Stress concentration at cruciform welded joints under axial and bending loading modes

The paper is concerned with the problem of stress concentration in cruciform fillet welded joints subjected to axial and bending load. Extended numerical analyses were carried out with the help of the finite element method. It made it possible to estimate stress concentration factors Kt for a variety of geometrical parameters defining the geometry of cruciform welded joints. It has been found that approximate Kt formulas, available in the literature, have two disadvantages, i.e. an unknown accuracy and small range of application with respect to geometrical parameters defining the weld shape. For these reasons, more general and accurate new formulas for stress concentration factors Kt have been derived. Even though the present approach is applicable to all types of welded joints, the analysis presented below has been conducted for a cruciform joint with the weld flank angle of θ = 45°. Final solutions have been given in the form of polynomial expressions, and they can be easily used in computer-aided design procedures.


List of symbols a
Weld throat thickness K t Theoretical elastic stress concentration factor K tb Pure bending stress concentration factor K tt Pure axial stress concentration factor h Attachment weld leg length h p Main plate leg length n=λ−1 Stress field exponent for a sharp corner N Number of loading cycles S Cyclic stress t Thickness of the main plate T Thickness of the attachment plate X=ρ/(ρ+a) Normalised weld toe radius parameter Y=a/(a+t) Normalised weld thickness parameter 2α Total angle of the sharp corner δ x Accuracy of the approximate K t value θ Weld flank angle κ Correction function for the relative attachment thickness T/a κ b Correction function for the relative attachment thickness T/a for pure bending load κ t Correction function for the relative attachment thickness T/a for pure axial load λ Eigenvalue of the characteristic equation ρ Weld toe radius σ b Nominal bending stress σ t Nominal axial stress

Introduction
The linear elastic stress concentration factor K t is one of the most important parameters used in predicting fatigue life of structural components with various types of stress raisers like holes, notches, grooves, and stiffeners. A variety of K t solutions and approximated formulas can be found in the literature (e.g. [1][2][3][4][5][6][7]). Welded joints are often used in engineering practice, but apart from their economic advantages, the Recommended for publication by Commission XV -Design, Analysis, and Fabrication of Welded Structures technological process involved possesses several additional problems which have to be considered in design procedures. The heat-affected zone, residual stresses, microstructural changes of the material and geometry of the weld-producing local non-uniform stress distribution are of great importance while predicting the fatigue strength and durability of such joints. The damage process usually initiates at the weld toe [8,9], especially in the case of cruciform joints without the lack of penetration [10] or when the fillet weld size is sufficiently large [11]. Therefore, this region is of special interest to designers. From the point of view of the joint quality, it is important to predict the maximum permissible stresses at critical locations of the joint as well as to verify design requirements by means of measurable geometry parameters. The general form of a cruciform welded joint and its basic geometrical dimensions are shown in Fig. 1.
It is generally accepted [12,13] that the overall load applied to the main plate of thickness t may be regarded as the superposition of the axial and bending load. For this reason, the maximum stress, appearing at critical locations, shown in Fig. 1 as circles, depends on both the axial and bending load magnitude, and it is described by two stress concentration factors K tt and K tb , respectively. These factors are usually defined as the ratio of the maximum peak stress to the nominal stress, often called the remote stress [14,15] or as the average stress in the notch root cross section. Sometimes, the hot spot stress [12] is also used to define the stress concentration factor as well. Despite the fact that maximum stresses due to tension and bending are not located at the same point on the small curvilinear surface of the weld toe, they are close enough to consider the peak stress as being the sum of those two independent contributions. Since the stress concentration phenomenon depends on several geometrical parameters of the joint, the determination of the stress concentration factor K t is a troublesome and time-consuming process. For practical engineering applications, various approximate formulas of K t are used.
2 Available stress concentration factor formulas for cruciform joints A variety of approximate K t formulas for welded joints can be found in the literature [12,[14][15][16]. Examples of such formulas, presented by Ushirokawa and Nakayama (U&N) [14] and Tsuji [16] for a cruciform joint subjected to axial and bending load, are shown below, Eqs. (1)- (3):
Taking into account the typical shape and slope of the well-known fatigue S-N curve [17,18], it is clear that the predicted fatigue life of a welded joint depends strongly on the accuracy of estimated K t factors. Despite the fact that several authors were concerned mostly with  [11,19,20], the stress concentration factor K t is also useful for estimating stress distributions in critical cross sections needed for fracture mechanics-based analyses [9,13]. For this reason, extended numerical FEM-based analyses have been carried out to verify the range of validity and accuracy of available approximate K t formulas (1)- (3) and to develop extended K t formulas for a wider range of geometrical parameters.
3 Analysis of stress concentration factors by using the finite element method

General assumptions
The shapes and basic geometrical parameters of the full penetration cruciform welded joint are depicted in Fig. 1.
The following assumptions have been made: 1. Joint material is linear elastic, isotropic and homogeneous. 2. Small deformations occur due to external loadingtensile and bending. 3. Joint material is free from residual stresses, structural irregularities and imperfections including lack of penetration defects. 4. Weld faces are plane, and contour of the weldment is smooth, with a transition radius ρ > 0. 5. Weld flank angle θ = 45°. 6. Both attached plates are of the same thickness T and are co-linear. 7. Four welds are symmetrical. 8. K tt for tensile and K tb for bending loads are defined as σ 1max /σ t and σ 1max /σ b , respectively.

