕ sin 𝜓+𝑅w) sin 𝜓− cos 𝜓tan 𝜙w) (6) where 𝑡w is the thickness of the weld cross section, 𝜙w is the slope angle of the weld upper surface, 𝑅w is the gap in height between the chord and brace. Since the purpose of the model is to determine the local elastic stress fields along the brace-chord intersection as input to highcycle fatigue analyses, the finite element simulations will be performed considering linear elastic material behavior. It is fair to assume that the linear elastic material properties of the base and weld material are the same. In that case, only the convex envelope of the weld cross-section has to be modeled. The weld cross-section can thus be characterized by the vertices A, B, D and E (Fig. 4(b)). Such cross-section can then be meshed into a structured grid through the bilinear interpolation:  𝐫𝑖𝑗 =({𝑁1, 𝑁2, 𝑁3, 𝑁4}|||𝜉=𝜉𝑖,𝜂=𝜂𝑗 ⊗𝐈2×2) 𝐫v(7) where the interpolation functions, in terms of discrete (𝜉, 𝜂) points, are evaluated by: 𝑁1=1 4(1 − 𝜉)(1 − 𝜂)(8) 𝑁2=1 4(1 − 𝜉)(1 + 𝜂)(9) Advances in Engineering Software 211 (2025) 104046 5 S. Zhang et al. Fig. 5. The geometries of the weld cross-sections [28] with respect to the range of the dihedral angle 𝜓: (a) detail A; (b) detail B blunt (left) and B sharp (right); (c) detail C; (d) transition from C to D; (e) detail D. 𝑁3=1 4(1 + 𝜉)(1 − 𝜂)(10) 𝑁4=1 4(1 + 𝜉)(1 + 𝜂)(11) and the coordinates of the vertices are collected as:  𝐫v={𝑥A, 𝑦A, 𝑥B, 𝑦B, 𝑥D, 𝑦D, 𝑥E, 𝑦E}T(12) After obtaining the local coordinates of each node,   𝐫, the positions of the nodes within the global coordinate system,  𝐫, can be evaluated through a translational motion (ensuring that the point I1 is exactly at the inner intersection point  𝐫w0) and a 3D rotation (ensuring that all the unit vectors are pointing to the correct directions). This is implemented by the coordinate transformation:  𝐫= 𝐫w0 +𝐑  𝐫(13) where 𝐑={ 𝐞1, 𝐞2, 𝐞3}T(14) is the 3D rotation matrix from the global coordinate system to the local coordinate system. 2.2. Brace As mentioned in Fig. 2, the brace is divided into three regions: B-HS (hot-spot region), B-TR (transition region) and B-UN (uniformly meshed region). For the sake of convenience, a reference plane containing the interface between the B-TR and B-UN regions is defined by its normal vector  𝐧= (cos 𝜃, sin 𝜃, 0) and the point at the center of the cross section  𝐫c= (𝑥c, 𝑦c, 𝑧c) (Fig. 7). For each node at the top surface of the weld, denoted by  𝐫0= (𝑥0, 𝑦0, 𝑧0), the corresponding node at the reference plane is:  𝐫𝑐= 𝐫0+ (𝐿−𝑥0cos 𝜃−𝑦0sin 𝜃) 𝐧(15) In addition to the reference plane, the nodes at the interface between the B-HS and B-TR regions,  𝐫ht , as well as the nodes at the top surface of the B-UN region,  𝐫t, are:  𝐫ht = 𝐫0+𝐿h 𝐧(16)  𝐫t= 𝐫𝑐+𝐿bu  𝐧(17) where 𝐿h is the distance between the joint center (i.e., the origin of the global coordinate system) and the center of the reference plane, and 𝐿bu is the axial length of the B-UN region. Based on this, the nodes within the B-HS region and B-UN region can be obtained from uniform interpolations between  𝐫0 and  𝐫ht , and between  𝐫c and  𝐫t, respectively. The nodes within the B-TR region can be evaluated by using a linear mapping from a geometry sequence between 0 and 1: 𝛯𝑗=1 − 𝑞𝑗−1 1 − 𝑞𝑛(18) to the physical coordinates between  𝐫ht and  𝐫ref , where 𝑞 is the factor to describe the steepness of the changing element sizes which is usually taken between 1.05 and 1.50. Consider a case where the regions are respectively divided into 6, 15, 100 layers and the factor 𝑞= 1.20, an example of the brace mesh is shown in Fig. 8. It is noticed that the bottom nodes of the brace are exactly overlapped with the upper nodes of the weld. 2.3. Chord As mentioned in [27], among others, the geometry of the chord can be obtained by wrapping a rectangular plate, and the grid points of the plate can be simply generated through uniform discretizations Advances in Engineering Software 211 (2025) 104046 