## 1. Introduction

The welded joints have been widely used in thin-walled structures such as bridges, oil rigs, pressure vessels and ships, due to their low cost, superior performance, and high reliability [

1,

2], Fatigue cracks are prone to occur at the welded joint under cyclic loadings, which significantly threaten the safe operation of the thin-walled structures [

3,

4]. The fatigue behavior of welded joints is susceptible to several factors, including the residual stress, loading magnitude, loading type, and local stress concentration [

5,

6]. Fatigue cracks usually initiate in correspondence of stress localizations caused by geometric discontinuities and notches [

7]. Therefore, it is important to assess the stress concentration of welded joints for both accurate prediction of their fatigue life and to propose a geometrical shape of the weld (for example, by grinding [

8]) that could improve the fatigue performance.

The stress concentration effects can be expressed by the stress concentration factor (SCF), which has been widely applied in the engineering structures [

9]. Several methods are available to calculate the SCF including analytical, numerical, and experimental analyses. The analytical approach, based on elastic mechanics, is usually applied to some simple geometric shapes [

10], while it is difficult to use to assess the SCF of weld with sophisticated shapes of weld toe or root. The photo-elastic method [

11] is one of the experimental ways to assess the SCF by visualizing the elastic stress directly on the welded components. On the other hand, the accuracy of the estimation of the SCF with the photo-elastic method remains to be improved. In recent years, some new technologies, such as the fiber Bragg gratings [

12], have been applied to assess the stress concentration in fatigue experiments with promising results. A wider diffusion of these new technologies is expected in the future. Numerical methods, based on finite elements (FE) [

13], have usually been used to obtain parametric formulae of the SCF. This approach allows to calculate a SCF as a function of essential geometric parameters of the weld (weld toe radius

r, flank angle

α, etc.). Then parametric formulae can be fitted by selecting suitable numerical functions [

14]. The neural network method [

15], developed in recent years, is also available to ensure a high accuracy in the estimation of the SCF. A variety of formulae considering key geometric parameters have been proposed based on the numerical methods considering different types of welded joints, including the cruciform joints [

16], T-shape joints [

17], lap joints [

18], and butt joints [

19]. Modifications of the aforementioned formulae have also been carried out in order to address particular cases such as loading types (bending load and tensile load) [

20], weld shape after grinding [

21], additional weld [

22], and penetration ratio [

23].

An accurate weld model is necessary in the FE analyses for the precise computation of SCF. The existing parametric formulae are mostly based on the line model, i.e., the shape of the weld surface is usually approximated with a straight line connecting the arc of the weld toe and the upper weld end [

24]. The flank angle and weld leg length were considered while the streamline shape of the weld was neglected. The predicted results can be acceptable under some conditions if the weld shape resembles a straight line. However, the weld shape is usually arc-shaped and its approximation with a straight line can result in reduced accuracy of the SCF. The real shape of the weld can be obtained by 3D laser scanning technology. For instance, Hou [

25] applied this technology to acquire the real weld toe geometry of cruciform specimens and to calculate the SCF based on FE analyses. However, the relationship between the geometric parameters and the SCF has not been analyzed yet, and the parametric formula for general conditions is not proposed.

The novelty of the work consists in the formulation of a parametric formula for the fillet weld of the T-shape joint and cruciform joint based on the FEM and regression analysis. The geometric model with a spline curve of the weld is proposed to consider the real shape of the weld. The accuracy of the line model widely used in the existing parametric formulae is also discussed. Subsequently, based on a large set of FE analysis, the existing parametric formulae for the fillet weld are tested and discussed. The influence of key parameters on the SCF is investigated, and the parametric formulae for the T-shape joint and cruciform joint under the tensile stress and bending stress are proposed through the training data system and are examined by the testing data system. The parameters of a real specimen were also measured, and the severity of stress concentration is assessed by means of a probabilistic approach.

## 4. Probability Analysis of Fillet Weld Based on the Parametric Formulae

Section 3.3 dealt with the proposition of a parametric formula for the evaluation of the SCF for a fillet weld in T-shape joint and cruciform joint based on the spline model. The severity of the stress concentration of these two welded joints is compared in this section. Based on the data from the training system and testing system with broad application ranges, the mean values of the SCFs for the four investigated configurations were 2.134, 2.103, 2.420, and 1.908, respectively. The stress concentration, assessed by the mean values, is highest in the cruciform joint under the tensile stress. However, the application of mean values to assess the severity of stress concentration could not be representative. This is due to the fact that the mean values can be affected by the presence of high SCFs generated by geometrical parameters that lie within small ranges. For instance, as discussed in the previous section, the high SCFs obtained for small values of the

r/t ratio can influence the mean value significantly.