The numerical model and boundary conditions
Numerical models of a variety of cruciform joint geometries have been developed using the ANSYS FEM program. Due to the double symmetry of the joint under axial load and symmetric-anti-symmetric deformation under bending load, only one quarter of the total joint was modelled. The geometry of the joint and the boundary conditions are schematically shown in Fig. 2. The weld throat thickness parameter a was introduced for convenience, and it represents, in this case (for θ = 45°), the weld throat thickness, as shown in Figs. 1 and 2. The attachment leg length h and the main plate leg length h p , shown in Fig. 1 and appearing in Eqs.
(1)- (3), are related to the weld throat thickness a in the form of Eq. (4).
Equation (4) makes it possible to reduce the number of independent variables used in the analysis. In the present case (θ = 45°), the following relations hold a = 0.707 h = 0.707h p . Besides that, two non-dimensional parameters X and Y have been introduced in the form of expressions (5) and (6).
The introduction of parameters X and Y makes further analysis more convenient than when directly using parameters ρ/a and a/t. In this way, all possible values of parameters X and Y fall into the range of 0, 1. For each particular geometrical case analysed in plane stress, a very fine mesh composed of about 800,000 linear PLANE182 finite elements was used. PLANE182 finite element is defined by 4 nodes having 2 degrees of freedom at each node. The mesh density was increased in the weld toe region where the number of finite elements along the arc, described by the toe radius ρ, was about 15 to 60, depending on the radius magnitude. In all cases, the geometrical proportions were selected in a way to obtain an unperturbed nominal stress field in a cross section located sufficiently far from the weld zone. At the initial stage of building the FEM model, the finite element mesh density was successively increased to obtain a stable numerical solution with the maximum stress value staying constant. As a result of gathering a significant number of cases and corresponding numerical solutions, a special mesh-generating procedure was developed. Moreover, the size of the finite elements changed smoothly the further from the zone of maximum stress concentration. These assumptions made the finite element mesh very fine.
The weld toe radius, ρ 1 , along the attachment weld line was arbitrarily chosen as 0.1a, and it had no influence on stress concentration factors K tt and K tb . Relative lengths of the main and attachment plate, as well as the type of load and "rolling support" displacement boundary conditions, are shown schematically in Fig. 2. The thickness of the attachment plate, denoted as T in the present analysis, was normalised with respect to the weld throat thickness parameter a and varied within the range of 1 ≤ T/a ≤ 4. The non-dimensional parameters X and Y have fallen into the range of 0.01 to 0.57 each. Particular values of stress concentration factors K tt and K tb were obtained by dividing the maximum peak stress at the weld toe by the nominal remote stress.
Number of finite elements, nodes and the minimum size of the element depended on the proportions between geometrical parameters of the joint represented by X, Y and T/a. For example, the model shown in

Accuracy of approximate K t formulas
After performing the numerical calculations, approximate K tt and K tb values, given by formulas (1)-(3), were compared to those ones obtained with the help of the FEM method. Two examples of such comparisons obtained for the tensile and bending load are shown in Table 1 and  Table 2, respectively. The accuracy δ x is defined as the difference (%) between approximate K t formulas (Eqs. (1)-(3)) and those obtained from the present FEM analysis and relative to the FEM data.
From the set of results presented in Table 1 and Table 2, one may conclude that the accuracy of stress concentration factors K tt and K tb obtained from approximate formulas (2) and (3) varies within the range of + 12.3 to − 22% and would give rather poor predictions of fatigue life. It appears that the approximate formulas have two important disadvantages: undetermined accuracy and narrow range of application with respect to geometrical parameters defining the weld geometry. For this reason, it seems necessary to elaborate on a slightly updated approach to the problem of stress concentration factors K tt and K tb in cruciform welded joints.

The weld toe radius and the plate thickness effects
It is important, while analysing the stress concentration phenomenon in welded joints with fillet welds, to know the effect of particular geometrical parameters on stress concentration factors K tt and K tb . Two typical situations appearing while designing welded structures are considered below. The first one is when only the weld toe radius ρ varies, and the second one is when only the main plate thickness t varies, while in both cases, all remaining parameters stay the same. It is worth noting that in both cases, the classical most important nondimensional parameter ρ/t changes as well, and it tends to the limit zero when ρ → 0 or t → ∞. As a result, two different limit stress concentration factors occur for the value of ρ/t → 0.

