## 4 Limit Load Analysis of Straight Pipes with Local Wall-Thinning Defects under Complex Loads
### Finite Element Analysis Results
The limit load of a straight pipe with an axial local wall-thinning defect can be calculated by Equation (10): \[n_{\rm l}^{1.6} = 0.958-1.016m_{\rm l}^{2}-1.284p_{\rm l}^{2}-1.028f_{\rm l}^{2}+0.054 m_{\rm l}+0.322p_{\rm l}+0.028f_{\rm l}\] (10) where the applicable range of defect size is \(a/b\geq 10\), \(5\geq a\geq 1\), \(b\leq 0.3\), and the least square error \(R^{2}\) of the equation is 0.999. Within the applicable range of defect size, the upper boundary point (\(a=5\), \(b=0.3\), \(c=0.5\)) and the lower boundary point (\(a=1\), \(b=0.05\), \(c=0.5\)). Figure 7: The load vs. plastic strain curves of a straight pipe with a local wall-thinning defect under complex loads: (**a**) internal pressure; (**b**) bending moment; (**c**) torque; (**d**) axial force. \begin{table}
\begin{tabular}{c c c c c c c} \hline \multirow{2}{*}{\begin{tabular}{c} **Internal** \\ **Pressure** \\ **\(p_{1}\)** \\ \end{tabular} } & \multirow{2}{*}{\
\begin{tabular}{c} **Bending** \\ **Moment** \\ **\(m_{\rm l}\)** \\ \end{tabular} } & \multirow{2}{*}{
\begin{tabular}{c} **Torque** \\ **\(f_{\rm l}\)** \\ \end{tabular} } & \multicolumn{4}{c}{**Limit Torque/kNm**} \\ \cline{3-6}  & & & & **Axial** & **Circumferential** & **Large-Area** & **Small-Area** \\  & & & **Defect** & **Defect** & **Defect** & **Defect** \\ \hline
0.1 & 0.1 & 0.1 & 18.10 & 13.09 & 14.67 & 18.43 \\
0.1 & 0.2 & 0.2 & 17.31 & 12.28 & 14.57 & 17.88 \\
0.1 & 0.4 & 0.4 & 14.93 & 9.33 & 13.49 & 15.01 \\
0.2 & 0.1 & 0.2 & 17.61 & 12.49 & 14.49 & 17.85 \\
0.2 & 0.2 & 0.4 & 15.53 & 10.67 & 13.75 & 15.72 \\
0.2 & 0.4 & 0.1 & 16.33 & 11.43 & 13.79 & 16.83 \\
0.4 & 0.1 & 0.4 & 13.73 & 10.03 & 12.75 & 13.52 \\
0.4 & 0.2 & 0.1 & 16.58 & 11.79 & 13.46 & 16.82 \\
0.4 & 0.4 & 0.2 & 14.95 & 9.78 & 12.89 & 14.64 \\ \hline \end{tabular}
\end{table} 
Table 9: The calculation results of the limit load of straight pipes with local wall-thinning defects under complex loads. \(c=0.5\)) were selected for finite element calculation; the error between the results of the finite element calculation and that of Equation (10) was 7.56% and 5.08%, respectively. 2. The limit load of a straight pipe with a circumferential local wall-thinning defect can be calculated by Equation (11): \[n_{1}^{1.5}=0.989-1.251m_{1}^{2}-1.104p_{1}^{2}-0.588f_{1}^{2}+0.238m_{1}+0.072p_ {1}-0.407f_{1}\] (11) where the applicable range of defect size is \(a/b\leq 1/2\), \(a\leq 0.3\), \(0.6\geq b\geq 0.3\), and the least square error \(R^{2}\) of the equation is 0.974. Within the applicable range of defect size, the upper boundary point (\(a=0.3\), \(b=0.6\), \(c=0.5\)) and the lower boundary point (\(a=0.1\), \(b=0.3\), \(c=0.5\)) were selected for finite element calculation; the error between the results of the finite element calculation and that of Equation (11) was 5.78% and 4.61%, respectively. 3. The limit load of a straight pipe with a large-area local wall-thinning defect can be calculated by Equation (12): \[n_{1}^{2}=0.856-0.863m_{1}^{2}-0.983p_{1}^{2}-1.332f_{1}^{2}+0.023m_{1}+0.137p_ {1}+0.424f_{1}\] (12) where the applicable range of defect size is \(10>a/b>1/2\), \(0.6\geq b\geq 0.3\), \(5\geq a\geq 1\), and the least square error \(R^{2}\) of the equation is 0.995. Within the applicable range of defect size, the upper boundary point (\(a=5\), \(b=0.6\), \(c=0.5\)) and the lower boundary point (\(a=1\), \(b=0.3\), \(c=0.5\)) were selected for finite element calculation; the error between the results of the finite element calculation and that of Equation (12) was 8.46% and 4.32% respectively. 4. The limit load of a straight pipe with a small-area local wall-thinning defect can be calculated by Equation (13): \[n_{1}^{1.3}=0.989-1.120m_{1}^{2}-1.315p_{1}^{2}-1.023f_{1}^{2}+0.134m_{1}+0. 308p_{1}-0.048f_{1}\] (13) where the applicable range of defect size is \(b<0.3\), \(a<1\), and the least square error \(R^{2}\) of the equation is 0.999.