# Safety Analysis and Condition Assessment of Corroded Energy Pipelines under Landslide Disasters
## 5 Equation Prediction  
When no landslide displacement acts on the pipeline, the specific weakening effect of the corrosion size on the pipeline is studied, as shown in Figure 20.  
As the depth of the corrosion increases, the stress of the pipeline continuously grows at an accelerating rate. However, with the extension of the corrosion length, while the stress still rises, the growth rate gradually decreases. Thus, it can be inferred that the depth of the corrosion defect is the primary factor affecting the pipeline stress without the influence of a landslide. Based on the stress results of the corroded pipeline under internal pressure, using the Marquardt method in nonlinear multivariate regression and the general global optimization technique, a multivariate fit was performed with the maximum equivalent stress as the dependent variable and the corrosion length and depth as independent variables. \[\sigma=E\times a(\frac{x_{c}}{t}+b)^{2}\frac{x_{s}}{t}+\sigma_{0} \tag{8}\]  
Figure 19: Limit landslide displacement for different axial corrosion positions. Figure 20: Stress in corroded pipeline without landslide impact. where \(x_{c}\) is the length of the corrosion defect, mm; \(x_{s}\) is the depth of the corrosion defect, mm; \(\sigma\) is the equivalent stress of the pipeline, MPa; \(t\) is the wall thickness of the pipeline, 18.2 mm; \(E\) is the elastic modulus, 2.07 \(\times\) 10\({}^{5}\) MPa; \(\sigma_{0}\) is the stress value of the intact pipeline under normal operation, MPa; and \(a\) and \(b\) are unknown coefficients. \(\sigma_{0}\) was determined to be 252.5 MPa following the numerical simulation, and the coefficients \(a\) and \(b\) were found to be 7.25 \(\times\) 10\({}^{-7}\) and 43.526, respectively, through the fitting analysis. Thus, the relationship describing the impact of the corrosion size on stress without landslide influence can be represented by the ensuing equation:  
\[\sigma=E\times 7.25\times 10^{-7}(\frac{x_{c}}{t}+43.526)^{2}\frac{x_{s}}{t}+252.5 \tag{9}\]  
Table 3 shows that the fitted expression aligns closely with the original data. Given the high unpredictability of corrosion in pipelines, to ensure their safe operation, the most unfavorable position in both the axial and circumferential directions is analyzed. Hence, based on the corrosion defect located axially in the middle of the landslide and circumferentially at 120\({}^{\circ}\), the trend with displacement is shown in Figure 21.  
The fitted relationship between the pipeline stress under the landslide and the soil displacement is as follows:  
\[\sigma=E\times(0.002\frac{x}{L_{c}})^{0.527}+\sigma_{1} \tag{10}\]  
where \(\sigma_{1}\) is the stress of the pipeline without landslide, which is determined to be 366.5 MPa through numerical simulation; \(x\) is the landslide displacement, m; and \(L_{c}\) is the distance of the pipeline from the landslide crack, set as 3 m. The correlation coefficients for the fitted pipeline stress and displacement load equation are shown in Table 4, suggesting a satisfactory regression outcome that aligns closely with the original data. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Model** & \(R\) & \(R^{2}\) & **MSE** & **F-Test** \\ \hline Test value & 0.9928 & 0.9679 & 9.88 & 321.95 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Model verification. Figure 21: Limiting landslide displacement for specific corrosion dimensions. Combining Formulas (9) and (10), the stress equation for the pipeline under the influence of various displacements with a corrosion defect of 20 cm length and 0.25\(t\) depth is derived:  
\[\sigma=E\bigg{(}(0.002\frac{x}{L_{c}})^{0.527}+7.25\times 10^{-7}(\frac{x_{c}}{t}+43.526)^{2}\frac{x_{s}}{t}\bigg{)}+\sigma_{0} \tag{11}\]  
To investigate the relationship between full-scale corroded pipeline stress and landslide displacement, laying the groundwork for the warning model, it is assumed that the corrosion defect is at its riskiest position (axially central and circumferentially facing the slide).