## 2 Equivalent Structural Stress Approach  
### Stress-Based Integration Equivalent Structural Stress Approach  
Figure 2 shows the stress distribution and decomposition at the weld toe section [13]. The actual stress is composed of three parts: The membrane stress \(\sigma_{m}\), bending stress \(\sigma_{b}\), and nonlinear peak stress \(\sigma_{nlp}\). Among them, the nonlinear peak stress \(\sigma_{nlp}\) caused by the weld notch effect is a type of self-equilibrium stress. The structural stress \(\sigma_{s}\) is the stress in equilibrium with external forces, resolved by taking into account the effects of a structural discontinuity, and consisting of membrane stress \(\sigma_{m}\) and bending stress \(\sigma_{b}\), as shown in Equation (1) [12]. The structural hot-spot stress \(\sigma_{ls}\) is a value of structural stress on the surface at the hot spot. Therefore, \(\sigma_{ls}\) equals to \(\sigma_{s}\) at the weld toe. \[\sigma_{s}=\sigma_{m}+\sigma_{b} \tag{1}\]  
Figure 3 shows the stress distribution of the 3D solid element isolated body for the structural stress calculation, based on stress integration [16]. In Figure 3, AA'D'D is the crack propagation surface,  
Figure 2: Stress distribution and decomposition at the weld toe section. BB'C'C is the reference surface, and AA'B'B and DD'C'C are side surfaces 1 and 2, respectively. Taking the shear stresses \(\tau_{zx}\), \(\tau_{zy}\) on both side surfaces into consideration, the force equilibrium equation in the x-direction and the moment equilibrium equation with respect to the AD axis are listed, respectively, as shown in Equations (2) and (3). \[\sigma_{m}=\frac{1}{wt}\iint\limits_{BB^{\prime}CC}\sigma_{x}(y,z)dzdy+\iint \limits_{AA^{\prime}B^{\prime}}\tau_{zx}(x,y)dxdy+\iint\limits_{DD^{\prime}CC }\tau_{zx}(x,y)dxdy \tag{2}\]  
\[ \begin{array}{c}\sigma_{m}\frac{t^{2}}{2}+\sigma_{b}\frac{t^{2}}{6}=\frac{1} {w}\iint\limits_{BB^{\prime}C^{\prime}C}\sigma_{x}(y,z)ydzdy+\frac{\delta}{w} \iint\limits_{BB^{\prime}CC}\tau_{xy}(y,z)dzdy\\ +\frac{t}{2w}\iint\limits_{AA^{\prime}BB}\tau_{zx}(x,y)dxdy+\frac{t}{2w} \iint\limits_{DD^{\prime}CC}\tau_{zx}(x,y)dxdy\\ +\frac{1}{w}\iint\limits_{AA^{\prime}B^{\prime}}\tau_{zy}(x,y)xdxdy+\frac{1} {w}\iint\limits_{DD^{\prime}CC}\tau_{zy}(x,y)xdxdy\end{array}  \tag{3}\]  
where \(w\) is the width of the isolated body along the weld length direction, \(t\) is the plate thickness, and \(\delta\) is the distance between the crack propagation surface and the reference surface. If the shear stresses on both side surfaces are ignored, the simplified stress-based integration equivalent structural stress approach for 3D solid elements is obtained, as shown in Equations (4) and (5). \[\sigma_{m}=\frac{1}{wt}\iint\limits_{BB^{\prime}C^{\prime}C}\sigma_{x}(y,z) dzdy \tag{4}\]  
\[\sigma_{m}\frac{t^{2}}{2}+\sigma_{b}\frac{t^{2}}{6}=\frac{1}{w}\iint\limits_ {BB^{\prime}C^{\prime}C}\sigma_{x}(y,z)ydzdy+\frac{\delta}{w}\iint\limits_{ BB^{\prime}C^{\prime}C}\tau_{xy}(y,z)dzdy \tag{5}\]  
Simultaneously, solving Equations (1), (4) and (5) gives Equation (6) for calculating the structural stress:  
\[\sigma_{s}=\frac{6}{wt^{2}}\times\left(\iint\limits_{BB^{\prime}CC}\sigma_{x} (y,z)ydzdy+\delta\iint\limits_{BB^{\prime}CC}\tau_{xy}(y,z)dzdy\right)-\frac{ 2}{wt}\iint\limits_{BB^{\prime}C^{\prime}C}\sigma_{x}(y,z)dzdy \tag{6}\]  
Equations (4) and (5) are similar to the equivalent structural stress equations of the 2D plane solid element [15], which takes into account the spatial effect of the BB'C'C reference surface. In this approach, the stress components are first extracted by using the commands of surface mapping and surface operation in the finite element software, and then the structural stress is calculated by Equation (6). For simplicity, this stress-based integration structural stress approach is called the stress integration approach in the present study. There are two uncertain parameters in Equation (6): The width \(w\) of the isolated body, and the distance \(\delta\) between the crack propagation surface and the reference surface.