The weld toe radius effect
It is well-known that the weld toe radius ρ is the most important parameter influencing the stress concentration in a welded joint. Therefore, there are design requirements available preventing using notch tip radii smaller than the recommended limit.
Let us assume that the weld toe radius ρ varies, while all remaining geometrical parameters stay the same. The FEM-based stress concentration factors K tt obtained in due course are shown in Fig. 4 in terms of the normalised parameter X. They were determined for several levels of parameter Y.
All curves (Fig. 4) are singular, which means that stress concentration factors increase rapidly when the weld toe root radius ρ approaches zero, while all other geometrical parameters of the joint do not vary. Such a behaviour needs special mathematical care when using approximate fitted expressions to capture variations of the stress concentration factor. In such cases, Eqs. (1) and (2) can be presented in the form of Eqs. (7) and It is worth noting that the exponents in Eqs. (7) and (8) are different, and this means that the stress concentration factors K tt will vary with different rates when the weld toe radius decreases, i.e. when ρ → 0.

The plate thickness effect
Another important case occurs when only the thickness t of the main plate varies, while other geometrical parameters of the joint remain unchanged. Such a situation takes place when all weld dimensions depend on constant thickness T of the attachment plate, while the thickness t of the main plate varies. The K tt data obtained for such a case are presented in Fig. 5, where the axial stress concentration factor was obtained for a range of values of parameter X, and it is given as a function of the relative weld throat thickness Y.
In all those cases, the K tt vs. Y relationship can be approximated by a smooth and regular function tending  Fig. 4 Influence of the relative weld toe radius X on the axial stress concentration factor K tt obtained for a variety of Y ratios while T = 1.4a Fig. 5 The effect of the relative thickness Y on the axial stress concentration factor K tt for various X ratios, while T = 1.4a to a finite limit when the relative weld throat thickness Y tends to zero. When the main plate thickness t increases and the weld toe radius ρ stays constant, the ratio ρ/t → 0, but the stress concentration factor K tt tends to a finite value and not to infinity, as in the previous case of ρ → 0.

Conclusions drawn from finite element method solutions
The analysis of the two cases named as the "radius effect" and the "thickness effect", presented above, indicates that the ρ/t ratio is a very important parameter influencing stress concentration factors. However, the stress concentration factor K tt depends also on other parameters like the weld throat thickness a and the attachment thickness T. The data presented in Figs. 4 and 5 indicate that when the weld toe radius ρ tends to zero, while the plate thickness t stays constant, the singularity effect occurs, resulting in theoretically infinite stresses at the weld toe. However, when the weld toe radius ρ remains constant, while the main plate thickness t tends to infinity, the maximum stress at the weld toe tends to a finite value. The dual tendency concerning stress concentration factors in welds needs appropriate treatment.
Analogous data and tendencies concerning stress concentration factors under axial load (Figs. 4 and 5) were also obtained under pure bending load. The phenomenon discussed above is illustrated in Fig. 6 for both pure tension (a) and bending load (b), where at least two parameters describing the variation of stress concentration factors in weldments are necessary because the description in terms of only one parameter ρ/t is insufficient.
The data presented in Fig. 6 were obtained with the finite element method by varying the parameters ρ/a and a/t within the range of 0.01 to 1.3.

General form of the analytical stress concentration factor formula
The data presented and discussed above indicate that any general mathematical formula for the weldment stress concentration factor requires using an expression capable of modelling the singularity when the weld toe radius ρ tends to zero. It is also necessary to approximate a potential stress concentration factor expression with respect to at least two geometrical parameters such as X and Y defined earlier.

The sharp corner configuration as the limiting case of the weld toe radius
When the weld toe radius ρ → 0, the weld toe region converts into a sharp corner with its characteristic angle 2α, as shown in Fig. 7.
The fundamental solution to the case [21,22] shown in Fig.  7 must satisfy the compatibility of the displacement field given by Eq. (9). This means that Eq. (9) must be satisfied for characteristic values of the parameter λ.
Since the complimentary angle in the analysed case (weld angle θ = 45°) is 2α = 5π/4, the first characteristic exponent of the solution is λ = 0.6736. Thus, the degree of the singularity of the fundamental solution is n = λ − 3264. This solution means that as the weld toe radius ρ becomes very small, the stress concentration factor K tt or K tb should be described by the power law function of ρ with the exponent n = − 0.3264. The singularity of the solution may be subsequently removed by using the K t /X n parameter instead of K tt or K tb . As a consequence, a smooth and regular function dependent on parameters X and Y can be obtained for simulating variations of stress concentration factor K tt or K tb . In other words, the stress concentration factor domain can be represented by a surface being the function of parameters X and Y.

