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Buckling Behavior of Loosely Fitted Formed-In-Place Pipe Liner in Circular Host Pipe under External Pressure

Buckling Behavior of Loosely Fitted Formed-In-Place Pipe Liner in Circular Host Pipe under... applied sciences Article Buckling Behavior of Loosely Fitted Formed-In-Place Pipe Liner in Circular Host Pipe under External Pressure 1 1 2 1 3 , Zheng Zeng , Xuefeng Yan , Weigang Xiang , Yahong Zhao and Samuel T. Ariaratnam * Faculty of Engineering, China University of Geosciences—Wuhan, Wuhan 430074, China Wuhan Metro Bridge and Tunnel Management Co., Ltd., Wuhan 430077, China School of Sustainable Engineering and the Built Environment, Arizona State University, Tempe, AZ 85259, USA * Correspondence: ariaratnam@asu.edu Abstract: The formed-in-place pipe (FIPP) is a trenchless technology used for pipeline rehabilitation. It is a folded PVC pipe that expands through thermoforming to fit tightly inside the host pipe. However, the deficiencies during the construction of FIPP liners such as insufficient inflation, pipe misalignment and initial deformation will lead to elliptical deformation of the FIPP liner, which affects the load-bearing performance of the liner and makes it susceptible to buckling failure. In this paper, the buckling behavior of loosely fitted FIPP liners under uniform external pressure was investigated by the external pressure resistance test and finite element model. The pre- and post- buckling equilibrium paths verified the finite element model. The results indicated that the value of the dimension ratio will significantly reduce the critical buckling pressure. With the increasing value of liner major axis ratio to host pipe, the reduction effect on the critical buckling pressure caused by the increase in the ovality will diminish. Different values of liner major axis ratio to host pipe and ovality changed the range of the detached portion, which affected the critical buckling pressure. The parametric studies modified the design model from ASTM F1216, which was established to predict the critical buckling pressure of a loosely fitted FIPP liner and reduced the average difference rate from 23.43% to 5.52%. Keywords: FIPP; buckling; ovality; FEM Citation: Zeng, Z.; Yan, X.; Xiang, W.; Zhao, Y.; Ariaratnam, S.T. Buckling Behavior of Loosely Fitted Formed- In-Place Pipe Liner in Circular Host Pipe under External Pressure. Appl. 1. Introduction Sci. 2023, 13, 679. https://doi.org/ In recent years, as the demand for energy transportation has increased, the construction 10.3390/app13020679 of pipeline systems has increased to keep cities functioning properly. The pipeline system Academic Editor: José contains multiple types of pipe [1–4] and water supply pipelines and drainage pipelines are António Correia 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 Received: 7 November 2022 under the effect of soil and groundwater. The pipe defects will eventually cause the bearing Revised: 29 December 2022 capacity reduction of the pipe and the leakage of pipe contents. Hence, the working Accepted: 31 December 2022 conditions of the water supply and drainage pipelines need to be assessed and the defect Published: 4 January 2023 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 Copyright: © 2023 by the authors. technologies are gradually replacing traditional excavation repair technologies as the main Licensee MDPI, Basel, Switzerland. technology for urban pipeline rehabilitation due to the ability to repair the pipe with minor This article is an open access article excavation faces. Formed-in-place pipe (FIPP) technology repairs pipelines by installing a distributed under the terms and thermoplastic polyvinyl chloride (PVC) pipe into host pipe, which expands to fit tightly conditions of the Creative Commons with host pipe by thermoforming [8]. The FIPP liner has high tensile and bending strength Attribution (CC BY) license (https:// and is suitable for water supply pipelines and drainage pipelines of different types of creativecommons.org/licenses/by/ sections, diameters and materials [9]. However, the FIPP liners will fail under high water 4.0/). Appl. Sci. 2023, 13, 679. https://doi.org/10.3390/app13020679 https://www.mdpi.com/journal/applsci Appl. Sci. 2023, 13, 679 2 of 15 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. Timoshenko 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, Glock [11] proposed a model for calculating the critical buckling pressure of the liner constrained by a rigid host pipe. Aggarwal and Cooper [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, Lo and Zhang [14] established an analytical model to predict the critical buckling pressure of the liner with an initial annular gap. EI-Sway [15] established an approximate solution for enhancement factors for loosely fitted liners. Zhao [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. El-Sway [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. Dong [18] proposed a model for calculating the critical buckling pressure of the liner in the presence of multiple partial gaps. Wang [19] found that the annular gap affects the buckling behavior of the liner. Li [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. El-Sawy [21] concluded that an increase in the partial buckling range of the liner leads to a decrease in the critical buckling pressure. Treitz et al. [22] designed a PVC pipe test system to study the effect of buoyancy on the buckling behavior of PVC liners by applying hydrostatic pressure to simulate grouting pressure. Results showed that the presence of buoyancy reduced the critical flexural strength of the liner. In the existing studies, only the initial gap formed by the partial detachment of the liner from the host pipe is considered, ignoring the effect of the annular gap on the buckling behavior of the elliptical liner under loosely fitted conditions and the change of the contact conditions between the liner and host pipe under the action of water pressure, resulting in a specification deviation of the critical buckling pressure calculated by the prediction model in ASTM F1216, which reduces the reliability of the liner designs. This study designed an experimental system to investigate the buckling behavior of the loosely fitted FIPP liner under external pressure. The deformation mode of the buckling lobes can be directly captured by the camera and the critical buckling pressure could be measured by pressure transducer when the liner had a buckling failure. The geometric parameters of the liner were indicated as the dimension ratio and ratio of liner major axis to host pipe radius and ovality, and the 2D finite element models were set to analyze the effect of different geometric parameters of the liners. This study aimed at obtaining the different buckling behaviors of the FIPP liner by changing different geometric parameters and analyzing the sensitivity of different parameters on critical buckling pressure. The results of the study were used to improve the accuracy of the critical buckling pressure prediction model in ASTM F1216 for the design of the FIPP liner by modifying the enhancement factor. 2. Materials and Methods 2.1. Test Design The experimental system is shown in Figure 1, the geometry dimension of the FIPP liner and the host pipe are shown in Table 1. During the installation of the host pipe, the outlet valve of the steel pipe was set at the bottom, and the inlet valves were set at the top of the host pipe. Considering the gravity of the water body itself and the buoyancy Appl. Sci. 2023, 13, x FOR PEER REVIEW 3 of 16 2. Materials and Methods 2.1. Test Design The experimental system is shown in Figure 1, the geometry dimension of the FIPP Appl. Sci. 2023, 13, 679 3 of 15 liner and the host pipe are shown in Table 1. During the installation of the host pipe, the outlet valve of the steel pipe was set at the bottom, and the inlet valves were set at the top of the host pipe. Considering the gravity of the water body itself and the buoyancy effect effect on the liner pipe, the pressure sensor interface is arranged at the spring line of the on the liner pipe, the pressure sensor interface is arranged at the spring line of the steel steel pipe and the displacement and deformation of the middle section of the FIPP liner are pipe and the displacement and deformation of the middle section of the FIPP liner are monitored using strain gauges and a PIV system. monitored using strain gauges and a PIV system. Figure 1. Experimental system design and geometric dimensions of equipment. Figure 1. Experimental system design and geometric dimensions of equipment. Table 1. Geometrical dimensions of FIPP liner and host pipe. Table 1. Geometrical dimensions of FIPP liner and host pipe. Internal Radius/mm Wall Internal Radius/mm Length/mm Length/mm Wall Thickness/mm Thickness/mm MaMajor jor Axis Axis Minor A Minor xis Axis FIPP 2000 309.85 287.15 7.95 FIPP 2000 309.85 287.15 7.95 The host pipe 2000 318 10 The host pipe 2000 318 10 Firstly, the FIPP liner was installed into the host pipe after grinding the ports and the Firstly, the FIPP liner was installed into the host pipe after grinding the ports and the axis of the liner was kept coincident with the host pipe. Considering the gravity of the axis of the liner was kept coincident with the host pipe. Considering the gravity of the liner and the buoyancy, the pressure transducer interface was set at the spring line of the liner and the buoyancy, the pressure transducer interface was set at the spring line of the steel pipe and strain gauges and PIV monitored the displacement and deformation of the steel pipe and strain gauges and PIV monitored the displacement and deformation of the middle section of the FIPP liner. The latex membrane was used to seal the port so that a middle section of the FIPP liner. The latex membrane was used to seal the port so that a closed cavity was formed between the liner and the host pipe. Secondly, water was slowly closed cavity was formed between the liner and the host pipe. Secondly, water was slowly injected into the closed chamber through the test pressure pump until the air in the annu- injected into the closed chamber through the test pressure pump until the air in the annular lar space gap was completely discharged and the test was started by maintaining a con- space gap was completely discharged and the test was started by maintaining a continuous tinuous pressurization rate of 12 kPa/min. The hydrostatic pressure acts directly on the pressurization rate of 12 kPa/min. The hydrostatic pressure acts directly on the outer wall outer wall of the liner, causing the liner tube to buckle and deform and when the liner of the liner, causing the liner tube to buckle and deform and when the liner buckles and buckles and deforms to the limit, the pressurization is stopped for 90 s and then unloaded deforms to the limit, the pressurization is stopped for 90 s and then unloaded and the test and the test is completed. is completed. 