## 1 Introduction  
In recent years, as the demand for energy transportation has increased, the construction of pipeline systems has increased to keep cities functioning properly. The pipeline system contains multiple types of pipe [1; 2; 3; 4] and water supply pipelines and drainage pipelines are the key to satisfying the daily water requirements of residents [5] and avoiding pollution of the city environment [6], which are easily deformed, cracked or suffer from erosion under the effect of soil and groundwater. The pipe defects will eventually cause the bearing capacity reduction of the pipe and the leakage of pipe contents. Hence, the working conditions of the water supply and drainage pipelines need to be assessed and the defect pipelines need to be repaired. Under the constraints of traffic conditions and the urban utility density in China, it is difficult to replace or repair defective pipes by excavation [7]. Trenchless rehabilitation technologies are gradually replacing traditional excavation repair technologies as the main technology for urban pipeline rehabilitation due to the ability to repair the pipe with minor excavation faces. Formed-in-place pipe (FIPP) technology repairs pipelines by installing a thermoplastic polyvinyl chloride (PVC) pipe into host pipe, which expands to fit tightly with host pipe by thermoforming [8]. The FIPP liner has high tensile and bending strength and is suitable for water supply pipelines and drainage pipelines of different types of sections, diameters and materials [9]. However, the FIPP liners will fail under high water pressure, especially in coastal cities, so it is necessary to study the buckling behavior of FIPP liners. Scholars have had certain research results for this thin-walled cylindrical structure stability problem. [PERSON] et al. [10] first proposed the free ring buckling theory and derived a model for calculating the critical buckling pressure of an unconstrained circular thin-walled pipe under uniform external pressure. Based on that, [PERSON] [11] proposed a model for calculating the critical buckling pressure of the liner constrained by a rigid host pipe. [PERSON] and [PERSON] [12] conducted an experimental study to consider the variation law of the support coefficient, they proposed to take the value of the enhancement factor as 7 to compensate for the deviation between the experimental results and the theory, and the enhancement factor was adopted by ASTM F1216 [13]. The liners might have initial defects during installation, [PERSON] and [PERSON] [14] established an analytical model to predict the critical buckling pressure of the liner with an initial annular gap. [PERSON] [15] established an approximate solution for enhancement factors for loosely fitted liners. [PERSON] [16] studied the buckling behavior of reinforced steel liners under negative pressure loading and proposed a design model to predict the critical buckling pressure of the liners. However, annular gaps might occur when the liner is installed or deformed after prolonged exposure to water pressure, which will affect the bearing capacity of the liner. [PERSON] [17] analyzed the effect of annular gaps on critical buckling pressure by combining neural networks and found that the effect of critical buckling pressure caused by an annular gap would decrease when the wall thickness increased. [PERSON] [18] proposed a model for calculating the critical buckling pressure of the liner in the presence of multiple partial gaps. [PERSON] [19] found that the annular gap affects the buckling behavior of the liner. [PERSON] [20] investigated the stability of the liner under uniform external pressure after grouting the annular space gap and found that the change in the contact conditions between the liner and the original pipe affected the pressure equilibrium path. [PERSON] [21] concluded that an increase in the partial buckling range of the liner leads to a decrease in the critical buckling pressure. [PERSON] et al.