6 S. Zhang et al. Fig. 6. Example of a structured mesh of a tubular joint weld profile using tri-quadratic hexahedron elements (𝑛𝜌= 24, 𝑛𝜉= 1, 𝑛𝜂= 1): (a) Global shape of the weld; (b) a close-up view of the weld cross section at 𝑧= 0 with the unit vectors of the local coordinate system. Fig. 7. Node mapping from the top surface of the weld to the reference plane at the top surface of the transition region. with respect to the length (axial), width (circumferential) and thickness directions. For the modeling of tubular joints, specifically, the nodes at the bottom surface of the weld should also be present on the chord. This is difficult to perfectly fulfill from such a uniform discretization with hexahedron elements. Therefore, the chord is partitioned into a number of regions/pieces (recall Fig. 2. Each of the regions/pieces will next be discretized by a structured mesh using Hexa27n elements. 2.3.1. Chord hot-spot region (C-HS) Attached to the bottom of the weld, the C-HS zone will be created with a refined element mesh, in order to allow extraction of the HSS stress near the weld toe. At each polar angle, the nodes at the bottom of the weld are taken. Based on this, another two reference points Pin and Pout are created, representing the inner and outer boundaries of the C-HS zone. The coordinates of Pin and Pout can be evaluated from Eq. (13) where the brace radius 𝑟 is replaced by 𝑟−𝑡−𝑏in and 𝑟+𝑏out , respectively. The parameters 𝑏in and 𝑏out characterizing the width of the C-HS zone are automatically determined as 𝑏in =𝑏out = 2𝑡chord by default, which are sufficient to include the local stress points for the hot-spot stress extrapolation in most of the cases. But still, these are included in the optional parameters where the users are allowed to select their specified values. A total number of 𝑛in uniformly discrete points are then generated between Pin and 𝐸 (weld), as well as 𝑛out points between 𝐴 (weld) and Pout . The inside and outside parts are then discretized into 𝑛in and 𝑛out elements along the radial direction. For a given node  𝐫t at the outer surface of the chord (which has been generated from the above), the coordinates of the corresponding node at the inner surface can be calculated by:  𝐫cb = 𝐫ct −(√𝑦2 ct +𝑧2 ct −𝑅+𝑇) 𝐧(19) where  𝐧={0, 𝑦𝑡, 𝑧𝑡}T √𝑦2 𝑡+𝑧2 𝑡 (20) is the unit vector pointing in the outward radial direction of the chord tubular. Based on the weld model shown in Fig. 6, an example of the attached connection zone is shown in Fig. 9. 2.3.2. Chord plug region (C-PL) For the plug zone, the grid points are first generated on a flat plane. The exact spatial points in the 𝑋-𝑌-𝑍 coordinate system are subsequently obtained from the coordinate transformation from the flat plane into a cylinder. Given that the geometry of the plug zone takes the same topological feature of a circular plate, the structured mesh cannot be made by directly using hexahedron elements for a uniform grid. In such case, the plug zone has to be further divided into the central part (C-PL-CT) and the transition part (C-PL-TR). The C-PL-CT part having the same topological feature as a rectangular plate is created through a uniform grid. The C-PL-TR part can be further divided into four meshes. As can be seen from Fig. 10, the geometry of each piece is a curvilinear trapezium, while it should be noticed that one of the edges is curved because the outer edge takes the shape of the intersection line. Therefore, each mesh can be generated based on a number of quadrilaterals, where the coordinates of the grid points are evaluated from the transfinite interpolation:  𝐫= (1−𝜉) 𝐫l+𝜉 𝐫r+(1−𝜂) 𝐫b+𝜂 𝐫t−(1−𝜉)(1−𝜂) 𝐫1−𝜂(1−𝜉) 𝐫2−𝜉𝜂 𝐫3−𝜉(1−𝜂) 𝐫4 (21) based on a structured grid on the 𝜉-𝜂 plane, where  𝐫1,  𝐫2,  𝐫3 and  𝐫4 are the 𝑥-𝑦 coordinates of the corner points, and the coordinates of the edge points are calculated by:  𝐫t=𝑓(𝜌, 𝑟, 𝑅)(22)  𝐫b= (1 − 𝜉) 𝐫1+𝜉 𝐫4(23)  𝐫l= (1 − 𝜂) 𝐫1+𝜂 𝐫2(24) Advances in