2.2. Finite Element Model Set-Up To quantify the buckling equilibrium paths of the thin-walled structures, the stress distribution and deformation extent of the structure must be calculated. Jiao [23] used the Differential Quadrature Method (DQM) to determine the pre-buckling in-plane stress distribution of thin rectangular FG-CNTRC plate. Kabir and Aghdam [24] developed an accurate Bézier-based multi-step method and implemented it to find the nonlinear vibration Appl. Sci. 2023, 13, x FOR PEER REVIEW 4 of 16 2.2. Finite Element Model Set-Up To quantify the buckling equilibrium paths of the thin-walled structures, the stress distribution and deformation extent of the structure must be calculated. Jiao [23] used the Differential Quadrature Method (DQM) to determine the pre-buckling in-plane stress dis- tribution of thin rectangular FG-CNTRC plate. Kabir and Aghdam [24] developed an ac- Appl. Sci. 2023, 13, 679 4 of 15 curate Bézier-based multi-step method and implemented it to find the nonlinear vibration and post-buckling configurations of Euler–Bernoulli composite beams reinforced with graphene nano-platelets (GnP). Both DQM and the Bézier method divide the structure and post-buckling configurations of Euler–Bernoulli composite beams reinforced with into a single element to calculate the stress and deformation during the buckling prosses, graphene nano-platelets (GnP). Both DQM and the Bézier method divide the structure into but those require extensive calculations computation to improve the precision of the cal- a single element to calculate the stress and deformation during the buckling prosses, but culation and the calculation results must be processed to be visualized. Thus, using the those require extensive calculations computation to improve the precision of the calculation finite element analysis will be a good solution to quantify the buckling equilibrium paths and the calculation results must be processed to be visualized. Thus, using the finite element of the FIPP liners. analysis will be a good solution to quantify the buckling equilibrium paths of the FIPP liners. ABAQUS is used for finite element simulation to study the effects of the dimension ABAQUS is used for finite element simulation to study the effects of the dimension ratio (DR), ratio of liner major axis to host pipe radius (a/R) and ovality (q) on the buckling ratio (DR), ratio of liner major axis to host pipe radius (a/R) and ovality (q) on the buckling behavior. The longitudinal deformation and the boundary effect caused by the end ports behavior. The longitudinal deformation and the boundary effect caused by the end ports of of the liner are neglected and this paper only considers the buckling cross-section of the the liner are neglected and this paper only considers the buckling cross-section of the liner. liner. Using 2D instead of 3D analysis effectively reduces simulation time while maintain- Using 2D instead of 3D analysis effectively reduces simulation time while maintaining ing calculation accuracy. The model uses an eight-node reduced-integration plane strain calculation accuracy. The model uses an eight-node reduced-integration plane strain element (CPE8R) and the liner elements are divided into three layers totaling 1200 ele- element (CPE8R) and the liner elements are divided into three layers totaling 1200 elements. ments. A total of 200 elements for steel host pipe are shown in Figure 2. Twenty-three A total of 200 elements for steel host pipe are shown in Figure 2. Twenty-three groups of groups of finite element model are shown in Table 2, Group A is set for the validation test, finite element model are shown in Table 2, Group A is set for the validation test, Group B Group B to Group E are distinguished by the value of a/R to analyze the buckling behavior to Group E are distinguished by the value of a/R to analyze the buckling behavior under under the different values of a/R. the different values of a/R. Figure 2. The element mesh of FIPP and steel pipe model. Figure 2. The element mesh of FIPP and steel pipe model. Table 2. Geometric parameters of the numerical model. Table 2. Geometric parameters of the numerical model. N No. o. a/aR/R q q DR DR A 0.974 3.8 75 A 0.974 3.8 75 B-1 0.999 5 100 B-1 0.999 5 100 B B-2 -2 0.999 0.999 0 0 100 100 C-1 0.99 5 100 C-1 0.99 5 100 C-2 0.99 5 150 C-2 0.99 5 150 C-3 0.99 5 200 C-3 0.99 5 200 C-4 0.99 5 250 C-4 0.99 5 250 C-5 0.99 5 300 C-5 0.99 5 300 C-6 0.99 0 100 C-6 0.99 0 100 C-7 0.99 2.5 100 C-7 0.99 2.5 100 C-8 0.99 7.5 100 C-8 0.99 7.5 100 C-9 0.99 10 100 C-9 0.99 10 100 D-1 0.98 5 100 D D-2 -1 0.98 0.98 5 5 150 100 D-3 0.98 5 200 D-4 0.98 5 250 D-5 0.98 5 300 D-6 0.98 0 100 D-7 0.98 2.5 100 D-8 0.98 7.5 100 D-9 0.98 10 100 E-1 0.97 5 100 E-2 0.97 0 100 Appl. Sci. 2023, 13, 679 5 of 15 The host pipe material adopts the elastic constitutive model, the mechanical parame- ters of FIPP liner and steel are shown in Table 3. The outer surface of the steel pipe is fully restrained from displacement and rotation in all directions. The uniform external pressure is applied to the outer surface of the liner. The property of the contact surface between the liner and host pipe is defined as “surface to surface” in ABAQUS and allows the liner and host pipe to separate from each other when the normal pressure is zero or negative. The friction coefficient is 0.4 for the tangential contact, considering the friction between the liner and the pipe [9]. The displacement of the steel pipe is not considered during the liner buckling process and the boundary condition of the outer surface of the steel pipe is set to limit its displacement in the x and y axes, as shown in Figure 3a. A static general analysis step is used to apply the displacement of the liner along the y-axis and control the displacement of the liner on the x-axis and the rotation along the z-axis to zero, which simulates the process of the liner floating and contacting with the inner wall of the steel pipe under the action of buoyancy. The details about the boundary and load distribution are shown in Appendix A. The Riks step in ABAQUS is used to track the pre-buckling and post-buckling of the liner, which needs to assume the existence of the initial defect which will cause the buckling of the liner. In this model, a concentrated force is applied to the bottom surface of the liner to simulate the initial defect occurring at the bottom part of the pipe under the action of buoyancy, while the hydrostatic pressure acts on the outer surface of the liner, as shown in Figure 3b. The non-linear geometry (NIgeom) switch needs to be turned on in the ABAQUS simulation step settings to simulate the large deflection deformation during the buckling process of the liner. Table 3. Mechanical properties of FIPP liner and host pipe. Materials Modulus of Elasticity (MPa) Poisson’s Ratio Bending Strength Appl. Sci. 2023, 13, x FOR PEER REVIEW 6 of 16 Steel pipe 190,000 0.3 / FIPP 2089.9 0.3 62 Figure 3. The condition of (a) boundary and (b) load distribution. Figure 3. The condition of (a) boundary and (b) load distribution. 2.3. Prediction Method of Critical Buckling Pressure 2.3. Prediction Method of Critical Buckling Pressure According to the designs of the semi-structural rehabilitation to flexible liner in ASTM 1216, A the cco critical rding buckling to the dpr esi essur gns e ois f th calculated e semi-st by ruc Equation tural reh (1), abili while tatiothe n to enhan flexible cement liner in factor ASTM value 1216is , th 7e and criti the cal safety bucklin factor g pres is sure 2. is calculated by Equation (1), while the enhance- ment factor value is 7 and the safety factor is 2. 2 3 P = 2KE C/[(1 m )(DR 1) N] (1) cr 2 3 Pcr = 2KELC/[(1 − μ )(DR − 1) N] (1) From Equation (1), it can be found that the influence of the annular gap is not con- sidered when calculating the critical buckling pressure. The enhancement factor K is de- signed for reducing the deviates of the critical buckling pressure value between the theo- retical condition and working condition. The enhancement factor value recommended by ASTM 1216 is unable to guarantee the accuracy of the critical buckling pressure value when the annular gap occurs. Therefore, the value of enhancement factor K will be quan- tified by analyzing the difference in the critical buckling pressure of finite element models which set the different geometric parameters according to Table 1. 3. Results and Discussion 3.1. Finite Element Model Validation The FIPP liner material in the finite element model adopts the elastic–plastic consti- tutive model tested according to ISO 178. The relationship between the true strain and stress in the plastic phase of the FIPP material is shown in Figure 4, which will be used to define the plastic constitutive model of the finite element model. The elastic limit of the FIPP material is 41.8 MPa. However, bending failure in the specimen did not occur during the test and the ultimate bending strength was considered 62.62 MPa in the configuration of the finite element model. Appl. Sci. 2023, 13, 679 6 of 15 From Equation (1), it can be found that the influence of the annular gap is not consid- ered when calculating the critical buckling pressure. The enhancement factor K is designed for reducing the deviates of the critical buckling pressure value between the theoretical condition and working condition. The enhancement factor value recommended by ASTM 1216 is unable to guarantee the accuracy of the critical buckling pressure value when the annular gap occurs. Therefore, the value of enhancement factor K will be quantified by analyzing the difference in the critical buckling pressure of finite element models which set the different geometric parameters according to Table 1. 3. Results and Discussion 3.1. Finite Element Model Validation The FIPP liner material in the finite element model adopts the elastic–plastic consti- tutive model tested according to ISO 178. The relationship between the true strain and stress in the plastic phase of the FIPP material is shown in Figure 4, which will be used to define the plastic constitutive model of the finite element model. The elastic limit of the FIPP material is 41.8 MPa. However, bending failure in the specimen did not occur during Appl. Sci. 2023, 13, x FOR PEER REVIEW 7 of 16 the test and the ultimate bending strength was considered 62.62 MPa in the configuration of the finite element model. Figure 4. True stress versus true strain curve of FIPP material. Figure 4. True stress versus true strain curve of FIPP material. The comparison of the pressure–displacement curve of the FIPP liner is shown in The comparison of the pressure–displacement curve of the FIPP liner is shown in Figure 5. According to the curve, the buckling behavior of the liner can be divided into Figure 5. According to the curve, the buckling behavior of the liner can be divided into three stages. At first (stage AB), the deformation displacement versus the pressure is linear three stages. At first (stage AB), the deformation displacement versus the pressure is linear during the increase of hydrostatic pressure. At this stage, the liner did not form a buckling during the increase of hydrostatic pressure. At this stage, the liner did not form a buckling lobe. Then (stage BC), as the pressure increases, the deformed part of liner formed a bulge, lobe. Then (stage BC), as the pressure increases, the deformed part of liner formed a bulge, which can be called the buckling lobe. When the pressure value approaches the critical which can be called the buckling lobe. When the pressure value approaches the critical buckling pressure, the deformation increases and rate of the buckling lobe is significantly incr buckli eased, ng while pressure, the pr th essur e def e incr orma ease tio rate n inrcr emains eases unchanged. and rate ofThe the buckling bucklinlobe g lobe continues is significantly deformation with no enhancement of the ability to withstand the external pressure, but the increased, while the pressure increase rate remains unchanged. The buckling lobe contin- liner still has the bearing capacity until the pressure reaches the critical buckling pressure ues deformation with no enhancement of the ability to withstand the external pressure, which will cause the buckling failure at the buckling lobe. The liner has a buckling failure at but the liner still has the bearing capacity until the pressure reaches the critical buckling the end (stage CD). During stage AB, the test result and simulation result both indicate that pressure which will cause the buckling failure at the buckling lobe. The liner has a buck- the linear relationship between pressure and displacement is ended when the displacement ling failure at the end (stage CD). During stage AB, the test result and simulation result reaches 12.53 mm, but the pressure of the test is 19.4% greater than simulation. At stage BC, both indicate that the linear relationship between pressure and displacement is ended the liner material may have had errors in the manufacturing process and the constraints when the displacement reaches 12.53 mm, but the pressure of the test is 19.4% greater than of the flange on the liner in the test limited the displacement of the liner at both ends of the sim liner ulatio , pr noducing . At stage end BC, effects the li af nfecting er mater the iabuckling l may hacr ve oss-section had errors inin the thtest. e ma When nufactur the ing pro- displacement of the buckling lobe was equal, the pressure of the test result was greater than cess and the constraints of the flange on the liner in the test limited the displacement of the liner at both ends of the liner, producing end effects affecting the buckling cross-sec- tion in the test. When the displacement of the buckling lobe was equal, the pressure of the test result was greater than the numerical result during the buckling process. Hence, the liner stiffness in