Graphical form of the regular two-dimensional function representing stress concentration factors
Graphical representations of stress concentration factors in the form of surfaces drawn in X and Y coordinates are shown, for axial and bending load, in Fig. 8 and Fig. 9, respectively. The data were obtained with the finite element method for the attachment thickness to the weld throat thickness ratio of T/ a = 1.
Both surfaces can be easily described by appropriate polynomials involving parameters X and Y varying in the range of 0 to 0.57. The resulting range of variation for parameters ρ/a and a/t was of 0 to 1.3. It has been concluded that assumed ranges of variation, for all parameters being discussed, were essential and sufficient for practical applications.
The closed form expression for convenient estimation of stress concentration factors in cruciform weldments was selected in the form of Eq. (10) The A i parameters depend only on the non-dimensional geometrical parameter Y. The formula (10) is valid for constant value of parameter T/a = 1 involving the thickness of the attachment plate and the weld throat thickness. Parameters A it for the axial and A ib for the bending load are given in the Appendix.

The effect of the attachment thickness T on stress concentration factors K tt and K tb
The weld throat thickness a is related to the thickness of one or both plates welded together, and therefore, additional analyses were carried out for relative attachment thicknesses varying in the range of 1 ≤ T/a ≤ 4.
The effect of the relative attachment thickness T/a on the stress concentration factors K tt and K tb can be accounted for by introducing two correction functions κ t (T/a,X,Y) and κ b (T/ a,X,Y) for tensile and bending load, respectively, given by Eqs. (11) and (12) Fig. 8 Regular surface representing the axial stress concentration factor K tt normalised by the singular term; cruciform joint, axial load, θ = 45°, T/ a = 1, n = − 0.3264 Fig. 7 Sharp corner (ρ → 0) in a semi-infinite plane and its characteristic angles of 2α and θ = 45°representing the sharp weld toe region Fig. 9 Regular surface representing the bending stress concentration factor K tb normalised by the singular term; cruciform joint, bending load, θ = 45°, T/a = 1, n = − 0.3264 The accuracy of each formula is higher than 99% comparing to FEM results.
A range of values of the correction functions κ t and κ b for arbitrarily chosen relative attachment thickness T/a = 2 and T/ a = 4 are shown in Fig. 10 a and b for the axial and bending load, respectively.
Thus, the final formulas for the stress concentration factors in cruciform weldments, accounting for the variation of the parameter T/a, can be written down in the form of Eq. (13) and Eq. (14).
Particular formulas for calculating numerical values of A it and A ib are given in Appendix.

Validation of approximating formulas
Numerical FEM K tt and K tb values have been compared to their equivalencies obtained by means of the approximating functions. Some examples of such comparisons are presented in Tables 3, 4, 5, and 6, for a cruciform joint subjected to tensile and bending loads.
Accuracy for all approximating formulae and variables in the range of validity, summarised in Appendix, is much better than 97.5%.

Conclusions
Extended numerical simulations that were carried out with the help of the FEM package ANSYS Mechanical APDL 16.2 have shown that the weld toe radius ρ, the weld throat thickness a and the main plate thickness t are the most important geometrical parameters affecting stress concentration factors Fig. 10 Correction functions κ t and κ b accounting for the effect of T/a for various X and Yparameters on the stress concentration factors K tt and K tb , respectively, for a axial load and b bending load K tt and K tb of cruciform welded joints having the weld angle θ = 45°. The appropriate description of the singularity appearing when the weld toe radius ρ tends to zero and the introduction of normalising parameters X and Y made the range of application of the final formulae significantly large, i.e. 0 < ρ/a ≤ 1.3, 0 < a/t ≤ 1.3 and 1 ≤ T/a ≤ 4 with the maximum percentage error less than 2.5%.
The effect of the relative attachment thickness T/a on K tt and K tb was accounted for by means of two correction functions κ t and κ b , given in a close form by Eqs. (11) and (12), which corrected K tt and K tb of a particular reference T/a value to a new actual one. The accuracy of each correction function is about 99% compared to FEM results.
In this way, one general formula (for each loading case) was obtained covering all proportions between geometrical parameters of the cruciform joint commonly used in engineering applications.
Since the final solution is given in the form of a polynomial expression, it can be easily applied in computer-aided design processes involving optimisation. The procedure described above may be easily repeated for other types of joints with fillet and butt welds and various weld angles θ.
Approximation accuracy: maximum percentage error much lower than 2.5% compared with FEM results General formulas: Cruciform joint, axial load, θ = 45°A Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.