Engineering Software 211 (2025) 104046 7 S. Zhang et al. Fig. 8. Example of a structured mesh of a brace (𝑛1= 3, 𝑛2= 7, 𝑛3= 6, 𝑞= 1.20): (a) overview of the brace connected to the weld; (b) a close-up view of the cross section at 𝑧= 0 with the interface to the weld. Fig. 9. Example of a structured mesh of a chord C-HS region (𝑛in = 2, 𝑛out = 3) attached to the underside of the weld: (a) global geometry of the chord connection zone; (b) cross section at the plane 𝑧= 0. Fig. 10. Subdivision of the C-PL-TR part into four transfinite meshes: (a) overview of the mesh strategy; (b) computational domain 𝜉-𝜂 for the transfinite interpolation of the transfinite mesh II as an example. Advances in Engineering Software 211 (2025) 104046 8 S. Zhang et al. Fig. 11. Example of a structured mesh of a chord connection zone (𝑛in = 2, 𝑛out = 3) attached below the weld: (a) overall geometry of the chord connection zone; (b) cross section at the plane 𝑧= 0. Fig. 12. Subdivision of the C-TR region into eight transfinite meshes: (a) overview of the mesh generation; (b) computational domain 𝜉-𝜂 for the transfinite interpolation of the mesh I as an example.  𝐫r= (1 − 𝜂) 𝐫4+𝜂 𝐫3(25) By uniformly discretizing the plug transition zone into three intervals in the computational domain, an example of the entire plug zone connecting with the connection zone is shown in Fig. 11. 2.3.3. Chord transition region (C-TR) For the chord, a transition region extending outward from the hotspot region is defined, where its edge gradually changes from a smooth curve (overlapping with the outer surface of the connection zone) into four distinct boundary curves. In order to generate a quadrilateral mesh on the reference flat plane, the transition zone is subdivided into eight transfinite meshes as shown in Fig. 12. For each mesh, the grid points of a structured mesh can be generated from the transfinite interpolation through Eqs. (21)–(25). As the main focus of the developed model is the stress field near the weld, the element sizes at the locations far away from the weld are allowed to be larger without loss of accuracy. Being different from the plug zone (see Fig. 10(b)), the grid points along the 𝜂 axis, i.e., the radial direction, are therefore generated as a geometry sequence (from Eq. (18)) instead of a uniform grid in the computational domain, while it should be noted that the number of elements along the circumferential direction 𝑛𝜉=𝑛𝜌∕8 is not independent. After mapping the grid points from the planar coordinate system 𝜉-𝜂 to the 3D spatial coordinate system 𝑋-𝑌-𝑍, the structured mesh of the transition zone is generated, as shown in Fig. 13. It has to be mentioned that the size of the C-TR region cannot be too large, in order to avoid the distortion of elements far away from the origin due to the coordinate transformation wrapping the reference flat plate into a cylinder. 2.3.4. Chord extended region (C-EXT) Based on the outer edges of the C-TR region as shown in Fig. 13, the regular structured meshes will be created by extending these edges outward with a certain space interval. First, the edges parallel to the X-axis will be extended along the positive and negative Z-directions, which forms the grid points along the circumferential direction (i.e., C-EXT-Zp and C-EXT-Zn parts, respectively). Similarly, the edges perpendicular to the X-axis, resulting from the C-TR, C-EXTZn and C-EXT-Zp, are combined and are extended along the positive and negative X-directions, which form the C-EXT-Xp and C-EXT-Xn parts, respectively. An example of the C-EXT region is shown in Fig. 14. 