the test was significantly greater than the simulation. At point C, the liner reached the critical buckling pressure of 38.08 kPa when the displacement of the buckling lobe was 52.77 mm in the test and the liner reached the critical buckling pressure of 35.94 kPa when the displacement of the buckling lobe reached 76.39 mm in the numerical re- sults. The critical buckling pressure in the test was 5.6% greater than the simulation and the displacement of buckling lobe in the simulation was 30.9% greater than in the test. In stage CD, when the displacement of the buckling lobe increased to 74.4 mm in the test, the buckling lobe began to move axially to the port along the liner and the liner eventually failed (Figure 6), while the buckling lobe continued to deform in the numerical results. This paper mainly studied the influence of changing geometric parameters on critical buckling pressure. Although the pressure of the simulation result was lower than the test result during the buckling process, the accuracy of the critical buckling pressure was still reliable. Appl. Sci. 2023, 13, x FOR PEER REVIEW 8 of 16 Appl. Sci. 2023, 13, 679 7 of 15 the numerical result during the buckling process. Hence, the liner stiffness in the test was significantly greater than the simulation. At point C, the liner reached the critical buckling pressure of 38.08 kPa when the displacement of the buckling lobe was 52.77 mm in the test and the liner reached the critical buckling pressure of 35.94 kPa when the displacement of the buckling lobe reached 76.39 mm in the numerical results. The critical buckling pressure in the test was 5.6% greater than the simulation and the displacement of buckling lobe in the simulation was 30.9% greater than in the test. In stage CD, when the displacement of the buckling lobe increased to 74.4 mm in the test, the buckling lobe began to move axially Figure 5. Comparing the pressure–displacement curves of test and simulation divided into float- to the port along the liner and the liner eventually failed (Figure 6), while the buckling lobe ing stage (AB), buckling stage (BC) and failure stage (CD). continued to deform in the numerical results. This paper mainly studied the influence of Appl. Sci. 2023, 13, x FOR PEER REVIEW 8 of 16 changing geometric parameters on critical buckling pressure. Although the pressure of the The deformation of the liner at points A, B, C and D was selected according to Figure simulation result was lower than the test result during the buckling process, the accuracy 5 to compare the test with the numerical results as shown in Figure 6. As shown in Figure of the critical buckling pressure was still reliable. 6a, the liner appears to float up under hydrostatic pressure, and the outer surface of the liner tube contacts the inner surface of the host pipe, while the liner undergoes slight ro- tation and the inner surface of liner had tensile deformation at monitoring point No. 2 and monitoring point No. 8. Compressive deformation occurred at monitoring points No. 5 and No. 11. The finite element model ignored the possible rotation of the liner under the condition of uniform external pressure. The numerical results show that the tensile defor- mation occurs at the invert of the inner surface of the liner. As shown in Figure 6b, the displacement of the liner was slightly elevated along the radial direction at monitoring point No. 2. With the continuous increase in hydrostatic pressure, the displacement of the liner at monitoring point No. 2 along the radial direction gradually increased. When the displacement of the buckling lobe reached point C, as shown in Figure 4, the deformation of the liner in the test and numerical result was shown in Figure 6c. After the pressure value reached the critical buckling pressure, the buckling lobe had deformation in the op- posite direction at monitoring point No. 2 and began to move to the port along the axial direction. Finally, the liner failed, while the buckling lobe continued to deform after the pressure value reached the critical buckling pressure, as shown in Figure 6d. The reason for this difference was that the 2D finite element model ignored the deformation in the Figure 5. Comparing the pressure–displacement curves of test and simulation divided into floating axial direction and only considered the deformation of the cross-section of the liner. Fig stage ure(AB), 5. Com buckling paring stage the pr (BC) essand ure– failur displ ea stage cemen (CD). t curves of test and simulation divided into float- ing stage (AB), buckling stage (BC) and failure stage (CD). The deformation of the liner at points A, B, C and D was selected according to Figure 5 to compare the test with the numerical results as shown in Figure 6. As shown in Figure 6a, the liner appears to float up under hydrostatic pressure, and the outer surface of the liner tube contacts the inner surface of the host pipe, while the liner undergoes slight ro- tation and the inner surface of liner had tensile deformation at monitoring point No. 2 and monitoring point No. 8. Compressive deformation occurred at monitoring points No. 5 and No. 11. The finite element model ignored the possible rotation of the liner under the condition of uniform external pressure. The numerical results show that the tensile defor- mation occurs at the invert of the inner surface of the liner. As shown in Figure 6b, the displacement of the liner was slightly elevated along the radial direction at monitoring point No. 2. With the continuous increase in hydrostatic pressure, the displacement of the liner at monitoring point No. 2 along the radial direction gradually increased. When the Figure 6. Cont. displacement of the buckling lobe reached point C, as shown in Figure 4, the deformation of the liner in the test and numerical result was shown in Figure 6c. After the pressure value reached the critical buckling pressure, the buckling lobe had deformation in the op- posite direction at monitoring point No. 2 and began to move to the port along the axial direction. Finally, the liner failed, while the buckling lobe continued to deform after the pressure value reached the critical buckling pressure, as shown in Figure 6d. The reason for this difference was that the 2D finite element model ignored the deformation in the axial direction and only considered the deformation of the cross-section of the liner. Appl. Sci. 2023, 13, x FOR PEER REVIEW 9 of 16 Appl. Sci. 2023, 13, 679 8 of 15 Figure 6. Comparing the deformation status of buckling liners at (a) point A; (b) point B; (c) point C; Figure 6. Comparing the deformation status of buckling liners at (a) point A; (b) point B; (c) point (d) point D marked in Figure 5. C; (d) point D marked in Figure 5. The deformation of the liner at points A, B, C and D was selected according to Figure 5 3.2. Sensitive Analysis of Parameters to compare the test with the numerical results as shown in Figure 6. As shown in Figure 6a, the liner appears to float up under hydrostatic pressure, and the outer surface of the liner 3.2.1. Effect of Dimension Ratio tube contacts the inner surface of the host pipe, while the liner undergoes slight rotation and The pressure versus displacement curves when the value of DR is changed are shown the inner surface of liner had tensile deformation at monitoring point No. 2 and monitoring in Figure 7. With the dimension ratio increase, the liner’s critical buckling pressure de- point No. 8. Compressive deformation occurred at monitoring points No. 5 and No. 11. creases and the displacement increases. Comparing the buckling equilibrium paths in Fig- The finite element model ignored the possible rotation of the liner under the condition of ure 7a,b, it can be found that the pressure increase rate of the liner is significantly greater uniform external pressure. The numerical results show that the tensile deformation occurs than the other groups when DR = 100, and the pressure reaches 15 kPa when the Appl. Sci. 2023, 13, 679 9 of 15 at the invert of the inner surface of the liner. As shown in Figure 6b, the displacement of the liner was slightly elevated along the radial direction at monitoring point No. 2. With the continuous increase in hydrostatic pressure, the displacement of the liner at monitoring Appl. Sci. 2023, 13, x FOR PEER REVIEW 10 of 16 point No. 2 along the radial direction gradually increased. When the displacement of the buckling lobe reached point C, as shown in Figure 4, the deformation of the liner in the test and numerical result was shown in Figure 6c. After the pressure value reached the critical buckling pressure, the buckling lobe had deformation in the opposite direction at displacement of the buckling lobe reaches 70 mm in Group C-1, while the pressure reaches monitoring point No. 2 and began to move to the port along the axial direction. Finally, the 4.76 kPa in Group C-2. The pressure does not increase significantly while the displacement liner failed, while the buckling lobe continued to deform after the pressure value reached of the buckling lobe increases in the pre-buckling equilibrium path when DR = 150, 200, the critical buckling pressure, as shown in Figure 6d. The reason for this difference was 250 and 300. The critical buckling pressure of Group D-1 compared to Group C-1 de- that the 2D finite element model ignored the deformation in the axial direction and only creased by 2.31 kPa, while the displacement increased 15.25 mm. Under the same condi- considered the deformation of the cross-section of the liner. tions, decreasing the value of a/R will decrease the critical buckling pressure and increase the deformation of the buckling lobe. 3.2. Sensitive Analysis of Parameters The comparison of the critical buckling pressure between the numerical result and 3.2.1. Effect of Dimension Ratio calculation results when the value of DR is changed under different values of a/R are The pressure versus displacement curves when the value of DR is changed are shown shown in Figure 8. The critical buckling pressure calculated by Equation (1) reduces by in Figure 7. With the dimension ratio increase, the liner ’s critical buckling pressure de- 7.5 kPa when DR changed from 100 to 150, while the critical buckling pressure of the sim- creases and the displacement increases. Comparing the buckling equilibrium paths in ulation result reduced by 10.25 kPa and 10.15 kPa when the a/R = 0.99 and a/R = 0.98. The Figure 7a,b, it can be found that the pressure increase rate of the liner is significantly increase in annular gaps causes an increase in the amount of reduction in critical buckling greater than the other groups when DR = 100, and the pressure reaches 15 kPa when the pressure. The critical buckling pressure in Group C-2 decreases by 68.18% compared to displacement of the buckling lobe reaches 70 mm in Group C-1, while the pressure reaches Group C-1 and Group D-2 decreases by 79.74% compared to Group D-1. The reduction of 4.76 kPa in Group C-2. The pressure does not increase significantly while the displacement the value of a/R enhances the decrease caused by the changing value of DR on the critical of the buckling lobe increases in the pre-buckling equilibrium path when DR = 150, 200, buckling pressure. When DR = 100, the critical buckling pressure of Group D-1 decreased 250 and 300. The critical buckling pressure of Group D-1 compared to Group C-1 decreased by 15.36% compared to Group C-1; when DR = 200, the critical buckling pressure of Group by 2.31 kPa, while the displacement increased 15.25 mm. Under the same conditions, D-3 decreased by 42.95% compared to Group C-3 and the effect caused by changing the decreasing the value of a/R will decrease the critical buckling pressure and increase the value of a/R on critical buckling pressure reduced as the value of DR increased. deformation of the buckling lobe. Figure 7. The pressure–displacement curves of FIPP liner with different DR under (a) a/R = 0.99; Figure 7. The pressure–displacement curves of FIPP liner with different DR under (a) a/R = 0.99; (b) (b) a/R = 0.98. a/R = 0.98. The comparison of the critical buckling pressure between the numerical result and calculation results when the value of DR is changed under different values of a/R are shown in Figure 8. The critical buckling pressure calculated by Equation (1) reduces by 7.5 kPa when DR changed from 100 to 150, while the critical buckling pressure of the simulation result reduced by 10.25 kPa and 10.15 kPa when the a/R = 0.99 and a/R = 0.98. The increase in annular gaps causes an increase in the amount of reduction in critical buckling pressure. The critical buckling pressure in Group C-2 decreases by 68.18% compared to Group C-1 and Group D-2 decreases by 79.74% compared to Group D-1. The reduction of the value of a/R enhances the decrease caused by the changing value of DR on the critical buckling Appl. Sci. 2023, 13, x FOR PEER REVIEW 10 of 16 displacement of the buckling lobe reaches 70 mm in Group C-1, while the pressure reaches 4.76 kPa in Group C-2. The pressure does not increase significantly while the displacement of the buckling lobe increases in the pre-buckling equilibrium path when DR = 150, 200, 250 and 300. The critical buckling pressure of Group D-1 compared to Group C-1 de- creased by 2.31 kPa, while the displacement increased 15.25 mm. Under the same condi- tions, decreasing the value of a/R will decrease the critical buckling pressure and increase the deformation of the buckling lobe. The comparison of the critical buckling pressure between the numerical result and calculation results when the value of DR is changed under different values of a/R are shown in Figure 8. The critical buckling pressure calculated by Equation (1) reduces by 7.5 kPa when DR changed from 100 to 150, while the critical buckling pressure of the sim- ulation result reduced by 10.25 kPa and 10.15 kPa when the a/R = 0.99 and a/R = 0.98. The increase in annular gaps causes an increase in the amount of reduction in critical buckling pressure. The critical buckling pressure in Group C-2 decreases by 68.18% compared to Group C-1 and Group D-2 decreases by 79.74% compared to Group D-1. The reduction of the value of a/R enhances the decrease caused by the changing value of DR on the critical buckling pressure. When DR = 100, the critical buckling pressure of Group D-1 decreased by 15.36% compared to Group C-1; when DR = 200, the critical buckling pressure of Group D-3 decreased by 42.95% compared to Group C-3 and the effect caused by changing the value of a/R on critical buckling pressure reduced as the value of DR increased. Appl. Sci. 2023, 13, 679 10 of 15 pressure. When DR = 100, the critical buckling pressure of Group D-1 decreased by 15.36% compared to Group C-1; when DR = 200, the critical buckling pressure of Group D-3 decreased by 42.95% compared to Group C-3 and the effect caused by changing the value Appl. Sci. 2023, 13, x FOR PEER REVIEW 11 of 16 Figure 7. The pressure–displacement curves of FIPP liner with different DR under (a) a/R = 0.99; (b) of a/R on critical buckling pressure reduced as the value of DR increased. a/R = 0.98. Figure 8. Comparing the influence of increasing DR for critical buckling pressure under different a/R. 3.2.2. Effect of Ovality The pressure versus displacement curves when the value of q is changed are shown in Figure 9. Similar to the effect caused by changing the value of DR on the buckling be- havior, as the value of q increases, the critical buckling pressure of the liner decreases and the displacement of buckling lobe increases. Unlike the effect caused by the changing value of DR, the curves of Group C-6 in Figure 9a and Group D-6 in Figure 9b have distinct peaks compared to the other curves and the increasing rate of pressure in the pre-buckling equilibrium path and the decreasing rate of pressure in the post-buckling equilibrium path are significantly greater than the other groups. As the value of q increases, the curve peak is gradually less obvious and the effect of changing the liner cross-sectional shape from circular to elliptical on the buckling equilibrium path is greater than the effect of Figure 8. Comparing the influence of increasing DR for critical buckling pressure under different a/R. continuing to increase the value of q for elliptical cross-sections. The decreasing of the 3.2.2. Effect of Ovality value of a/R has the greatest effect on the case where the liner cross-section is circular (q = The pressure versus displacement curves when the value of q is changed are shown 0); the critical buckling pressure decreases and the displacement of the buckling lobe in- in Figure 9. Similar to the effect caused by changing the value of DR on the buckling creases, the increasing rate of pressure in the pre-buckling equilibrium path and the de- behavior, as the value of q increases, the critical buckling pressure of the liner decreases creasing rate of pressure in the post- buckling equilibrium path decrease, but there are and the displacement of buckling lobe increases. Unlike the effect caused by the changing still obvious peaks in the curves. value of DR, the curves of Group C-6 in Figure 9a and Group D-6 in Figure 9b have distinct The comparison of the critical buckling pressure between the numerical result and peaks compared to the other curves and the increasing rate of pressure in the pre-buckling calculation results when the value of q is changed under a different value of a/R are shown equilibrium path and the decreasing rate of pressure in the post-buckling equilibrium path in Figure 10. The critical buckling pressure of the simulation results are significantly are significantly greater than the other groups. As the value of q increases, the curve peak is greater than the calculation results. The greatest discrepancies are 22.92 kPa and 7.68 kPa, gradually less obvious and the effect of changing the liner cross-sectional shape from circular which occur at q = 0 when the a/R = 0.99 and a/R = 0.98. The reduction of the value of a/R to elliptical on the buckling equilibrium path is greater than the effect of continuing to increase diminishes the reduction effect caused by the changed value of q on the critical buckling the value of q for elliptical cross-sections. The decreasing of the value of a/R has the greatest pressure; the decreased rate of critical buckling pressure gradually decreases as the value effect on the case where the liner cross-section is circular (q = 0); the critical buckling pressure of q is increasing. The critical buckling pressure of Group C-7 decreased 51.45% compared decreases and the displacement of the buckling lobe increases, the increasing rate of pressure to Group C-6, while Group D-7 decreased 33.72% compared to Group D-6; the decreasing in the pre-buckling equilibrium path and the decreasing rate of pressure in the post- buckling of the value of a/R will cause the reduction in the decreasing rate of critical buckling pres- equilibrium path decrease, but there are still obvious peaks in the curves. sure under the same increment of the value of q. Figure 9. The pressure–displacement curves of FIPP liner with different q under (a) a/R = 0.99; (b) Figure 9. The pressure–displacement curves of FIPP liner with different q under (a) a/R = 0.99; a/R = 0.98. (b) a/R = 0.98. Appl. Sci. 2023, 13, 679 11 of 15 The comparison of the critical buckling pressure between the numerical result and calculation results when the value of q is changed under a different value of a/R are shown in Figure 10. The critical buckling pressure of the simulation results are significantly greater than the calculation results. The greatest discrepancies are 22.92 kPa and 7.68 kPa, which occur at q = 0 when the a/R = 0.99 and a/R = 0.98. The reduction of the value of a/R diminishes the reduction effect caused by the changed value of q on the critical buckling pressure; the decreased rate of critical buckling pressure gradually decreases as the value of q is increasing. The critical buckling pressure of Group C-7 decreased 51.45% compared to Group C-6, while Group D-7 decreased 33.72% compared to Group D-6; the decreasing of Appl. Sci. 2023, 13, x FOR PEER REVI the EW value of a/R will cause the reduction in the decreasing rate of critical buckling pr12 essur of 16 e under the same increment of the value of q. Figure 10. Comparing the influence of increasing q for critical buckling pressure under different a/R. Figure 10. Comparing the influence of increasing q for critical buckling pressure under different a/R. 3.2.3. Effect of the Ratio of Liner Major Axis to Host Pipe Radius 3.2.3. Effect of the Ratio of Liner Major Axis to Host Pipe Radius The pressure versus displacement curves when the value of a/R is changed are shown The pressure versus displacement curves when the value of a/R is changed are shown in Figure 11. As the value of a/R increases, the critical buckling pressure of the liner in Figure 11. As the value of a/R increases, the critical buckling pressure of the liner de- decreases and the displacement of the buckling lobe increases. Comparing the curves in creases and the displacement of the buckling lobe increases. Comparing the curves in Fig- Figure 11a,b, it can be found that the critical buckling pressure of the circular liner (q = 0) is ure 11a,b, it can be found that the critical buckling pressure of the circular liner (q = 0) is significantly higher than the elliptical liner (q = 2.5). The critical buckling pressure in Group significantly higher than the elliptical liner (q = 2.5). The critical buckling pressure in C-6 decreased by 46.21% compared to Group B-2 and the displacement of the buckling lobe Group C-6 decreased by 46.21% compared to Group B-2 and the displacement of the buck- increased by 100.52%, while the critical buckling pressure in Group C-1 decreased by 10.64% ling lobe increased by 100.52%, while the critical buckling pressure in Group C-1 de- compared to Group B-1 and the displacement of the buckling lobe increased by 21.77%. creased by 10.64% compared to Group B-1 and the displacement of the buckling lobe in- Apparently, the increase rate and decrease rate caused by the decrease of a/R during the creased by 21.77%. Apparently, the increase rate and decrease rate caused by the decrease buckling equilibrium path is greater when the liner cross-section is circular (q = 0) than of a/R during the buckling equilibrium path is greater when the liner cross-section is cir- when the liner cross-section is elliptical (q = 5). The deformation of the buckling lobe when cular (q = 0) than when the liner cross-section is elliptical (q = 5). The deformation of the Group B-2 and Group C-6 reach the critical buckling pressure is shown in Figure 12 and buckling lobe when Group B-2 and Group C-6 reach the critical buckling pressure is the stress in the tensile part of the liner does not change significantly but the range of the shown in Figure 12 and the stress in the tensile part of the liner does not change signifi- detached portion increases when the value of a/R decreases, which leads to the decrease in cantly but the range of the detached portion increases when the value of a/R decreases, critical buckling pressure and increase in the displacement of the buckling lobe. which leads to the decrease in critical buckling pressure and increase in the displacement The comparison of the critical buckling pressure between numerical results and calcula- of the buckling lobe. tion results when the value of a/R is changed under different values of q is shown in Figure 13; The comparison of the critical buckling pressure between numerical results and cal- the effect of the ratio of liner major axis to host pipe radius (a/R) in Equation (1) is ignored. culation results when the value of a/R is changed under different values of q is shown in However, the critical buckling pressure of the simulation results reduce by 33.95 kPa and Figure 13; the effect of the ratio of liner major axis to host pipe radius (a/R) in Equation (1) 1.79 kPa under q = 0 and q = 5 when a/R changed from 0.999 to 0.99. The critical buckling is ignored. However, the critical buckling pressure of the simulation results reduce by pressure of the simulation results reduce by 3.77 kPa and 0.89 kPa under q = 0 and q = 5 when 33.95 kPa and 1.79 kPa under q = 0 and q = 5 when a/R changed from 0.999 to 0.99. The a/R changed from 0.99 to 0.98. The simulation results show that the critical buckling pressure critical buckling pressure of the simulation results reduce by 3.77 kPa and 0.89 kPa under of the liner decreases as the value of a/R is decreasing, while the reduction amount of the q = 0 and q = 5 when a/R changed from 0.99 to 0.98. The simulation results show that the critical buckling pressure of the liner decreases due to the value of q increasing. critical buckling pressure of the liner decreases as the value of a/R is decreasing, while the reduction amount of the critical buckling pressure of the liner decreases due to the value of q increasing. Appl. Sci. 2023, 13, x FOR PEER REVIEW 13 of 16 Appl. Sci. 2023, 13, x FOR PEER REVIEW 13 of 16 Appl. Sci. 2023, 13, x FOR PEER REVIEW 13 of 16 Appl. Sci. 2023, 13, 679 12 of 15 Figure 11. The pressure–displacement curves of FIPP liner with different a/R under (a) q = 0; (b) q = Figure 11. The pressure–displacement curves of FIPP liner with different a/R under (a) q = 0; (b) q = Figure 11. The pressure–displacement curves of FIPP liner with different a/R under (a) q = 0; (b) q = 5. Figure 11. The pressure–displacement curves of FIPP liner with different a/R under (a) q = 0; (b) q = Figure 12. The stress distribution of (a) Sample B-2; (b) Sample C-6 under the critical buckling pres- Figure 12. The stress distribution of (a) Sample B-2; (b) Sample C-6 under the critical buckling pres- Figure sure. 12. The stress distribution of (a) Sample B-2; (b) Sample C-6 under the critical buckling pressure. sure. Figure 12. The stress distribution of (a) Sample B-2; (b) Sample C-6 under the critical buckling pres- sure. Figure Figure 13. 