## 2 Theoretical Basis of Rigid-Flexible Composite Lining Structure Design  
The design theory of rigid-flexible composite lining structures is based on the buckling theory of rings under uniform compression. For the rigid-flexible composite lining structure, the rigid structure is formed by filling mortar in the gap between the flexible lining pipe and the original pipeline. However, due to the long longitudinal length of the pipeline, it is often difficult to ensure that the gap between the flexible lining pipe and the original pipeline is fully filled and compacted in the construction process of mortar filling. Therefore, there are often gaps between the flexible liner and the original pipeline, as shown in Figure 1. In this case, the two ends of the flexible liner at the gap can be regarded as fixed in the rigid filling slurry. Therefore, it is reasonable to consider that the theoretical basis of rigid-flexible composite lining structure design is the uniform compression arch buckling theory with fixed ends.  
The theoretical schematic diagram of uniform compression arch buckling with fixed ends is shown in Figure 2. Both ends of \(A\) and \(B\) are fixed ends, and the angle between \(A\) and \(B\) is \(2\alpha\). The arch is subjected to uniform pressure \(P\). Assuming that after buckling of the arch, there is not only horizontal pressure \(S\), but also direct shear force \(Q\) at the midpoint \(C\), and the arch is regarded as the cable curve of uniform pressure and \(w\) is used to represent the radial displacement of the center, then the bending moment \(M\) of any cross section m determined by angle \(\theta\) is as follows [2]:  
\[M=Sw-QRsin\theta=PRw-QRsin\theta \tag{2}\]  
Figure 1: Schematic diagram of rigid–flexible composite lining structure.  
The bending differential equation of the circular slender rod shown in Figure 2 is as follows:  
\[\frac{d^{2}w}{d\theta^{2}}+w=-\frac{MR^{2}}{EI} \tag{3}\]  
According to Formulas (2) and (3), Formula (4) can be obtained.  
\[\frac{d^{2}w}{d\theta^{2}}+w=-\frac{R^{2}}{EI}(PRw-QRsin\theta) \tag{4}\]  
Formula (4) can also be expressed as Formula (5).  
\[\frac{d^{2}w}{d\theta^{2}}+\bigg{(}1+\frac{PR^{3}}{EI}\bigg{)}w=-\frac{QR^{3} sin\theta}{EI} \tag{5}\]  
By replacing \(1+\frac{PR^{3}}{EI}\) with \(K_{1}\)\({}^{2}\), Formula (5) may be expressed as Formula (6).  
\[\frac{d^{2}w}{d\theta^{2}}+K_{1}^{2}w=\frac{QR^{3}sin\theta}{EI} \tag{6}\]  
The ordinary solution of the equation is shown as Formula (7).  
\[w=AsinK_{1}\theta+BcosK_{1}\theta+\frac{QR^{3}sin\theta}{\left(K_{1}^{2}-1 \right)EI} \tag{7}\]  
The conditions determining constants \(A\) and \(B\) and shear force \(Q\) are shown as Formulas (8) and (9).  
\[w=\frac{d^{2}w}{d\theta^{2}}=0,\;\theta=0 \tag{8}\]  
\[w=\frac{dw}{d\theta}=0,\;\theta=\alpha \tag{9}\]  
When \(B\) is zero, Formula (8) can be satisfied, so Formulas (10) and (11) can be obtained from Formula (9).  
\[AsinK_{1}\alpha+\frac{QR^{3}sin\alpha}{\left(K_{1}^{2}-1\right)EI}=0 \tag{10}\]  
\[AK_{1}cosK_{1}\alpha+\frac{QR^{3}cos\alpha}{\left(K_{1}^{2}-1\right)EI}=0 \tag{11}\]  
Through Formulas (10) and (11), Formula (12) can be obtained.  
\[sinK_{1}acosa-K_{1}sinacosaK_{1}\alpha=0 \tag{12}\]  
Figure 2: Schematic diagram of theoretical model of rigid–flexible composite lining structure design.  
From Formula (12), the value of \(K_{1}\) can be obtained, which is brought into formula \({K_{1}}^{2}=1+\frac{PR^{3}}{EI}\), and the buckling critical value of uniform compression arch with fixed ends is shown as Formula (13).  
\[P_{cr}=\frac{EI}{R^{3}}\Big{(}K_{1}^{2}-1\Big{)} \tag{13}\]