2.4. Merge multiple meshes Once all the meshed regions/parts have been generated (from Sections 2.1–2.3), the entire model of the elementary joint can be created by merging together all the meshes through node sharing at their interfaces. Such operation includes the following steps: 1. Model combination: For the main mesh and an added mesh, the nodal coordinates are collected in the 3-column matrices Advances in Engineering Software 211 (2025) 104046 9 S. Zhang et al. Fig. 21. Mapping from beam-element force to nodal forces at the loaded cross section: (a) axial force; (b) in-plane bending moment; (c) out-of-plane bending moment. Fig. 22. The signed von Mises stress field (unit: MPa) of the example T joints subjected to: (a) axial loading; (b) in-plane bending; (c) out-of-plane bending. Table 1 Convergence of the SCFs with respect to the mesh refinement for the outer side of the C-HS region. 𝑛out AX IPB OPB BS BC CS CC BC CC BS CS 1 8.12 1.53 11.26 4.34 1.52 3.17 8.10 12.16 2 8.07 1.52 14.16 4.84 1.51 3.52 8.06 15.27 3 8.06 1.52 15.27 4.52 1.51 3.28 8.05 16.61 4 8.06 1.52 15.02 4.49 1.51 3.25 8.05 16.36 (log10 𝑛DOF,log10 𝑇CPU), the complexity of the entire work flow is estimated as (𝑁1.89), which is slightly lower than (𝑁2) in most standard finite element analysis [32]. This is benefited from the lightweight feature of the HFJOINT solver and the efficient computation with csr matrices. Considering both the normal stress (after the coordinate transform using Eq. (59)) and the maximum principal stress, the resulting SCFs at the brace and chord sides are computed, as shown in Fig. 24. The results are further compared with a benchmark model developed in Abaqus version v2019 (having an unstructured mesh using 20-node brick elements, C3D20R) [27], where the SCF distributions are investigated (see the blue area) based on a group of parameter ranges. From the figures, it can be seen that the SCFs evaluated from the normal stress to the weld toe and the maximum principal stress are close to each other, because the stress is dominated by the bending normal stress. Exemptions are the SCFs in the weld crown regions for the in-plane bending case (Fig. 24(e)) due to the significant shear stress component which contributes to the maximum principal stress. It is also worth mentioning that, for the in-plane bending case, the normal stress is perfectly zero at the saddle (see the SCFs at 90◦ and 270◦ Advances in Engineering Software 211 (2025) 104046 16 S. Zhang et al. Fig. 23. Convergence of the SCFs with improving mesh refinement: (a) axial tension (brace side); (b) in-plane bending (brace side); (c) out-of-plane bending (brace side); (d) axial tension (chord side); (e) in-plane bending (chord side); (f) out-of-plane bending (chord side). Table 2 Summary of the computational cost for the axial loaded case. 𝑛𝜌Number of DOFs Sparsity Memory cost CPU time (s) 𝐊g𝐊g,bc of 𝐊g,bc for 𝐊g,bc (MB) 𝐊g𝐊g,bc 𝐔g𝝈n 32 26388 25 740 0.5505% 41.84 21.13 1.51 5.06 10.94 48 36360 35 640 0.3988% 58.11 50.32 3.92 9.14 27.47 64 46620 45 828 0.3108% 74.86 50.80 3.60 31.50 24.09 96 68 004 67 068 0.2128% 109.79 108.32 9.64 71.63 60.58 Fig. 24. The stress concentration factors along the weld circumference of the example T joint: (a) axial tension (brace side); (b) in-plane bending (brace side); (c) out-of-plane bending (brace side); (d) axial tension (chord side); (e) in-plane bending (chord side); (f) out-of-plane bending (chord side). Advances in Engineering Software 211 (2025) 104046 17 S. Zhang et al. Table 3 Calculation of mandatory modeling parameters for X and K example joints. Joint Parameters from Ref. HFJOINT input 𝐷 𝜏 𝛽 𝛾 𝛼 𝜃 𝑟brace 𝑡chord 𝑡brace 𝐿chord X (E30) 508 mm 0.79 0.38 20.7 9.8 90◦96.52 mm 12.27 mm 9.69 mm 2.489 m K (K-4) 216 mm 0.88 0.47 13.5 10.2 60◦50.76 mm 8.00 mm 7.04 mm 1.266 m Table 4 The SCFs of the Xand K-joints evaluated from HFJOINT (based on the transformed normal stresses, Eq. (59)) with comparison to the experimental results [33]. Joint Approach AX IPB OPB BC BS CC CS BC BS CC CS BC BS CC CS X (E30) HFJOINT 0.4 7.6 4.5 21.3 1.7 0.0 4.2 0.0 0.0 4.4 0.0 12.2 Experiment 1.5 10.6 3.3 21.8 2.5 – 4.4 – – 3.8 – 7.7 K (K-4) HFJOINT 1.0 2.2 6.5 5.6 1.7 0.0 3.4 0.7 0.0 3.3 0.0 8.6 Experiment – 2.3 5.4 3.6 – – – – – 2.9 – 6.8 in Figs. 24(b) and 