13. Comparing Comparingthe the influence influence of of incr inceasing reasing a/ aR /R for for critical criticabuckling l buckling pr pr essur essur e under e unde dif r d fer iffent eren q.t Figure 13. Comparing the influence of increasing a/R for critical buckling pressure under different q. 3.3. Quantification of Enhancement Factor q. 3.3. Q The uantifica parameters tion of E innh Table ancem 1 ar ent e br Fact ought or into Equation (1) to find that the critical buckling Figure 13. Comparing the influence of increasing a/R for critical buckling pressure under different 3.3. Quantification of Enhancement Factor pressure value deviates significantly. The sensitivity of different parameters to the critical q. The parameters in Table 1 are brought into Equation (1) to find that the critical buck- buckling pressure of the liner was analyzed in Section 3.2. Figures 8, 10 and 13 show the The parameters in Table 1 are brought into Equation (1) to find that the critical buck- ling pressure value deviates significantly. The sensitivity of different parameters to the deviates 3.3. Quantifica of critical tion o buckling f Enhancem pressur ent Fact e values or between the calculation results and simulation ling pressure value deviates significantly. The sensitivity of different parameters to the results, which indicate that the enhancement factor K is significantly relevant with the The parameters in Table 1 are brought into Equation (1) to find that the critical buck- values of DR, a/R and q. DR, a/R and q are used as variables and a polynomial fit is ling pressure value deviates significantly. The sensitivity of different parameters to the Appl. Sci. 2023, 13, x FOR PEER REVIEW 14 of 16 critical buckling pressure of the liner was analyzed in Section 3.2. Figures 8, 10 and 13 Appl. Sci. 2023, 13, 679 13 of 15 show the deviates of critical buckling pressure values between the calculation results and simulation results, which indicate that the enhancement factor K is significantly relevant with the values of DR, a/R and q. DR, a/R and q are used as variables and a polynomial fit performed based on the numerical results to obtain a prediction model for enhancement is performed based on the numerical results to obtain a prediction model for enhancement factor K as in Equation (2). For simplification of Equation (2), a/R is replaced by D, as factor K as in Equation (2). For simplification of Equation (2), a/R is replaced by Δ, as shown in Equation (3). shown in Equation (3). 1/2 3/2 1/2 1/2 2 2 2 1/2 1/2 KΔ = 37.94DR Δ − 170.9Δ q − 0.000069DR + 11561Δ + 0.0964q + 0.1068q − 36.522DR − 44831Δ + 1/2 3/2 1/2 1/2 2 2 2 K = 37.94DR D 170.9D q 0.000069DR + 11561D + 0.0964q + 0.1068q (2) 1/2 (2) 162.57q + 33287 1/2 1/2 1/2 36.522DR 44831D + 162.57q + 33287 Δ = a/R (3) D = a/R (3) The The comparison comparison between between th the e re results sults ca calculated lculated by byEEquation quation (1(1) ) an and d ththe e rer sul esult t cacal- lcu- culated after using the enhancement factor K is shown in Figure 14 and after revising lated after using the enhancement factor KΔ is shown in Figure 14 and after revising the the enhancement factor K, the calculation results are more accurate and fit better with the enhancement factor K, the calculation results are more accurate and fit better with the numerical results and the average difference rate between predicted results and numerical numerical results and the average difference rate between predicted results and numerical results was reduced from 23.43% to 5.52%. results was reduced from 23.43% to 5.52%. Figure 14. Comparing the critical buckling pressure between the simulation result and result calcu- Figure 14. Comparing the critical buckling pressure between the simulation result and result calcu- lated by the prediction model. lated by the prediction model. 4. Conclusions 4. Conclusions In this paper, the buckling behavior of a loosely fitted FIPP liner under external In this paper, the buckling behavior of a loosely fitted FIPP liner under external pres- pressure was investigated experimentally; the sensitivity of the critical buckling pressure of sure was investigated experimentally; the sensitivity of the critical buckling pressure of the loosely fitted FIPP liner to different parameters was investigated by 2D finite element the loosely fitted FIPP liner to different parameters was investigated by 2D finite element model and a prediction method of the critical buckling pressure was proposed. Based on model and a prediction method of the critical buckling pressure was proposed. Based on the results, the following conclusions can be obtained. the results, the following conclusions can be obtained. (1) A loosely fitted FIPP liner will float under the action of buoyancy. While rotation (1) A loosely fitted FIPP liner will float under the action of buoyancy. While rotation may occur, the buckling lobe of the initial elliptical liner is likely to appear in the direction may occur, the buckling lobe of the initial elliptical liner is likely to appear in the direction of the short axis of the liner cross-section when the pressure reaches the critical buckling of the short axis of the liner cross-section when the pressure reaches the critical buckling pressure, the buckling lobe will move in the axial direction, the buckling part has not had pressure, the buckling lobe will move in the axial direction, the buckling part has not had plastic deformation and the liner will rebound after the movement of the buckling lobe. plastic deformation and the liner will rebound after the movement of the buckling lobe. (2) Through test and simulation, results indicate that the value of critical buckling (2) Through test and simulation, results indicate that the value of critical buckling pressure calculated by the equation from ASTM 1216 is significantly lower than the results pressure calculated by the equation from ASTM 1216 is significantly lower than the results under working conditions. The difference in the critical buckling pressure between the under working conditions. The difference in the critical buckling pressure between the calculated result and actual result will increase when the installation condition of the liner calculated result and actual result will increase when the installation condition of the liner is closed to the tightly fitted status. is closed to the tightly fitted status. (3) The decrease in the value of a/R causes a change in the contact conditions between (3) The decrease in the value of a/R causes a change in the contact conditions between the liner and the host pipe under the loosely fitted condition, the increase in the gap between the liner and the host pipe under the loosely fitted condition, the increase in the gap be- the liner and the host pipe will lead to the movement of the contact point between the tween the liner and the host pipe will lead to the movement of the contact point between liner and the original pipe and the range of the buckling portion increases under external pressure conditions, resulting in a reduction in the critical buckling pressure. (4) The effects of changes in the dimensional parameters of the liner on the critical buckling pressure are not independent of each other. With the determined value of DR, a Appl. Sci. 2023, 13, 679 14 of 15 decrease in the value of a/R will diminish the effect of DR and q on the critical buckling pressure, while an increase in the value of q will diminish the effect of a/R on the critical buckling pressure with the determined value of DR. Considering the interaction between the dimensional parameters, a polynomial fit is used to propose the equation for the enhancement factor K and the average difference rate was reduced from 23.43% to 5.52%. The present study only considers 2D plane conditions; any possible deformation in the axial direction is ignored, while the possible rotation of the liner under the water pressure is also ignored and the results are idealized. Future research can focus on the buckling behavior of the FIPP liner which is suspended in the host pipe under water pressure and not in contact with the inner surface of the host pipe. Author Contributions: Conceptualization, Z.Z., X.Y. and S.T.A.; methodology, Z.Z. and Y.Z.; soft- ware, Z.Z. and Y.Z.; validation, X.Y. and W.X.; formal analysis, Z.Z.; investigation, X.Y. and S.T.A.; resources, X.Y. and S.T.A.; data curation, W.X. and Z.Z.; writing—original draft preparation, Z.Z.; writing—review and editing, S.T.A., Y.Z. and X.Y.; visualization, Z.Z.; supervision, X.Y.; project administration, X.Y.; funding acquisition, X.Y. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by the National Natural Science Foundation of China under grant (No. 42002284) and Zhejiang Provincial Natural Science Foundation of China under grant (No. LGG20E080005). Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: Data are contained within the article. Conflicts of Interest: The authors declare no conflict of interest. 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Buckling Behavior of Loosely Fitted Formed-In-Place Pipe Liner in Circular Host Pipe under External Pressure

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Abstract

applied sciences Article Buckling Behavior of Loosely Fitted Formed-In-Place Pipe Liner in Circular Host Pipe under External Pressure 1 1 2 1 3 , Zheng Zeng , Xuefeng Yan , Weigang Xiang , Yahong Zhao and Samuel T. Ariaratnam * Faculty of Engineering, China University of Geosciences—Wuhan, Wuhan 430074, China Wuhan Metro Bridge and Tunnel Management Co., Ltd., Wuhan 430077, China School of Sustainable Engineering and the Built Environment, Arizona State University, Tempe, AZ 85259, USA * Correspondence: ariaratnam@asu.edu Abstract: The formed-in-place pipe (FIPP) is a trenchless technology used for pipeline rehabilitation. It is a folded PVC pipe that expands through thermoforming to fit tightly inside the host pipe. However, the deficiencies during the construction of FIPP liners such as insufficient inflation, pipe misalignment and initial deformation will lead to elliptical deformation of the FIPP liner, which affects the load-bearing performance of the liner and makes it susceptible to buckling failure. In this paper, the buckling behavior of loosely fitted FIPP liners under uniform external pressure was investigated by the external pressure resistance test and finite element model. The pre- and post- buckling equilibrium paths verified the finite element model. The results indicated that the value of the dimension ratio will significantly reduce the critical buckling pressure. With the increasing value of liner major axis ratio to host pipe, the reduction effect on the critical buckling pressure caused by the increase in the ovality will diminish. Different values of liner major axis ratio to host pipe and ovality changed the range of the detached portion, which affected the critical buckling pressure. The parametric studies modified the design model from ASTM F1216, which was established to predict the critical buckling pressure of a loosely fitted FIPP liner and reduced the average difference rate from 23.43% to 5.52%. Keywords: FIPP; buckling; ovality; FEM Citation: Zeng, Z.; Yan, X.; Xiang, W.; Zhao, Y.; Ariaratnam, S.T. Buckling Behavior of Loosely Fitted Formed- In-Place Pipe Liner in Circular Host Pipe under External Pressure. Appl. 