## 3 Analysis
### Comparison with Theoretical Predictions  
The ratio \(l/R\) is used as the \(x\)-coordinate and the value \(\varphi\) as the \(y\)-coordinate. For each value of \(\alpha\), the part plotted by different value of n can be obtained. By substituting the performance parameters of the SSL (Elastic Modulus is 195 GPa; Poisson ration is 0.247; and \(t\) equals 1.61, 1.79 and 2.17 mm, respectively) into 22, the diagram shown in Figure 15 can be drawn. For the same value of \(\alpha\), we can finally get a line composed of the parts of the curve given for various values of n (see Figure 15a, red line). It shows that, for the cylindrical shell with short length, when the ratio \(l/R\) decreases, the critical load increases rapidly. For the long cylindrical thin shell, the critical buckling pressure is no longer related to the length but is equal to the cylinder's value under the condition of infinite length.  
The comparison between thin-shell theory and test results shows that the host pipe has a significant supporting effect on the lining. Based on Equation (22), the critical buckling pressure (\(p_{cr0}\)) of liner under ideal condition is calculated with the lobe number n equals 2. It then has the critical buckling pressure of this baseline, compared with the test results (\(p_{test}\)), the host pipe reinforcement coefficient \(k=p_{test}/p_{cr0}\) is given, as shown in Table 3, and the test data is also plotted in Figure 15b.  
Figure 15: The prediction model of single layer pipeline buckling based on thin-shell theory. (**a**) The buckling model of pipeline under different lobe parameters (T1 data). (**b**) Minimum lobe envelope of different \(\alpha\) corresponding to different test groups.  
It shows that, since to the thin-shell theory does not consider the reinforcement effect of HPs on lining, the result has a large deviation. In the data analysis, it is found that the enhancement effect of HPs can be divided into two parts: the first part is the raising of the threshold of lining lobes, and the second part shows the increase of critical buckling pressure for the lining. The \(l\)/\(R\) value is scaled to \(\sqrt{k}\), and the thin-shell buckling Equation (21) is revised to Equation (23), which can be regarded as the lining buckling Equation considering the reinforcement effect of the existing pipeline.  
\[p_{cr}=\frac{Et^{3}}{12R^{3}(1-\mu^{2})}\bigg{(}n^{2}-1+\frac{2n^{2}-1-\mu}{1+ n^{2}l^{2}/\kappa\cdot\pi^{2}R^{2}}\bigg{)}+\frac{Et}{(n^{2}-1)(1+n^{2}l^{2}/ \kappa\cdot\pi^{2}R^{2})R} \tag{23}\]  
According to the curves and test data in Figure 16, Equation (23) can better predict the critical buckling pressure of the lining.  
Figure 16 indicates that four lobes occur randomly in the lining when the vacuum degree of T4 group is between 33.4 and 48.5 kPa. However, five lobes occur randomly when the vacuum degree ranges from 46.5 to 70.0 kPa for T1 and T5, from 17.1 to 32.0 kPa for T2 and from 22.5 to 36.8 kPa for T3. Moreover, six lobes are easily generated between 36.8 and 54.8 kPa for T3. This further indicates that the thinner the wall thickness and the larger the pipe diameter, the lower the threshold for forming multiple lobes. Combined with the experimental results, Equation (23) is fully demonstrated.