24(e)), and for the out-of-plane bending case, the normal stress is perfectly zero at the crown (see the SCFs at 0◦ and 180◦ in Figs. 24(c) and 24(f)). This is caused by the neutral surfaces of the bending deformations given that the brace is rather slender. Moreover, the SCF curves (absolute values) are all symmetric about a 180◦ shift, since the model is geometrically symmetric about the plane 𝑥= 0. For the SCF distributions at the brace side, the results from HFJOINT fall well within the envelope of the benchmark model [27]. In [27], constrained optimizations were performed at each polar angle to achieve the minimum and maximum weld cross-section areas within the AWS recommendations. Such optimization is not included in this work. The weld features are taken 𝑅w= 0.6𝑇= 7.62 mm and 𝜙w= 45◦ for the entire circumference, yielding in the cross-section area overall close to (at places, even slightly higher than) the maximum case of the AWS recommendations. The SCFs of the HFJOINT model are therefore slightly closer to the lower boundaries of the benchmark model, especially for the IPB case. For the SCF distributions at the chord side, the blue values are constrained within narrow ranges due to their lower sensitivities to the weld geometric parameters. Considering the discrepancy resulting from the optimization, the SCF distributions evaluated from HFJOINT overall show good agreement with the benchmark model. From the above comparison, the reliability of the developed HFJOINT tool is confirmed. 5.2. Case example II - stress concentration factors of Xand K-joints In order to further validate the SCFs calculated using HFJOINT, Xand K-joints are selected from [33] where a database of experimental results is provided. From the chord diameter and the remaining nondimensional parameters characterizing the geometric features of the joints, the parameters required for HFJOINT modeling inputs are calculated and provided in Table 3. The material properties are the same as defined in Section 5.1. Except for the mandatory parameters, the chord length 𝐿chord is also specified since it has been provided in the literature. Additionally, the optional parameter 𝑛𝜌= 64 and 𝑛out = 3 are selected referring to the mesh convergence studied in Section 5.1. For the chord, the nodes on the end cross sections are fully constrained as the boundary conditions of the model. The load conditions axial tension (AX), in-plane bending moment (IPB) and out-of-plane bending moment (OPB) are considered. For the X joint, the loads are symmetrically applied on both the braces, in order to be consistent with the balanced load mentioned in the literature. The von Mises stress fields are then solved, as shown in Fig. 25. For each load case, the SCFs along the weld circumference, with respect to both the brace and chord sides, can be obtained by following the HFJOINT workflow, where the normal stresses after the coordinate transform (Eq. (59)) are taken. In Table 4, the SCF values at the crown and saddle points are listed, and are compared with test data provided in literature [33]. From Table 4, it can be concluded that the SCFs analyzed from HFJOINT align well with the experimental data for most loading cases and positions. For example, under AX and IPB loading, the SCF deviations at most of the positions are below 10%, and in several cases (e.g., AX-CC and IPB-CC), the differences are even below 5%. Meanwhile, a few discrepancies can still be noted. For instance, in OPB conditions, the SCFs at the chord (saddle side) are always higher, where the deviation may reach up to 40%. This may possibly be caused by inevitable deviations in positioning the strain gauges used to calculate the experimental SCF values, as well as due to the real geometry of the weld which has a sensitive effect on the local stress field. Despite this, the overall agreement confirms the reliability of HFJOINT for analyzing local stress distribution features in welded tubular joints, showing its potential for fatigue assessment of complex joints having diverse configurations. 