1. Introduction Sci. 2023, 13, 679. https://doi.org/ In recent years, as the demand for energy transportation has increased, the construction 10.3390/app13020679 of pipeline systems has increased to keep cities functioning properly. The pipeline system Academic Editor: José contains multiple types of pipe [1–4] and water supply pipelines and drainage pipelines are António Correia 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 Received: 7 November 2022 under the effect of soil and groundwater. The pipe defects will eventually cause the bearing Revised: 29 December 2022 capacity reduction of the pipe and the leakage of pipe contents. Hence, the working Accepted: 31 December 2022 conditions of the water supply and drainage pipelines need to be assessed and the defect Published: 4 January 2023 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 Copyright: © 2023 by the authors. technologies are gradually replacing traditional excavation repair technologies as the main Licensee MDPI, Basel, Switzerland. technology for urban pipeline rehabilitation due to the ability to repair the pipe with minor This article is an open access article excavation faces. Formed-in-place pipe (FIPP) technology repairs pipelines by installing a distributed under the terms and thermoplastic polyvinyl chloride (PVC) pipe into host pipe, which expands to fit tightly conditions of the Creative Commons with host pipe by thermoforming [8]. The FIPP liner has high tensile and bending strength Attribution (CC BY) license (https:// and is suitable for water supply pipelines and drainage pipelines of different types of creativecommons.org/licenses/by/ sections, diameters and materials [9]. However, the FIPP liners will fail under high water 4.0/). Appl. Sci. 2023, 13, 679. https://doi.org/10.3390/app13020679 https://www.mdpi.com/journal/applsci Appl. Sci. 2023, 13, 679 2 of 15 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. Timoshenko 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, Glock [11] proposed a model for calculating the critical buckling pressure of the liner constrained by a rigid host pipe. Aggarwal and Cooper [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, Lo and Zhang [14] established an analytical model to predict the critical buckling pressure of the liner with an initial annular gap. EI-Sway [15] established an approximate solution for enhancement factors for loosely fitted liners. Zhao [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. El-Sway [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. Dong [18] proposed a model for calculating the critical buckling pressure of the liner in the presence of multiple partial gaps. Wang [19] found that the annular gap affects the buckling behavior of the liner. Li [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. El-Sawy [21] concluded that an increase in the partial buckling range of the liner leads to a decrease in the critical buckling pressure. Treitz et al. [22] designed a PVC pipe test system to study the effect of buoyancy on the buckling behavior of PVC liners by applying hydrostatic pressure to simulate grouting pressure. Results showed that the presence of buoyancy reduced the critical flexural strength of the liner. In the existing studies, only the initial gap formed by the partial detachment of the liner from the host pipe is considered, ignoring the effect of the annular gap on the buckling behavior of the elliptical liner under loosely fitted conditions and the change of the contact conditions between the liner and host pipe under the action of water pressure, resulting in a specification deviation of the critical buckling pressure calculated by the prediction model in ASTM F1216, which reduces the reliability of the liner designs. This study designed an experimental system to investigate the buckling behavior of the loosely fitted FIPP liner under external pressure. The deformation mode of the buckling lobes can be directly captured by the camera and the critical buckling pressure could be measured by pressure transducer when the liner had a buckling failure. The geometric parameters of the liner were indicated as the dimension ratio and ratio of liner major axis to host pipe radius and ovality, and the 2D finite element models were set to analyze the effect of different geometric parameters of the liners. This study aimed at obtaining the different buckling behaviors of the FIPP liner by changing different geometric parameters and analyzing the sensitivity of different parameters on critical buckling pressure. The results of the study were used to improve the accuracy of the critical buckling pressure prediction model in ASTM F1216 for the design of the FIPP liner by modifying the enhancement factor. 2. Materials and Methods 2.1. Test Design The experimental system is shown in Figure 1, the geometry dimension of the FIPP liner and the host pipe are shown in Table 1. During the installation of the host pipe, the outlet valve of the steel pipe was set at the bottom, and the inlet valves were set at the top of the host pipe. Considering the gravity of the water body itself and the buoyancy Appl. Sci. 2023, 13, x FOR PEER REVIEW 3 of 16 2. Materials and Methods 2.1. Test Design The experimental system is shown in Figure 1, the geometry dimension of the FIPP Appl. Sci. 2023, 13, 679 3 of 15 liner and the host pipe are shown in Table 1. During the installation of the host pipe, the outlet valve of the steel pipe was set at the bottom, and the inlet valves were set at the top of the host pipe. Considering the gravity of the water body itself and the buoyancy effect effect on the liner pipe, the pressure sensor interface is arranged at the spring line of the on the liner pipe, the pressure sensor interface is arranged at the spring line of the steel steel pipe and the displacement and deformation of the middle section of the FIPP liner are pipe and the displacement and deformation of the middle section of the FIPP liner are monitored using strain gauges and a PIV system. monitored using strain gauges and a PIV system. Figure 1. Experimental system design and geometric dimensions of equipment. Figure 1. Experimental system design and geometric dimensions of equipment. Table 1. Geometrical dimensions of FIPP liner and host pipe. Table 1. Geometrical dimensions of FIPP liner and host pipe. Internal Radius/mm Wall Internal Radius/mm Length/mm Length/mm Wall Thickness/mm Thickness/mm MaMajor jor Axis Axis Minor A Minor xis Axis FIPP 2000 309.85 287.15 7.95 FIPP 2000 309.85 287.15 7.95 The host pipe 2000 318 10 The host pipe 2000 318 10 Firstly, the FIPP liner was installed into the host pipe after grinding the ports and the Firstly, the FIPP liner was installed into the host pipe after grinding the ports and the axis of the liner was kept coincident with the host pipe. Considering the gravity of the axis of the liner was kept coincident with the host pipe. Considering the gravity of the liner and the buoyancy, the pressure transducer interface was set at the spring line of the liner and the buoyancy, the pressure transducer interface was set at the spring line of the steel pipe and strain gauges and PIV monitored the displacement and deformation of the steel pipe and strain gauges and PIV monitored the displacement and deformation of the middle section of the FIPP liner. The latex membrane was used to seal the port so that a middle section of the FIPP liner. The latex membrane was used to seal the port so that a closed cavity was formed between the liner and the host pipe. Secondly, water was slowly closed cavity was formed between the liner and the host pipe. Secondly, water was slowly injected into the closed chamber through the test pressure pump until the air in the annu- injected into the closed chamber through the test pressure pump until the air in the annular lar space gap was completely discharged and the test was started by maintaining a con- space gap was completely discharged and the test was started by maintaining a continuous tinuous pressurization rate of 12 kPa/min. The hydrostatic pressure acts directly on the pressurization rate of 12 kPa/min. The hydrostatic pressure acts directly on the outer wall outer wall of the liner, causing the liner tube to buckle and deform and when the liner of the liner, causing the liner tube to buckle and deform and when the liner buckles and buckles and deforms to the limit, the pressurization is stopped for 90 s and then unloaded deforms to the limit, the pressurization is stopped for 90 s and then unloaded and the test and the test is completed. is completed. 2.2. Finite Element Model Set-Up To quantify the buckling equilibrium paths of the thin-walled structures, the stress distribution and deformation extent of the structure must be calculated. Jiao [23] used the Differential Quadrature Method (DQM) to determine the pre-buckling in-plane stress distribution of thin rectangular FG-CNTRC plate. Kabir and Aghdam [24] developed an accurate Bézier-based multi-step method and implemented it to find the nonlinear vibration Appl. Sci. 2023, 13, x FOR PEER REVIEW 4 of 16 2.2. Finite Element Model Set-Up To quantify the buckling equilibrium paths of the thin-walled structures, the stress distribution and deformation extent of the structure must be calculated. Jiao [23] used the Differential Quadrature Method (DQM) to determine the pre-buckling in-plane stress dis- tribution of thin rectangular FG-CNTRC plate. Kabir and Aghdam [24] developed an ac- Appl. Sci. 2023, 13, 679 4 of 15 curate Bézier-based multi-step method and implemented it to find the nonlinear vibration and post-buckling configurations of Euler–Bernoulli composite beams reinforced with graphene nano-platelets (GnP). Both DQM and the Bézier method divide the structure and post-buckling configurations of Euler–Bernoulli composite beams reinforced with into a single element to calculate the stress and deformation during the buckling prosses, graphene nano-platelets (GnP). Both DQM and the Bézier method divide the structure into but those require extensive calculations computation to improve the precision of the cal- a single element to calculate the stress and deformation during the buckling prosses, but culation and the calculation results must be processed to be visualized. Thus, using the those require extensive calculations computation to improve the precision of the calculation finite element analysis will be a good solution to quantify the buckling equilibrium paths and the calculation results must be processed to be visualized. Thus, using the finite element of the FIPP liners. analysis will be a good solution to quantify the buckling equilibrium paths of the FIPP liners. ABAQUS is used for finite element simulation to study the effects of the dimension ABAQUS is used for finite element simulation to study the effects of the dimension ratio (DR), ratio of liner major axis to host pipe radius (a/R) and ovality (q) on the buckling ratio (DR), ratio of liner major axis to host pipe radius (a/R) and ovality (q) on the buckling behavior. The longitudinal deformation and the boundary effect caused by the end ports behavior. The longitudinal deformation and the boundary effect caused by the end ports of of the liner are neglected and this paper only considers the buckling cross-section of the the liner are neglected and this paper only considers the buckling cross-section of the liner. liner. Using 2D instead of 3D analysis effectively reduces simulation time while maintain- Using 2D instead of 3D analysis effectively reduces simulation time while maintaining ing calculation accuracy. The model uses an eight-node reduced-integration plane strain calculation accuracy. The model uses an eight-node reduced-integration plane strain element (CPE8R) and the liner elements are divided into three layers totaling 1200 ele- element (CPE8R) and the liner elements are divided into three layers totaling 1200 elements. ments. A total of 200 elements for steel host pipe are shown in Figure 2. Twenty-three A total of 200 elements for steel host pipe are shown in Figure 2. Twenty-three groups of groups of finite element model are shown in Table 2, Group A is set for the validation test, finite element model are shown in Table 2, Group A is set for the validation test, Group B Group B to Group E are distinguished by the value of a/R to analyze the buckling behavior to Group E are distinguished by the value of a/R to analyze the buckling