## 3 Analysis
### Design Procedures  
Through an example analysis, this part introduces in detail the negative pressure design process by using SSL trenchless technology. It should be noted that one of the limitations of this article is that only the stability of the lining-water-supply pipeline system under negative pressure is considered. If the pipeline is buried in the ground, the soil pressure, hydrostatic pressure and traffic load on the pipeline must be considered comprehensively. Case parameters are as follows:  
* Pipe diameter: DN800 mm; Pipe length: 10 m;
* Lining material model: stainless steel 304;
* Elastic Modulus: 195,000 MPa;
* Poisson ratio: 0.247;
* Thickness: 1.2 mm.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**No.**} & **Theoretical Value of** & \(\mathbf{k_{1}}\) & \(\mathbf{k_{2}}\) & \(\mathbf{k_{3}}\) & \(\mathbf{k_{4}}\) & \multirow{2}{*}{\(\mathbf{k_{average}}\)} \\  & **Free Ring Model/kPa** & **Lobe 1** & **Lobe 2** & **Lobe 3** & **Lobe 4** & \\ \hline Test 1 & 4.65 & 11.85 & 11.27 & 11.70 & 14.80 & 12.40 \\ \hline Test 2 & 2.26 & 12.57 & 13.01 & 12.43 & NAN & 12.67 \\ \hline Test 3 & 2.46 & 14.02 & 21.82 & 19.42 & 19.48 & 14.02/20.24 \\ \hline Test 4 & 3.39 & 12.22 & 11.90 & NAN & NAN & 1 The design process is as follows:  
1. Using the experimental results (Table 3), we take a relatively conservative value of 14.02 (The larger value of \(k_{average}\) in T1-T5);
2. Then, according to the actual lining material performance parameters, the curves under different lobe number \(n\) are calculated and drawn;
3. Obtaining the envelope curve of the buckling value \(\varphi\) and \(l\)/\(R\) ratio (See figure line '\(t\) = 1.2' in Figure 17);  
Figure 16: The \(\varphi\)-\(l\)/\(R\) principles for T1 to T5.  
4. Taking the actual \(l\)/\(R\) ratio to determine the corresponding \(\varphi\) value and reverse calculating the critical buckling pressure; 5. If the calculated pressure does not meet the specified vacuum negative pressure value, several groups of large wall mass are used for the new drawing. In this case, several scenarios are considered (\(t\) = 1.2 mm, \(t\) = 1.4 mm, \(t\) = 1.6 mm, \(t\) = 1.8 mm, \(t\) = 2.0 mm and \(t\) = 3.0 mm); 6. So far, the ratio of \(l\)/\(R\) can be set as the actual value and plotted in the figure to obtain the corresponding value of \(\varphi\). Applying the definition of \(\varphi\), the critical buckling pressure can be derived reversely. The critical buckling values in this case are 14.33 kPa for \(t\) = 1.2 mm, 20.82 kPa for \(t\) = 1.4 mm, 28.86 kPa for \(t\) = 1.6 mm, 37.73 kPa for \(t\) = 1.8 mm, 50.70 kPa for \(t\) = 2.0 and 120.0 kPa for \(t\) = 3.0 mm, respectively; 7. Finally, \(t\) = 2.0 mm is taken as the repair wall thickness of the lining under the negative pressure of the water-supply pipeline (The standard requires it to withstand negative pressure of \(-\)0.05 MPa). Moreover, the lining generates four random lobes in this circumstance.  
This case is based on an experiment carried out in 2013. The continuous negative pressure test shows that the 1.2 mm lining can withstand a vacuum of about 10 kPa, which is consistent with the results of the enhancement model in this paper. It should be noted that, due to the conservative enhancement coefficient value (compared with other values in Table 3), the buckling pressure is smaller than the actual pressure in the case of \(t\) =1.6 mm and \(t\) =1.80 mm. Moreover, this case further proves that, with the increase of lining wall thickness, the number of lobes becomes smaller and smaller, and the lobe formation becomes more and more difficult. Moreover, when \(t\) = 3.0 mm, the critical buckling pressure reached 0.120 MPa, which exceeded the vacuum degree range, indicating that, in this case, negative pressure is no longer a limiting factor for the lining with a wall thickness of 3.0 mm (considered negative pressure separately).