6. Conclusions In this study, a high-fidelity numerical modeling tool, HFJOINT, is developed for the SCF analysis of welded tubular joints. Compared with general finite element analysis tools (and softwares), the developed tool supports for the structured mesh of elementary joints, the creation of complex 3D joints, the mapping from beam-element forces to Hexa27n nodal forces, the extraction of local stress at an arbitrary point and the evaluation of weld SCFs at both the chord and brace sides, from a semiautomatic and user-friendly approach. Especially, the following points are remarked: 1. The Hexa27n mesh of an elementary joint can be generated automatically as long as the mandatory parameters (the radius and thickness of chord and brace, as well as the intersection angle) are defined. The values of the remaining optional parameters will be taken by default rules if not specified. 2. For the load mapping from beam-element force to the Hexa27n nodal forces, the element nodal force vectors are evaluated by considering the exact distributions of axial, shear and bending stresses acting on the loaded cross section. 3. The SCFs can be evaluated from either the maximum principal stress or the transformed normal stress perpendicular to the weld toe. The local stress points are automatically selected according to the DNV [29] recommendations. 4. The developed HFJOINT tool is validated through comparison with the SCFs of the benchmark T-, Kand Xjoints obtained from simulation results and test data reported in literature, Advances in Engineering Software 211 (2025) 104046 18 S. Zhang et al. Fig. 25. The von Mises stress fields (unit: MPa) of the example joints (where the deformations are largely scaled in order to be visible): (a)(b)(c) X joint; (d)(e)(f) K joint; (a)(d) axial tension; (b)(e) in-plane bending; (c)(f) out-of-plane bending. confirming its feasibility and potential in fatigue assessment of broader practical welded tubular joints. Based on the developed GUI, a beta version of HFJOINT will soon be available online. The documentation has been made available on the Github website [34] and will be periodically updated. It should be noted that the current version of HFJOINT is developed under the assumption of linear elastic analysis, without considering plastic deformation, crack propagation or geometric imperfections. The weld geometry is idealized, ignoring as-built weld profile variations. The linear extrapolation of stress as recommended by standards may result in underestimated HSS and SCFs in cases with steep stress gradients. Future developments will aim to enrich HFJOINT by including the multi-scale modeling techniques [35] for weld profile irregularities [36], fatigue crack growth analysis [37] and lifetime assessment [13]. The tool will also be regularly updated to be compatible with state-of-the-art standards and methods. All of this will enable more accurate modeling and reliable fatigue assessment of welded tubular joints in broader practical applications. CRediT authorship contribution statement Songhan Zhang: Writing – original draft, Visualization, Validation, Software, Methodology, Formal analysis, Data curation. Wim De Waele: Writing – review & editing, Supervision, Resources, Funding acquisition, Conceptualization, Data curation, Investigation. Kris Hectors: Writing – review & editing, Validation, Project administration, Investigation, Data curation, Methodology, Resources. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments This work is part of the FLOATFARM project, funded by the European Union’s Horizon Europe research and innovation programme under grant agreement No. 101136091. Advances in Engineering Software 211 (2025) 104046 19 S. Zhang et al. Table A.5 Natural coordinates of the nodes in a Hexa27n element. Nodes 1–9 Nodes 10–18 Nodes 19–27 No. 