behavior under under the different values of a/R. the different values of a/R. Figure 2. The element mesh of FIPP and steel pipe model. Figure 2. The element mesh of FIPP and steel pipe model. Table 2. Geometric parameters of the numerical model. Table 2. Geometric parameters of the numerical model. N No. o. a/aR/R q q DR DR A 0.974 3.8 75 A 0.974 3.8 75 B-1 0.999 5 100 B-1 0.999 5 100 B B-2 -2 0.999 0.999 0 0 100 100 C-1 0.99 5 100 C-1 0.99 5 100 C-2 0.99 5 150 C-2 0.99 5 150 C-3 0.99 5 200 C-3 0.99 5 200 C-4 0.99 5 250 C-4 0.99 5 250 C-5 0.99 5 300 C-5 0.99 5 300 C-6 0.99 0 100 C-6 0.99 0 100 C-7 0.99 2.5 100 C-7 0.99 2.5 100 C-8 0.99 7.5 100 C-8 0.99 7.5 100 C-9 0.99 10 100 C-9 0.99 10 100 D-1 0.98 5 100 D D-2 -1 0.98 0.98 5 5 150 100 D-3 0.98 5 200 D-4 0.98 5 250 D-5 0.98 5 300 D-6 0.98 0 100 D-7 0.98 2.5 100 D-8 0.98 7.5 100 D-9 0.98 10 100 E-1 0.97 5 100 E-2 0.97 0 100 Appl. Sci. 2023, 13, 679 5 of 15 The host pipe material adopts the elastic constitutive model, the mechanical parame- ters of FIPP liner and steel are shown in Table 3. The outer surface of the steel pipe is fully restrained from displacement and rotation in all directions. The uniform external pressure is applied to the outer surface of the liner. The property of the contact surface between the liner and host pipe is defined as “surface to surface” in ABAQUS and allows the liner and host pipe to separate from each other when the normal pressure is zero or negative. The friction coefficient is 0.4 for the tangential contact, considering the friction between the liner and the pipe [9]. The displacement of the steel pipe is not considered during the liner buckling process and the boundary condition of the outer surface of the steel pipe is set to limit its displacement in the x and y axes, as shown in Figure 3a. A static general analysis step is used to apply the displacement of the liner along the y-axis and control the displacement of the liner on the x-axis and the rotation along the z-axis to zero, which simulates the process of the liner floating and contacting with the inner wall of the steel pipe under the action of buoyancy. The details about the boundary and load distribution are shown in Appendix A. The Riks step in ABAQUS is used to track the pre-buckling and post-buckling of the liner, which needs to assume the existence of the initial defect which will cause the buckling of the liner. In this model, a concentrated force is applied to the bottom surface of the liner to simulate the initial defect occurring at the bottom part of the pipe under the action of buoyancy, while the hydrostatic pressure acts on the outer surface of the liner, as shown in Figure 3b. The non-linear geometry (NIgeom) switch needs to be turned on in the ABAQUS simulation step settings to simulate the large deflection deformation during the buckling process of the liner. Table 3. Mechanical properties of FIPP liner and host pipe. Materials Modulus of Elasticity (MPa) Poisson’s Ratio Bending Strength Appl. Sci. 2023, 13, x FOR PEER REVIEW 6 of 16 Steel pipe 190,000 0.3 / FIPP 2089.9 0.3 62 Figure 3. The condition of (a) boundary and (b) load distribution. Figure 3. The condition of (a) boundary and (b) load distribution. 2.3. Prediction Method of Critical Buckling Pressure 2.3. Prediction Method of Critical Buckling Pressure According to the designs of the semi-structural rehabilitation to flexible liner in ASTM 1216, A the cco critical rding buckling to the dpr esi essur gns e ois f th calculated e semi-st by ruc Equation tural reh (1), abili while tatiothe n to enhan flexible cement liner in factor ASTM value 1216is , th 7e and criti the cal safety bucklin factor g pres is sure 2. is calculated by Equation (1), while the enhance- ment factor value is 7 and the safety factor is 2. 2 3 P = 2KE C/[(1 m )(DR 1) N] (1) cr 2 3 Pcr = 2KELC/[(1 − μ )(DR − 1) N] (1) From Equation (1), it can be found that the influence of the annular gap is not con- sidered when calculating the critical buckling pressure. The enhancement factor K is de- signed for reducing the deviates of the critical buckling pressure value between the theo- retical condition and working condition. The enhancement factor value recommended by ASTM 1216 is unable to guarantee the accuracy of the critical buckling pressure value when the annular gap occurs. Therefore, the value of enhancement factor K will be quan- tified by analyzing the difference in the critical buckling pressure of finite element models which set the different geometric parameters according to Table 1. 3. Results and Discussion 3.1. Finite Element Model Validation The FIPP liner material in the finite element model adopts the elastic–plastic consti- tutive model tested according to ISO 178. The relationship between the true strain and stress in the plastic phase of the FIPP material is shown in Figure 4, which will be used to define the plastic constitutive model of the finite element model. The elastic limit of the FIPP material is 41.8 MPa. However, bending failure in the specimen did not occur during the test and the ultimate bending strength was considered 62.62 MPa in the configuration of the finite element model. Appl. Sci. 2023, 13, 679 6 of 15 From Equation (1), it can be found that the influence of the annular gap is not consid- ered when calculating the critical buckling pressure. The enhancement factor K is designed for reducing the deviates of the critical buckling pressure value between the theoretical condition and working condition. The enhancement factor value recommended by ASTM 1216 is unable to guarantee the accuracy of the critical buckling pressure value when the annular gap occurs. Therefore, the value of enhancement factor K will be quantified by analyzing the difference in the critical buckling pressure of finite element models which set the different geometric parameters according to Table 1. 3. Results and Discussion 3.1. Finite Element Model Validation The FIPP liner material in the finite element model adopts the elastic–plastic consti- tutive model tested according to ISO 178. The relationship between the true strain and stress in the plastic phase of the FIPP material is shown in Figure 4, which will be used to define the plastic constitutive model of the finite element model. The elastic limit of the FIPP material is 41.8 MPa. However, bending failure in the specimen did not occur during Appl. Sci. 2023, 13, x FOR PEER REVIEW 7 of 16 the test and the ultimate bending strength was considered 62.62 MPa in the configuration of the finite element model. Figure 4. True stress versus true strain curve of FIPP material. Figure 4. True stress versus true strain curve of FIPP material. The comparison of the pressure–displacement curve of the FIPP liner is shown in The comparison of the pressure–displacement curve of the FIPP liner is shown in Figure 5. According to the curve, the buckling behavior of the liner can be divided into Figure 5. According to the curve, the buckling behavior of the liner can be divided into three stages. At first (stage AB), the deformation displacement versus the pressure is linear three stages. At first (stage AB), the deformation displacement versus the pressure is linear during the increase of hydrostatic pressure. At this stage, the liner did not form a buckling during the increase of hydrostatic pressure. At this stage, the liner did not form a buckling lobe. Then (stage BC), as the pressure increases, the deformed part of liner formed a bulge, lobe. Then (stage BC), as the pressure increases, the deformed part of liner formed a bulge, which can be called the buckling lobe. When the pressure value approaches the critical which can be called the buckling lobe. When the pressure value approaches the critical buckling pressure, the deformation increases and rate of the buckling lobe is significantly incr buckli eased, ng while pressure, the pr th essur e def e incr orma ease tio rate n inrcr emains eases unchanged. and rate ofThe the buckling bucklinlobe g lobe continues is significantly deformation with no enhancement of the ability to withstand the external pressure, but the increased, while the pressure increase rate remains unchanged. The buckling lobe contin- liner still has the bearing capacity until the pressure reaches the critical buckling pressure ues deformation with no enhancement of the ability to withstand the external pressure, which will cause the buckling failure at the buckling lobe. The liner has a buckling failure at but the liner still has the bearing capacity until the pressure reaches the critical buckling the end (stage CD). During stage AB, the test result and simulation result both indicate that pressure which will cause the buckling failure at the buckling lobe. The liner has a buck- the linear relationship between pressure and displacement is ended when the displacement ling failure at the end (stage CD). During stage AB, the test result and simulation result reaches 12.53 mm, but the pressure of the test is 19.4% greater than simulation. At stage BC, both indicate that the linear relationship between pressure and displacement is ended the liner material may have had errors in the manufacturing process and the constraints when the displacement reaches 12.53 mm, but the pressure of the test is 19.4% greater than of the flange on the liner in the test limited the displacement of the liner at both ends of the sim liner ulatio , pr noducing . At stage end BC, effects the li af nfecting er mater the iabuckling l may hacr ve oss-section had errors inin the thtest. e ma When nufactur the ing pro- displacement of the buckling lobe was equal, the pressure of the test result was greater than cess and the constraints of the flange on the liner in the test limited the displacement of the liner at both ends of the liner, producing end effects affecting the buckling cross-sec- tion in the test. When the displacement of the buckling lobe was equal, the pressure of the test result was greater than the numerical result during the buckling process. Hence, the liner stiffness in the test was significantly greater than the simulation. At point C, the liner reached the critical buckling pressure of 38.08 kPa when the displacement of the buckling lobe was 52.77 mm in the test and the liner reached the critical buckling pressure of 35.94 kPa when the displacement of the buckling lobe reached 76.39 mm in the numerical re- sults. The critical buckling pressure in the test was 5.6% greater than the simulation and the displacement of buckling lobe in the simulation was 30.9% greater than in the test. In stage CD, when the displacement of the buckling lobe increased to 74.4 mm in the test, the buckling lobe began to move axially to the port along the liner and the liner eventually failed (Figure 6), while the buckling lobe continued to deform in the numerical results. This paper mainly studied the influence of changing geometric parameters on critical buckling pressure. Although the pressure of the simulation result was lower than the test result during the buckling process, the accuracy of the critical buckling pressure was still reliable. Appl. Sci. 2023, 13, x FOR PEER REVIEW 8 of 16 Appl. Sci. 2023, 13, 679 7 of 15 the numerical result during the buckling process. Hence, the liner stiffness in the test was significantly greater than the simulation. At point C, the liner reached the critical buckling pressure of 38.08 kPa when the displacement of the buckling lobe was 52.77 mm in the test and the liner reached the critical buckling pressure of 35.94 kPa when the displacement of the buckling lobe reached 76.39 mm in the numerical results. The critical buckling pressure in the test was 5.6% greater than the simulation and the displacement of buckling lobe in the simulation was 30.9% greater than in the test. In stage CD, when the displacement of the buckling lobe increased to 74.4 mm in the test, the buckling lobe began to move axially Figure 5. Comparing the pressure–displacement curves of test and simulation divided into float- to the port along the liner and the liner eventually failed (Figure 6), while the buckling lobe ing stage (AB), buckling stage (BC) and failure stage (CD). continued to deform in the numerical results. This paper mainly studied the influence of Appl. Sci. 2023, 13, x FOR PEER REVIEW 8 of 16 changing geometric parameters on critical buckling pressure. Although the pressure of the The deformation of the liner at points A, B, C and D was selected according to Figure simulation result was lower than the test result during the buckling process, the accuracy 5 to compare the test with the numerical results as shown in Figure 6. As shown in Figure of the critical buckling pressure was still reliable. 