## 2 An Improved RPIM for Stress Concentration Evaluation of Welded Component
### Improved RPIM Based Stress Solutions for Stress Concentration Evaluation at Weld Toe  
As shown in Figure 1, the through-thickness stress distribution at the weld toe can be decomposed into the equilibrium-equivalent stress part and a self-equilibrating notch stress part [8; 19]. The equilibrium-equivalent stress part can be then decomposed into a membrane component (characterized by \(\sigma_{m}\)) and a bending component (characterized by \(\sigma_{b}\)). In present discussions, the transverse shear stress component is not considered. In fact, if the remote loading produces a significant shear force at section, the shear stress will have a greater effect on crack growth at the weld toe, a situation not considered in this paper. An improved RPIM is presented below that realizes the consistent stress solutions at and near weld toe. As shown in Figure 2, the section A-A at the location of interest and the reference section B-B with an arbitrary distance \(\zeta\) from section A-A were introduced. Taking the continuum between A-A and B-B as the research object, the following equation must be satisfied according to the equilibrium conditions:  
\[\sigma_{m}\cdot t=\int\limits_{-t/2}^{t/2}\sigma_{x}(y)\cdot dy \tag{35}\]  
\[\sigma_{b}\cdot\frac{t^{2}}{6}=\int\limits_{-t/2}^{t/2}\sigma_{x}(y)\cdot y \cdot dy+\xi\cdot\int\limits_{-t/2}^{t/2}\tau_{xy}(y)\cdot dy \tag{36}\]  
where Equation (35) is the force balances equation in \(x\) direction, and Equation (36) is the moment balances equation with respect to section A-A at \(y=0\). The physical significance of \(\int\limits_{-t/2}^{t/2}\sigma_{x}(y)\cdot dy\) is the internal force in the x direction on section B-B and the physical significance of \(\int\limits_{-t/2}^{t/2}\sigma_{x}(y)\cdot y\cdot dy\) is the internal moment on section B-B. Figure 1: Decomposition of through-thickness stress. As the above integral terms can be interpreted as the internal force and internal moment between the right part and left part of continuum separated by section B-B, these integral terms can be obtained directly from the force of field nodes of the background cells through the following steps. Step (1): Find all the field nodes within \(\pm\mathrm{d}_{\mathrm{ix}}\) (\(\mathrm{d}_{\mathrm{ix}}\) is the size of influence domain in the \(x\) direction) from the section B-B. These field nodes were marked green as shown in Figure 3. The number of these field nodes is written as \(n^{\prime}\). Step (2): In the right part of the continuum separated by section B-B, find all the background cells in which the support domain of Gauss points involves the field nodes mentioned in step (1). The number of these background cells is written as \(n_{k^{\prime}}\). Step (3): The resultant force of the field nodes described in step (1) given by the background cells described in step (2) can be calculated by the following formula:  
\[\mathbf{F}^{internal}=\left\{ \begin{array}{c}F_{\mathrm{ix}}^{internal}\\ \frac{F_{\mathrm{internal}}^{internal}}{1y}\\ \vdots\\ F_{\mathrm{ix}}^{internal}\\ \frac{F_{\mathrm{irx}}^{internal}}{1y}\\ \vdots\\ F_{\mathrm{irx}}^{internal}\\ \frac{F_{\mathrm{iry}}^{internal}}{1y}\end{array} \right\}=\sum_{k^{\prime}}^{ n_{k^{\prime}}}\sum_{i^{\prime}=1}^{n_{k}}\widehat{w}_{i^{\prime}}\mathbf{B}^{T}(x_{ \mathrm{Q}^{\prime}})\mathbf{D}\mathbf{B}(x_{\mathrm{Q}^{\prime}})\big{|}y_{ i^{\prime}k^{\prime}}^{D}\big{|}\cdot U \tag{37}\]  
Figure 3: Illustration of field nodes within \(\pm d_{ix}\) from the section B-B. Figure 2: Through-thickness stress distributions in section B-B. where \(x_{Qi^{r}}\) represents the Gauss points described in step (2). \(\mathbf{F}^{internal}\) is the \(2n\) rows column vector representing the field node force in domain \(\Omega\) given by the background cells described in step (2). Note that if field node \(i\) is in the support domain of Gauss point of the background cell describe in step (2), the field node force \(F_{ix}^{internal}\), \(F_{iy}^{internal}\) denotes the field node force given by the background cells described in step (2).