As mentioned earlier, the fatigue performance of fillet weld is strongly related to the SCF. As well, the SCF is a function of the geometric parameters that can be affected during the welding process. Even in the same welding seam, the shape of the fillet weld could be quite different. It is important to evaluate the SCF of a certain fillet weld to be able to apply some treatments and improve the shape in case the SCF is particularly high. Employing the probability analysis, it is possible to obtain a probabilistic distribution of the SCFs under certain welding conditions. On the other hand, since the parameters in the training and testing systems were randomly generated following a uniform distribution, they are not suitable for probability analyses. Actual measures should be used instead.

Here, the geometric data of the cruciform joint specimen mentioned in

Section 2.3 were measured for the probability analysis of fillet weld. The welding conditions were listed in

Table 2. It is assumed the same welding conditions were applied to a T-shape joint. A total of 50 sections were cut by the 3D Scanner VL-300. Thus, 200 cases for the fillet weld could be obtained, and parameters based on the spline model were measured.

The measured parameters used in the parametric formulae to obtain the SCFs of the T-shape joint and cruciform joint under the tension and bending stresses. The SCFs of 200 cases for each configuration are displayed in

Figure 8. It is believed the cases are sufficient to achieve the distribution rules of SCF for different fillet welds. The SCF is reported in the abscissa, whereas in the ordinate it is reported the corresponding number of cases distributed within a certain range. It could be easily observed that the histograms of the SCFs resemble the normal distribution, even if a perfect normal distribution could not be achieved due to the limited number of data. The lognormal density function, applicable to statistical data greater than 0 (see Equation (5)), was applied to assess the probability distribution since the SCF is always greater than unity.

where,

K_{t} denoted the SCF in the probability analysis,

μ denoted the mean value of SCF, and

σ denoted the standard deviation.

The probability analysis was conducted on the four geometrical configurations of

Section 2.2. The characteristic values of

μ and

σ, together with the probability distribution curve were also added in

Figure 8. Considering the mean value

μ, it could be concluded that worst scenario case, in terms of stress concentration, is represented by the cruciform joint under the tensile stress, followed by the T-shape joint under the bending stress, the cruciform joint under the bending stress, and lastly by the T-shape joint under the tensile stress. Moreover, the calculated mean values of the SCFs using the measured geometrical parameters were quite different from those assessed by the weld parameters generated randomly. This result indicated that it is crucial to assess the stress concentration based on the real welding conditions.

The possible ranges where the SCF might be distributed were also essential to judge the possible SCFs for the fillet welds. Considering a survival probability of 95%, the confidence intervals for the four configurations were also calculated. In detail, the values of probable SCFs are within the range of [2.032, 2.348] for T-shape joint under the tension stress, [2.350, 2.678] under the bending stress, [2.589, 2.913] for cruciform joint under tensile stress, and [2.051, 2.376] under the bending stress. The range of overlap of the confidence intervals for the T-shape joint under the tensile stress and cruciform joint under the bending stress is quite wide, indicating a similar response in terms of stress concentration. The SCF of the cruciform joint under tensile stress is the highest even under the probability approach. This conclusion is valid for the welding condition used in this research; moreover, the same method could be used based on the parametric formulae proposed in this study for assessment of the SCF under other welding conditions.

## 5. Conclusions

The parametric studies on the notch stress concentration factor at the weld toe of fillet weld were carried by a new weld model. The parametric formulae for SCF of T-shape joints and cruciform joints under tensile and bending stress were proposed based on a large set of FE analysis. A wide application range of parameters was considered as follows:

- (1)
Stiffener thickness T/t: 0.3–2.0;

- (2)
Bottom weld toe radius r_{1}/t: 0.003–0.36;

- (3)
Top weld toe radius r_{2}/t: 0.003–0.36;

- (4)
Bottom flank angle θ_{1}: 20°–90°

- (5)
Top flank angle θ_{2}: 20°–90°

- (6)
Bottom weld leg length L_{1}/t: 0.3–2.0;

- (7)
Top weld leg length L_{2}/t: 0.3–2.0;

- (8)
Salient point position 1/n: 0.2–0.9; and

- (9)
Hump height H/t: 0.0–0.3.

The spline model was proposed in this research to achieve higher precision for the assessment of SCF since it considers a better approximation of the real fillet weld shape. The line model was proved to give an inaccurate prediction of the SCF at weld toe of the fillet weld if the streamline shape of real fillet weld was considered. All the parametric formulae were proposed based on the regression analyses of numerous training data. The deviation ratio of the SCF evaluated by the proposed parametric formula and compared with the FE analyses results were proved to be less than 5%, according to the testing data system. Considering the welding condition in this research, the stress concentration of the cruciform joint under tensile stress conditions represents the worst case scenario if assessed by the confidence interval of 95% survival probability. Further research will be carried out on the fillet weld with initial angular distortion, which is inevitable in the welded structures. Moreover, the 3D spline model is going to be considered as the SCF on different positions of the weld might have interactive influence on each other.