## 3 Limit Load Analysis of Straight Pipes with Local Wall-Thinning Defects under Single Loads
### Empirical Limit-Load Equations of Straight Pipes with Local Wall-Thinning Defects  
Conveniently, the limit loads of \(\mathrm{SP_{LWT}}\) under single loads were normalized. The normalized limit internal pressure, the normalized limit bending moment, the normalized limit torque, and the normalized limit axial force are expressed by \(p_{\mathrm{LS}}=P_{\mathrm{LS}}/P_{\mathrm{L0}}\), \(m_{\mathrm{LS}}=M_{\mathrm{LS}}/M_{\mathrm{L0}}\), \(n_{\mathrm{LS}}=N_{\mathrm{LS}}/N_{\mathrm{L0}}\), and \(f_{\mathrm{LS}}=F_{\mathrm{LS}}/F_{\mathrm{L0}}\), respectively, where \(P_{\mathrm{L0}}\), \(N_{\mathrm{L0}}\), \(M_{\mathrm{L0}}\), and \begin{table}
\begin{tabular}{c c c c} \hline
**Level** & \(a\) & \(b\) & \(c\) \\ \hline
1 & 0.2 & 0.2 & 0.2 \\
2 & 1 & 0.4 & 0.4 \\
3 & 3 & 0.6 & 0.6 \\
4 & 5 & 0.8 & 0.8 \\ \hline \end{tabular}
\end{table} 
Table 4: Orthogonal test levels under single loads. \begin{table}
\begin{tabular}{c c c c} \hline
**Test Number** & \(a\) & \(b\) & \(c\) \\ \hline
1 & 0.2 & 0.2 & 0.2 \\
2 & 0.2 & 0.4 & 0.4 \\
3 & 1.0 & 0.8 & 0.6 \\
4 & 0.2 & 0.6 & 0.6 \\
5 & 5.0 & 0.6 & 0.4 \\
6 & 0.2 & 0.8 & 0.8 \\
7 & 5.0 & 0.2 & 0.8 \\
8 & 3.0 & 0.8 & 0.4 \\
9 & 3.0 & 0.6 & 0.2 \\
10 & 5.0 & 0.8 & 0.2 \\
11 & 3.0 & 0.4 & 0.8 \\
12 & 1.0 & 0.4 & 0.2 \\
13 & 1.0 & 0.6 & 0.8 \\
14 & 1.0 & 0.2 & 0.4 \\
15 & 5.0 & 0.4 & 0.6 \\
16 & 3.0 & 0.2 & 0.6 \\ \hline \end{tabular}
\end{table} 
Table 5: Calculation scheme of orthogonal tests under single loads. [PERSON] [16] proposed the calculation equations of limit internal pressure \(P_{\rm L0}\), limit bending moment \(M_{\rm L0}\), and limit axial force \(F_{\rm L0}\) for defect-free straight pipes. \[P_{\rm L0}\ =\ \frac{2}{\sqrt{3}}\sigma_{\rm y}\ln\frac{R_{\rm o}}{R_{\rm i}} \tag{2}\]  
\[M_{\rm L0}=4r^{2}T\sigma_{\rm y} \tag{3}\]  
\[F_{\rm L0}=\pi\Big{(}R_{\rm o}^{2}-R_{\rm i}^{2}\Big{)}\sigma_{\rm y} \tag{4}\]  
where \(r\) denotes the average radius of the straight pipe, and \(\sigma_{\rm y}\) denotes the yield strength of the material. [PERSON] [27] gave the calculation equation of the limit torque \(N_{\rm L0}\) of defect-free straight pipes. \[N_{\rm L0}\ =\ \frac{2}{\sqrt{3}}\sigma_{\rm y}r^{2}T \tag{5}\]  
The equations between the limit load and the size of the local wall-thinning defect were obtained using SPSS software. The relative error between the equation calculation results and the finite element calculation results was less than 5%. The dimensionless limit load empirical equations of SP\({}_{\rm LWT}\) under different single loads are as follows:  
(1) The limit load of SPLWT under internal pressure can be calculated by Equation (6):  
\[p_{\rm LS}=-0.517G_{1}+0.995 \tag{6}\]  
where \(G_{1}\) denotes the geometric parameters of the local wall-thinning defect, \(G_{1}=a^{0.5}b^{0.1}c\). The least square error \(R^{2}\) of the equation is 0.948. (2) The limit load of SPLWT under bending moment can be calculated by Equation (7):  
\[m_{\rm LS}=-0.033G_{2}^{2}-0.386G_{2}+1.039 \tag{7}\]  
where \(G_{2}\) denotes the geometric parameters of the local wall-thinning defect, \(G_{2}=a^{0.01}b^{0.1}c\). The least square error \(R^{2}\) of the equation is 0.957. (3) The limit load of SPLWT under torque can be calculated by Equation (8):