## 6 Fatigue Assessment  
Regarding three-dimensional structures, due to the three-dimensional solid elements involved, a modification of the formulae stated above is necessary, which is mainly based on two dimensional planar circumstances. A three-dimensional isolated body is taken from the weld toe, where two reference sections are defined. They are AA'D'D along the thickness and BB'C'C defined \(\delta\) away from AA'D'D. A local coordinate is established as shown in Figure 7 and the origin O is defined at the bottom of AA'D'D. By the space effect being considered, the modified formulae can be obtained by imposing equilibrium conditions that the force balances in \(x\) direction evaluated along BB'C'C and moment balances at Point O, as shown in Equations (10) and (11). The membrane stress \(\sigma_{m3D}\) and bending stress \(\sigma_{b3D}\) can be got and the structural stress \(\sigma_{s3D}\) is obtained as Equation (12). \[\sigma_{m3D}=\frac{1}{wt}(\iint\limits_{BB^{\prime}C^{\prime}C}\sigma_{x}(y) dzdy+\iint\limits_{ABB^{\prime}A^{\prime}}\tau_{zx}dxdy+\iint\limits_{DC^{ \prime}A^{\prime}}\tau_{zx}dxdy \tag{10}\]  
Figure 6: Pressure regions. \[ \begin{array}{l}\sigma_{m3D}\frac{t^{2}}{\gamma}+\sigma_{b3D}\frac{t^{2}}{\gamma} \\ =\frac{1}{\omega}\iint_{BB^{\prime}C^{\prime}C}\sigma_{x}(y)ydzdy+\frac{\delta}{ \omega}\iint_{BB^{\prime}C^{\prime}C}\tau_{xy}(x,y)dzdy\\ +\frac{1}{2\omega}\iint\limits_{ABB^{\prime}A^{\prime}}\tau_{xx}(x,y)dxdy+ \frac{1}{2\omega}\iint\limits_{DCC^{\prime}A^{\prime}}\tau_{xx}(x,y)dxdy\\ +\frac{1}{\omega}\iint\limits_{ABB^{\prime}A^{\prime}}\tau_{zy}x(x,y)dxdy+ \frac{1}{\omega}\iint\limits_{DCC^{\prime}A^{\prime}}\tau_{zy}x(x,y)dxdy\\ \sigma_{s3D}=\sigma_{m3D}+\sigma_{b3D}\end{array}  \tag{11}\]  
\[ \begin{array}{l}\sigma_{s3D}=\sigma_{m3D}+\sigma_{b3D}\end{array}  \tag{12}\]  
Based on the structural stress obtained, an equivalent structural stress parameter is formulated and proven to be effective in unifying a great number of fatigue test results into a single narrow band, called the master S-N curve. The equivalent structural stress \(\Delta S_{e}\) is defined as:  
\[\Delta S_{e}=\frac{\Delta\sigma_{s3D}}{{t^{s^{(2-m)/2m}}}\cdot I(r)^{1/m}} \tag{13}\]  
where \(t^{*}\) is a relative thickness, which is defined as the division between plate thickness t and a unit thickness (1 mm) as \(t^{*}=t/1\) mm. m is selected to be 3.6 based on a two stage crack growth model. The life integral \(I(r)\) is a function of bending ratio \(r\). They are defined as:  
\[r=\frac{|\sigma_{b3D}|}{|\sigma_{b3D}|+|\sigma_{b3D}|} \tag{14}\]  
\[I(r)^{\frac{1}{m}}=0.0011r^{6}+0.0767r^{5}+0.0988r^{4}+0.0946r^{3}+0.0221r^{2} +0.014r+1.2223 \tag{15}\]  
After defining the equivalent structural stress, the master S-N curve is defined in the form of:  
\[\Delta S_{e}=C\cdot N^{h} \tag{16}\]  
where \(C\) and \(h\) are constants, as shown in Table 1. \(\sigma\) represents the standard derivation defined with respect to cycle to failure in log scale. Different curves based on the corresponding statistical basis are illustrated in Figure 8. Here, the mean curve is selected, so \(C\) is 19930.2 and \(h\) is 0.32. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Statistical Basis** & \(C\) & \(h\) \\ \hline Mean & 19930.2 & \\ +2\(\sigma\) & 28626.5 & \\ \(-2\sigma\) & 13875.8 & 0.32 \\ +3\(\sigma\) & 31796.1 & \\ \(-3\sigma\) & 12492.6 & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters for master S-N curve. Figure 7: 3D isolation body.