The free ring buckling theory has been widely used in the structural design of flexible pipes, and the formula is as follows:  
\[P_{\sigma}=\frac{24 EI}{(1-\mu^{2})D^{3}} \tag{1}\]  
where \(D\) is the average pipe diameter; \(I\) is the moment of inertia of the cross section; \(\mu\) is Poisson's ratio; \(P_{\sigma}\) is the critical buckling pressure; \(E\) is the elastic modulus. [PERSON] [3; 4; 5] assumes that the total potential energy of the lining during buckling is equal to the work conducted by the strain energy of the lining and the external hydrostatic pressure. At the same time, it is assumed that the circumferential axial force on the whole ring is a constant and the shape of the deformation section of the lining is a chord function. Then, the semi-analytical solution of critical buckling of the lining with annular clearance is obtained. [PERSON] [6] studied the buckling performance of the lining with reduced local wall thickness due to corrosion under uniform external pressure, and established an analytical formula for the buckling strength of the lining with uneven thickness under external pressure, according to the classical Glock theory. [PERSON] et al. [7] studied the stress-strain relationship curve of the material after repairing the pipeline with high-density polyethylene (HDPE) lining, and studied the buckling failure characteristics through numerical simulation. On the basis of this, the corresponding calculation formula of buckling strength was proposed. [PERSON] and [PERSON] [8; 9] used numerical simulation to consider the influence of the initial annular gap and fold on the buckling strength of the lining. Finally, the calculation method of buckling strength of thin-walled lining was proposed. They also carried out a bending test and a finite element simulation of the lining with wrinkles, and found that the shape of the bending failure of the lining is almost identical. [PERSON] and [PERSON] [10] also proposed a plastic hinge model, which assumes that (1) the thin shell is an elastic body, and the buckling deformation occurs at a certain arc position without shrinkage or elongation, and (2) the two endpoints of the arc are moving plastic hinges, and the center points of the two are static plastic hinges. According to the above assumptions, they put forward the corresponding theoretical calculation formula. [PERSON] et al. [11] studied the buckling behavior of stainless steel thin lining under negative pressure after trenchless repair of an urban water supply network. Finally, an improved lining buckling prediction model (enhanced model) is proposed. In addition, the design of flexible solid-wall liner is stipulated in the ASTM series standards [12; 13; 14; 15; 16; 17], which is divided into structural repair and semi-structural repair. For the study of rigid lining structures, [PERSON] [18] proposed the basic principle of rigid lining structure design according to the different stress states of the old pipeline, and established the theoretical model of lining design. [PERSON] [19] carried out design research according to the limit state design method, according to different failure modes and structural models of pipelines. [PERSON] et al. [20] found that after the interior of the concrete pipe is repaired by cement mortar, if the shear strength of the old and new interface is low, the old and new interface of the composite structure will strip under the external loads, which will affect the bearing capacity of the structure. In the literature [21; 22; 23; 24], the bearing capacity of the pipeline repaired by mortar spraying lining was studied through experiments, indicating that the mortar spraying rigid lining significant improved the bearing capacity of the original pipeline. For rigid-flexible composite lining structure design, [PERSON] et al. considered the hydrostatic buckling failure of a noncircular composite liner with an arched invert installed in the sewer pipe. An equation was derived for the buckling of the invert liner, and its application procedure was proposed [25].