## 4 Limit Load Analysis of Straight Pipes with Local Wall-Thinning Defects under Complex Loads
### Finite Element Analysis Results  
To save calculation time, only the typical defects were selected for limit load analysis: axial defects (\(a=3\), \(b=0.15\), \(c=0.5\)), circumferential defects (\(a=0.2\), \(b=0.4\), \(c=0.5\)), large-area defects (\(a=3\), \(b=0.4\), \(c=0.5\)), and small-area defects (\(a=0.2\), \(b=0.05\), \(c=0.4\)). The case of \(p_{1}=0.4\), \(m_{1}=0.4\), and \(f_{1}=0.2\) is taken as an example to illustrate the calculation results. \begin{table}
\begin{tabular}{c c c c} \hline
**Level** & \(p_{1}\) & \(m_{1}\) & \(f_{1}\) \\ \hline
1 & 0.1 & 0.1 & 0.1 \\
2 & 0.2 & 0.2 & 0.2 \\
3 & 0.4 & 0.4 & 0.4 \\ \hline \end{tabular}
\end{table} 
Table 7: Orthogonal test levels under complex loads. \begin{table}
\begin{tabular}{c c c c} \hline
**Test Number** & \(p_{1}\) & \(m_{1}\) & \(f_{1}\) \\ \hline
1 & 0.1 & 0.1 & 0.1 \\
2 & 0.1 & 0.2 & 0.2 \\
3 & 0.1 & 0.4 & 0.4 \\
4 & 0.2 & 0.1 & 0.2 \\
5 & 0.2 & 0.2 & 0.4 \\
6 & 0.2 & 0.4 & 0.1 \\
7 & 0.4 & 0.1 & 0.4 \\
8 & 0.4 & 0.2 & 0.1 \\
9 & 0.4 & 0.4 & 0.2 \\ \hline \end{tabular}
\end{table} 
Table 8: Calculation scheme of orthogonal tests under complex loads. In the first load step, the internal pressure, bending moment, and axial force corresponding to \(p_{1}=0.4\), \(m_{1}=0.4\), and \(f_{1}=0.2\) are proportionally loaded, and the load step time is 1.0. In the second load step, the value of applied torque is 30 kNm, and the load step time is 2.0. Figure 6 shows the plastic strain contours of \(\text{SP}_{\text{LWT}}\) corresponding to the last convergence point of numerical calculation under complex loads. It can be seen that the failure position of \(\text{SP}_{\text{LWT}}\) varies with the shape of the defect. For the axial defect, the failure position is at the corner of the defect, because complex loads cause the stress to be concentrated at the corner of the defect. For the circumferential defect, the failure position is at the defect position. For the large-area defect, the failure position is at the corner of the defect, but the other positions of the defect and the surrounding position also produce stress concentration and plastic strain. For the small-area defect, the failure position is at the right end of the straight pipe, indicating that small-area defects have little effect on the failure of the straight pipe, and plastic strain also occurs at the defect location, because stress concentration occurs at the defect position. Figure 7 shows the load vs. plastic strain curves at the maximum plastic strain point of \(\text{SP}_{\text{LWT}}\) under complex loads. It can be seen that the plastic strain of \(\text{SP}_{\text{LWT}}\) does not change in the first load step. This is because the applied internal pressure, bending moment, and axial force are low. However, the internal pressure, bending moment, and axial force change the working conditions of the second load step calculation. The torque corresponding to the last convergence point is taken as the limit torque, and the value of limit torque is the applied torque multiplied by (TIME\(-1.0\)). Table 9 shows the calculation results of the limit load of \(\text{SP}_{\text{LWT}}\) under complex loads. Figure 6: The plastic strain contours of a straight pipe with a local wall-thinning defect under complex loads: (**a**) axial defect; (**b**) circumferential defect; (**c**) large-area defect; (**d**) small-area defect. Empirical Limit-Load Equations of Straight Pipes with Local Wall-Thinning Defects under Complex Loads  
The calculation results are expressed in dimensionless form, and the treatment method is shown in Section 2.4. The empirical equations of the relationship between non-dimensional torque \(n_{\rm l}\) and non-dimensional internal pressure \(p_{\rm l}\), non-dimensional bending moment \(m_{\rm l}\), and non-dimensional axial force \(f_{\rm l}\) were fitted by SPSS23 software. Although the equations were obtained from typical dimensionless defects results, they were verified within the range of defect size. 1.