## 1 Introduction
### Determination Method of Limit Load  
Usually, there are two algorithms to determine the limit load of the structure: limit load analysis, and elastic-plastic stress analysis. Limit load analysis is usually based on the elastic-perfectly plastic constitutive model with small deformation assumption, which  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Material** & **Elastic Modulus, GPa** & **Poisson’s Ratio** & **Yield Strength, MPa** \\ \hline
20 g & 207 & 0.3 & 245 \\ \hline \hline \end{tabular}
\end{table}
Table 2: The material parameters of the straight pipe.  
Figure 3: The direction of pipe load.  
often obtains a conservative result. Elastic-plastic stress analysis is usually based on the elastic-plastic constitutive model with nonlinear geometry, which obtains more reasonable results but requires more material information. According to the numerical results, the limit load can be determined by several methods [25], such as the double-tangent criterion, double-elastic-deformation method, double-elastic-slope criterion, 0.2% residual strain criterion, zero-slope criterion, zero-curvature criterion, etc. The limit load analysis is used conservatively in this paper [26], and the load corresponding to the last convergence point of the numerical calculation is taken as the limit load. This algorithm is used to determine the limit load of SP\({}_{\text{LWT}}\) under single and complex loads.  
When determining the limit load of SP\({}_{\text{LWT}}\) under a single load, the load step time is 1.0, and a sufficiently large load (\(P,M,N,F\)) is applied. The load of the last convergence point of numerical calculation corresponds to the limit load (\(P_{\text{LS}}\), \(M_{\text{LS}}\), \(N_{\text{LS}}\), \(F_{\text{LS}}\)) of SP\({}_{\text{LWT}}\). When determining the limit load of the pipeline under complex loads, such as the combination of internal pressure and bending moment, the following method is often adopted: the internal pressure load is fixed, and the limit bending moment of the pipeline is calculated to analyze the limit load of the pipeline with a local wall-thinning defect [7]. [PERSON] [16] further extended this method to the limit load analysis of straight, pipes with defects under the combination of loads of internal pressure, bending moment and torque. In this paper, the loading method for nonlinear finite element analysis of SP\({}_{\text{LWT}}\) under complex loads is as follows: In the first calculation step, the internal pressure \(P\), bending moment \(M\), and axial force \(F\) are applied in different proportions, and the load step time is 1.0. After the first calculation converges, the second step of nonlinear calculation is performed. Meanwhile, a sufficiently large torque is applied, and the load step time is 2.0. The torque of the last convergence point of numerical calculation corresponds to the limit torque of SP\({}_{\text{LWT}}\) under loads (\(P\), \(M\), and \(F\)), and the limit torque of SP\({}_{\text{LWT}}\) under complex loads is defined as \(N_{\text{LC}}\).  
Using the limit internal pressure \(P_{\text{LS}}\), limit bending moment \(M_{\text{LS}}\), limit torque \(N_{\text{LS}}\), and limit axial force \(F_{\text{LS}}\) of SP\({}_{\text{LWT}}\) under different single loads, the applied internal pressure \(P\), bending moment \(M\), axial force \(F\), and calculated limit torque \(N_{\text{LC}}\) can be expressed in dimensionless form. The dimensionless internal pressure, the bending moment, the axial force, and the torque are represented by \(p_{1}=P/P_{\text{LS}}\), \(m_{1}=M/M_{\text{LS}}\), \(f_{1}=F/F_{\text{LS}}\), and \(n_{1}=N_{\text{LC}}/N_{\text{LS}}\) respectively. Then, a dimensionless limit state point (\(p_{1}\), \(m_{1}\), \(n_{1}\), \(f_{1}\)) under complex loads is obtained. Through parametric calculation, the dimensionless limit state response hypersurface under complex loads can be obtained.