𝜉 𝜂 𝜁 type No. 𝜉 𝜂 𝜁 type No. 𝜉 𝜂 𝜁 type 1−1−1−1 corner 10 1 0 −1 edge center 19 1 1 0 edge center 2 1 −1−1 corner 11 0 1 −1 edge center 20 −1 1 0 edge center 3 1 1 −1 corner 12 −1 0 −1 edge center 21 −1 0 0 face center 4−1 1 −1 corner 13 0 −1 1 edge center 22 1 0 0 face center 5−1−1 1 corner 14 1 0 1 edge center 23 0 −1 0 face center 6 1 −1 1 corner 15 0 1 1 edge center 24 0 1 0 face center 7 1 1 1 corner 16 −1 0 1 edge center 25 0 0 −1 face center 8−1 1 1 corner 17 −1−1 0 edge center 26 0 0 1 face center 9 0 −1−1 edge center 18 1 −1 0 edge center 27 0 0 0 element center [𝐓(𝜎)]−1 = ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣ 3.234 −1.458 0.411 −1.458 0.657 −0.185 0.411 −0.185 0.052 −1.458 0.657 −0.185 0.657 −0.296 0.083 −0.185 0.083 −0.024 0.411 −0.185 0.052 −0.185 0.083 −0.024 0.052 −0.024 0.007 0.411 −0.185 0.052 −0.185 0.083 −0.024 0.052 −0.024 0.007 −1.458 0.657 −0.185 0.657 −0.296 0.083 −0.185 0.083 −0.024 3.234 −1.458 0.411 −1.458 0.657 −0.185 0.411 −0.185 0.052 0.052 −0.024 0.007 −0.185 0.083 −0.024 0.411 −0.185 0.052 −0.185 0.083 −0.024 0.657 −0.296 0.083 −1.458 0.657 −0.185 0.411 −0.185 0.052 −1.458 0.657 −0.185 3.234 −1.458 0.411 0.411 −0.185 0.052 −1.458 0.657 −0.185 3.234 −1.458 0.411 −0.185 0.083 −0.024 0.657 −0.296 0.083 −1.458 0.657 −0.185 0.052 −0.024 0.007 −0.185 0.083 −0.024 0.411 −0.185 0.052 0.411 −1.458 3.234 −0.185 0.657 −1.458 0.052 −0.185 0.411 −0.185 0.657 −1.458 0.083 −0.296 0.657 −0.024 0.083 −0.185 0.052 −0.185 0.411 −0.024 0.083 −0.185 0.007 −0.024 0.052 0.052 −0.185 0.411 −0.024 0.083 −0.185 0.007 −0.024 0.052 −0.185 0.657 −1.458 0.083 −0.296 0.657 −0.024 0.083 −0.185 0.411 −1.458 3.234 −0.185 0.657 −1.458 0.052 −0.185 0.411 0.007 −0.024 0.052 −0.024 0.083 −0.185 0.052 −0.185 0.411 −0.024 0.083 −0.185 0.083 −0.296 0.657 −0.185 0.657 −1.458 0.052 −0.185 0.411 −0.185 0.657 −1.458 0.411 −1.458 3.234 0.052 −0.185 0.411 −0.185 0.657 −1.458 0.411 −1.458 3.234 −0.024 0.083 −0.185 0.083 −0.296 0.657 −0.185 0.657 −1.458 0.007 −0.024 0.052 −0.024 0.083 −0.185 0.052 −0.185 0.411 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 2.187 −0.986 0.278 −0.986 0.444 −0.125 0.278 −0.125 0.035 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.278 −0.125 0.035 0.000 0.000 0.000 0.000 0.000 0.000 −0.986 0.444 −0.125 0.000 0.000 0.000 0.000 0.000 0.000 2.187 −0.986 0.278 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.278 −0.125 0.035 −0.986 0.444 −0.125 2.187 −0.986 0.278 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 2.187 −0.986 0.278 0.000 0.000 0.000 0.000 0.000 0.000 −0.986 0.444 −0.125 0.000 0.000 0.000 0.000 0.000 0.000 0.278 −0.125 0.035 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.278 −0.986 2.187 −0.125 0.444 −0.986 0.035 −0.125 0.278 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.035 −0.125 0.278 0.000 0.000 0.000 0.000 0.000 0.000 −0.125 0.444 −0.986 0.000 0.000 0.000 0.000 0.000 0.000 0.278 −0.986 2.187 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.035 −0.125 0.278 −0.125 0.444 −0.986 0.278 −0.986 2.187 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.278 −0.986 2.187 0.000 0.000 0.000 0.000 0.000 0.000 −0.125 0.444 −0.986 0.000 0.000 0.000 0.000 0.000 0.000 0.035 −0.125 0.278 0.000 0.000 0.000 0.000 2.187 0.000 0.000 −0.986 0.000 0.000 0.278 0.000 0.000 −0.986 0.000 0.000 0.444 0.000 0.000 −0.125 0.000 0.000 0.278 0.000 0.000 −0.125 0.000 0.000 0.035 0.000 0.000 0.278 0.000 0.000 −0.125 0.000 0.000 0.035 0.000 0.000 −0.986 0.000 0.000 0.444 0.000 0.000 −0.125 0.000 0.000 2.187 0.000 0.000 −0.986 0.000 0.000 0.278 0.000 0.000 0.035 0.000 0.000 −0.125 0.000 0.000 0.278 0.000 0.000 −0.125 0.000 0.000 0.444 0.000 0.000 −0.986 0.000 0.000 0.278 0.000 0.000 −0.986 0.000 0.000 2.187 0.000 0.000 0.278 0.000 0.000 −0.986 0.000 0.000 2.187 0.000 0.000 −0.125 0.000 0.000 0.444 0.000 0.000 −0.986 0.000 0.000 0.035 0.000 0.000 −0.125 0.000 0.000 0.278 0.000 0.000 0.000 0.000 0.000 1.479 