## 2 An Improved RPIM for Stress Concentration Evaluation of Welded Component
### Notch Stress Intensity Factors Using Improved RPIM Based Stress Solutions  
From the previous study [8], the notch stress effect at welds can be calculated by introducing characteristic depth \(t_{1}\) (\(t_{1}\) is taken as 0.1) in FEM model. The discussion on that the selection of \(t_{1}\) should be selected as 0.1 was given in [8,17]. In this paper, the same value of \(t_{1}\) (\(t_{1}=0.1\)) was introduced in meshfree model as shown in Figure 5a. The following Equation (46) must be satisfied by applying Equations (42) and (43) within region 1 and region 2, respectively (enforcing the equilibrium conditions and continuity condition within region 1 and region 2).  
\[\begin{array}{l}\sum\limits_{region1+region2}F_{ix}^{internal}=\sigma_{1}t_{1 }+\frac{1}{2}(\sigma_{2}-\sigma_{1})t_{1}+\sigma_{2}(t-t_{1})+\frac{1}{2}( \sigma_{3}-\sigma_{2})(t-t_{1})\\ \sum\limits_{region1}(F_{ix}^{internal}\cdot y_{i})=(\sigma_{1}t_{1})\cdot\frac {t_{1}}{2}+\left[\frac{1}{2}(\sigma_{1}-\sigma_{2})t_{1}\right]\cdot\frac{2}{ 3}t_{1}\\ \sum\limits_{region2}(F_{ix}^{internal}\cdot y_{i})=[\sigma_{3}(t-t_{1})]\cdot \frac{t-t_{1}}{2}+\left[\frac{1}{2}(\sigma_{3}-\sigma_{2})(t-t_{1})\right] \cdot\frac{2}{3}(t-t_{1})\end{array} \tag{46}\]  
Figure 4: Flowchart of obtaining the stress solutions of improved radial point interpolation method (RPIM).  
The membrane and bending components within region 1 and region 2 are as follows:  
\[\begin{array}{l}\sigma_{m(1)}=\frac{\sigma_{1}+\sigma_{2}}{2},\sigma_{p(1)}= \frac{\sigma_{1}-\sigma_{2}}{2}\\ \sigma_{m(1)}=\frac{\sigma_{2}+\sigma_{3}}{2},\sigma_{b(2)}=\frac{\sigma_{2}- \sigma_{3}}{2}\end{array} \tag{47}\]  
With \(\sigma_{m(2)}\), \(\sigma_{b(2)}\) defined in Equation (47) and \(\sigma_{m}\), \(\sigma_{b}\) defined in Equations (44) and (45), the distributions of notch stresses can then be obtained in terms of \(p_{m}\) and \(p_{b}\) expressed as Equation (48) as shown in Figure 5b. The detailed discussion on deriving the relationship between \(p_{m}\), \(p_{m}\) and \(\sigma_{m(2)},\sigma_{b(2)},\sigma_{m},\sigma_{b}\) were given in [2].  
\[\begin{array}{l}p_{m}=(\frac{t^{2}-3 lt+2t^{2}}{2 lt})\sigma_{m(2)}+(\frac{t ^{2}-3 lt+2t^{2}}{2 lt})\sigma_{b(2)}-(\frac{t^{2}-5 lt+2t^{2}}{2 lt})\sigma_{m}+ (\frac{t-t}{2l})\sigma_{b}\\ p_{b}=(\frac{t^{2}+lt-2t^{2}}{2 lt})\sigma_{m(2)}+(\frac{t^{2}+lt-2t^{2}}{2 lt })\sigma_{b(2)}-(\frac{t^{2}+lt-2t^{2}}{2 lt})\sigma_{m}+(\frac{t+t}{2l})\sigma _{b}\end{array} \tag{48}\]  
Then, the stress intensity factor can be obtained based on \(p_{m},p_{b}\). For example, with applying superposition principles, the mode \(I\) stress intensity factor for edge cracks with notch effect is as follows:  
\[K=p_{s}\sqrt{t}\left[f_{m}-r\frac{t}{a}(f_{m}-f_{b})\right] \tag{49}\]  
where \(p_{s}=p_{m}+p_{b}\), \(r=p_{b}/p_{s}a=l\). \(f_{m}(a/t)\) and \(f_{b}(a/t)\) denotes the dimensionless functions documented in [17].  
Figure 5: The distributions of notch stresses in terms of \(p_{m}\), \(p_{m}\) in Equation (48): (**a**) illustration of considering notch stress distribution as two linear distributions in region 1 and region 2; (**b**) relationship between \(p_{m},p_{m}\) in Equation (48) and \(\sigma_{m(2)},\sigma_{b(2)},\sigma_{m},\sigma_{b}\) in Equations (44), (45) and (47).