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 −0.667 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.188 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.188 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 −0.667 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 1.479 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 1.479 0.000 0.000 −0.667 0.000 0.000 0.188 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.188 0.000 0.000 −0.667 0.000 0.000 1.479 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 1.479 −0.667 0.188 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.188 −0.667 1.479 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 1.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 ⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (A.2) Box I. Appendix A. The constant matrix for stress extrapolation The natural coordinates of the nodes in each Hexa27n element are ordered as shown in Table A.5. Given the stresses at the Gaussian points 𝝈(G), the nodal stresses 𝝈(n) can then be evaluated from the extrapolation: 𝝈(n) = [𝐓(𝜎)]−1𝝈(G) (A.1) By substituting the shape functions (associated with the natural coordinates listed in Table A.5) into the stress interpolation formulated in Eq. (44), the stress extrapolation matrix [𝐓(𝜎)]−1 is explicitly solved as the equation given in Box I: for evaluating the Hexa27n nodal stresses from the stresses at the Gaussian points. Appendix B. Parameters for the mesh generation of an elementary joint When creating the elementary joint, the mandatory parameters must be manually provided from the user. The optional parameters can either be taken by default rules or be specified by the user. The physical meanings, recommended units and the default values are listed in Table B.1. Data availability Data will be made available on request. Advances in Engineering Software 211 (2025) 104046 20 S. Zhang et al. Table B.1 Geometric parameters (in alphabetical order) defined for the mesh generation of an elementary joint (Note that the mandatory parameters marked with ∗ do not have default values and must therefore be provided from the users). Variable description Default value Unit *Brace inclination angle (required) – deg Edge length of the C-PL-CT part (circumferential) 2𝑟∕3 m Transit factor of the B-TR region 1.20 – Transit factor of the C-TR region 1.15 – Node interval along the axis of the B-HS region 3𝑡∕𝑛bhs m Edge length of the C-PL-CT part (axial) 2𝑟∕3 m Edge length of the C-TR region (axial) 3.5 (𝑟+𝑡)m Edge length of the C-TR region (circumferential) 3.5 (𝑟+𝑡)m Length of the B-UN region 0.25 𝐿m Axial length of the B-TR region 0.25 𝐿m Total length of the chord 2𝑏cts m Number of intervals in the B-UN region 6 – Number of node intervals in the C-HS region (outward) 6 – Number of node intervals in the C-HS region (interior) 4 – Number of nodes along the radius of the C-TR region 5 – Number of nodes along the radius of the C-PL-TR part 5 – Number of nodes along the axis of the C-EXT-Xn/Xp part 5 – Number of nodes along the circumference of the C-EXT-Zn/Zp part 5 – Number of node intervals along the weld circumference 32 – Number of nodes through the chord thickness 2 – Number of intervals along the axis of the B-TR region 10 – Number of intervals along the radial direction of the weld 2 – Number of intervals along the upward direction of the weld 2 – Number of intervals along the axis of the B-HS region 6 – Inclined angle for the upper edge of the weld cross section 𝜋∕6 rad The gap between the C-PL region and the weld toe 2.0𝑇m The gap in height between the chord and the brace 0.6𝑡m *Outer radius of the brace (required) – m *Outer radius of the chord (required) – m *Wall thickness of the brace (required) – m *Wall thickness of the chord (required) – m The gap between the C-TR region and the weld toe 2.0𝑇m References [1] Saini DS, Karmakar D, Ray-Chaudhuri S. 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