6a, the liner appears to float up under hydrostatic pressure, and the outer surface of the liner tube contacts the inner surface of the host pipe, while the liner undergoes slight ro- tation and the inner surface of liner had tensile deformation at monitoring point No. 2 and monitoring point No. 8. Compressive deformation occurred at monitoring points No. 5 and No. 11. The finite element model ignored the possible rotation of the liner under the condition of uniform external pressure. The numerical results show that the tensile defor- mation occurs at the invert of the inner surface of the liner. As shown in Figure 6b, the displacement of the liner was slightly elevated along the radial direction at monitoring point No. 2. With the continuous increase in hydrostatic pressure, the displacement of the liner at monitoring point No. 2 along the radial direction gradually increased. When the displacement of the buckling lobe reached point C, as shown in Figure 4, the deformation of the liner in the test and numerical result was shown in Figure 6c. After the pressure value reached the critical buckling pressure, the buckling lobe had deformation in the op- posite direction at monitoring point No. 2 and began to move to the port along the axial direction. Finally, the liner failed, while the buckling lobe continued to deform after the pressure value reached the critical buckling pressure, as shown in Figure 6d. The reason for this difference was that the 2D finite element model ignored the deformation in the Figure 5. Comparing the pressure–displacement curves of test and simulation divided into floating axial direction and only considered the deformation of the cross-section of the liner. Fig stage ure(AB), 5. Com buckling paring stage the pr (BC) essand ure– failur displ ea stage cemen (CD). t curves of test and simulation divided into float- ing stage (AB), buckling stage (BC) and failure stage (CD). The deformation of the liner at points A, B, C and D was selected according to Figure 5 to compare the test with the numerical results as shown in Figure 6. As shown in Figure 6a, the liner appears to float up under hydrostatic pressure, and the outer surface of the liner tube contacts the inner surface of the host pipe, while the liner undergoes slight ro- tation and the inner surface of liner had tensile deformation at monitoring point No. 2 and monitoring point No. 8. Compressive deformation occurred at monitoring points No. 5 and No. 11. The finite element model ignored the possible rotation of the liner under the condition of uniform external pressure. The numerical results show that the tensile defor- mation occurs at the invert of the inner surface of the liner. As shown in Figure 6b, the displacement of the liner was slightly elevated along the radial direction at monitoring point No. 2. With the continuous increase in hydrostatic pressure, the displacement of the liner at monitoring point No. 2 along the radial direction gradually increased. When the Figure 6. Cont. displacement of the buckling lobe reached point C, as shown in Figure 4, the deformation of the liner in the test and numerical result was shown in Figure 6c. After the pressure value reached the critical buckling pressure, the buckling lobe had deformation in the op- posite direction at monitoring point No. 2 and began to move to the port along the axial direction. Finally, the liner failed, while the buckling lobe continued to deform after the pressure value reached the critical buckling pressure, as shown in Figure 6d. The reason for this difference was that the 2D finite element model ignored the deformation in the axial direction and only considered the deformation of the cross-section of the liner. Appl. Sci. 2023, 13, x FOR PEER REVIEW 9 of 16 Appl. Sci. 2023, 13, 679 8 of 15 Figure 6. Comparing the deformation status of buckling liners at (a) point A; (b) point B; (c) point C; Figure 6. Comparing the deformation status of buckling liners at (a) point A; (b) point B; (c) point (d) point D marked in Figure 5. C; (d) point D marked in Figure 5. The deformation of the liner at points A, B, C and D was selected according to Figure 5 3.2. Sensitive Analysis of Parameters to compare the test with the numerical results as shown in Figure 6. As shown in Figure 6a, the liner appears to float up under hydrostatic pressure, and the outer surface of the liner 3.2.1. Effect of Dimension Ratio tube contacts the inner surface of the host pipe, while the liner undergoes slight rotation and The pressure versus displacement curves when the value of DR is changed are shown the inner surface of liner had tensile deformation at monitoring point No. 2 and monitoring in Figure 7. With the dimension ratio increase, the liner’s critical buckling pressure de- point No. 8. Compressive deformation occurred at monitoring points No. 5 and No. 11. creases and the displacement increases. Comparing the buckling equilibrium paths in Fig- The finite element model ignored the possible rotation of the liner under the condition of ure 7a,b, it can be found that the pressure increase rate of the liner is significantly greater uniform external pressure. The numerical results show that the tensile deformation occurs than the other groups when DR = 100, and the pressure reaches 15 kPa when the Appl. Sci. 2023, 13, 679 9 of 15 at the invert of the inner surface of the liner. As shown in Figure 6b, the displacement of the liner was slightly elevated along the radial direction at monitoring point No. 2. With the continuous increase in hydrostatic pressure, the displacement of the liner at monitoring Appl. Sci. 2023, 13, x FOR PEER REVIEW 10 of 16 point No. 2 along the radial direction gradually increased. When the displacement of the buckling lobe reached point C, as shown in Figure 4, the deformation of the liner in the test and numerical result was shown in Figure 6c. After the pressure value reached the critical buckling pressure, the buckling lobe had deformation in the opposite direction at displacement of the buckling lobe reaches 70 mm in Group C-1, while the pressure reaches monitoring point No. 2 and began to move to the port along the axial direction. Finally, the 4.76 kPa in Group C-2. The pressure does not increase significantly while the displacement liner failed, while the buckling lobe continued to deform after the pressure value reached of the buckling lobe increases in the pre-buckling equilibrium path when DR = 150, 200, the critical buckling pressure, as shown in Figure 6d. The reason for this difference was 250 and 300. The critical buckling pressure of Group D-1 compared to Group C-1 de- that the 2D finite element model ignored the deformation in the axial direction and only creased by 2.31 kPa, while the displacement increased 15.25 mm. Under the same condi- considered the deformation of the cross-section of the liner. tions, decreasing the value of a/R will decrease the critical buckling pressure and increase the deformation of the buckling lobe. 3.2. Sensitive Analysis of Parameters The comparison of the critical buckling pressure between the numerical result and 3.2.1. Effect of Dimension Ratio calculation results when the value of DR is changed under different values of a/R are The pressure versus displacement curves when the value of DR is changed are shown shown in Figure 8. The critical buckling pressure calculated by Equation (1) reduces by in Figure 7. With the dimension ratio increase, the liner ’s critical buckling pressure de- 7.5 kPa when DR changed from 100 to 150, while the critical buckling pressure of the sim- creases and the displacement increases. Comparing the buckling equilibrium paths in ulation result reduced by 10.25 kPa and 10.15 kPa when the a/R = 0.99 and a/R = 0.98. The Figure 7a,b, it can be found that the pressure increase rate of the liner is significantly increase in annular gaps causes an increase in the amount of reduction in critical buckling greater than the other groups when DR = 100, and the pressure reaches 15 kPa when the pressure. The critical buckling pressure in Group C-2 decreases by 68.18% compared to displacement of the buckling lobe reaches 70 mm in Group C-1, while the pressure reaches Group C-1 and Group D-2 decreases by 79.74% compared to Group D-1. The reduction of 4.76 kPa in Group C-2. The pressure does not increase significantly while the displacement the value of a/R enhances the decrease caused by the changing value of DR on the critical of the buckling lobe increases in the pre-buckling equilibrium path when DR = 150, 200, buckling pressure. When DR = 100, the critical buckling pressure of Group D-1 decreased 250 and 300. The critical buckling pressure of Group D-1 compared to Group C-1 decreased by 15.36% compared to Group C-1; when DR = 200, the critical buckling pressure of Group by 2.31 kPa, while the displacement increased 15.25 mm. Under the same conditions, D-3 decreased by 42.95% compared to Group C-3 and the effect caused by changing the decreasing the value of a/R will decrease the critical buckling pressure and increase the value of a/R on critical buckling pressure reduced as the value of DR increased. deformation of the buckling lobe. Figure 7. The pressure–displacement curves of FIPP liner with different DR under (a) a/R = 0.99; Figure 7. The pressure–displacement curves of FIPP liner with different DR under (a) a/R = 0.99; (b) (b) a/R = 0.98. a/R = 0.98. The comparison of the critical buckling pressure between the numerical result and calculation results when the value of DR is changed under different values of a/R are shown in Figure 8. The critical buckling pressure calculated by Equation (1) reduces by 7.5 kPa when DR changed from 100 to 150, while the critical buckling pressure of the simulation result reduced by 10.25 kPa and 10.15 kPa when the a/R = 0.99 and a/R = 0.98. The increase in annular gaps causes an increase in the amount of reduction in critical buckling pressure. The critical buckling pressure in Group C-2 decreases by 68.18% compared to Group C-1 and Group D-2 decreases by 79.74% compared to Group D-1. The reduction of the value of a/R enhances the decrease caused by the changing value of DR on the critical buckling Appl. Sci. 2023, 13, x FOR PEER REVIEW 10 of 16 displacement of the buckling lobe reaches 70 mm in Group C-1, while the pressure reaches 4.76 kPa in Group C-2. The pressure does not increase significantly while the displacement of the buckling lobe increases in the pre-buckling equilibrium path when DR = 150, 200, 250 and 300. The critical buckling pressure of Group D-1 compared to Group C-1 de- creased by 2.31 kPa, while the displacement increased 15.25 mm. Under the same condi- tions, decreasing the value of a/R will decrease the critical buckling pressure and increase the deformation of the buckling lobe. The comparison of the critical buckling pressure between the numerical result and calculation results when the value of DR is changed under different values of a/R are shown in Figure 8. The critical buckling pressure calculated by Equation (1) reduces by 7.5 kPa when DR changed from 100 to 150, while the critical buckling pressure of the sim- ulation result reduced by 10.25 kPa and 10.15 kPa when the a/R = 0.99 and a/R = 0.98. The increase in annular gaps causes an increase in the amount of reduction in critical buckling pressure. The critical buckling pressure in Group C-2 decreases by 68.18% compared to Group C-1 and Group D-2 decreases by 79.74% compared to Group D-1. The reduction of the value of a/R enhances the decrease caused by the changing value of DR on the critical buckling pressure. When DR = 100, the critical buckling pressure of Group D-1 decreased by 15.36% compared to Group C-1; when DR = 200, the critical buckling pressure of Group D-3 decreased by 42.95% compared to Group C-3 and the effect caused by changing the value of a/R on critical buckling pressure reduced as the value of DR increased. Appl. Sci. 2023, 13, 679 10 of 15 pressure. When DR = 100, the critical buckling pressure of Group D-1 decreased by 15.36% compared to Group C-1; when DR = 200, the critical buckling pressure of Group D-3 decreased by 42.95% compared to Group C-3 and the effect caused by changing the value Appl. Sci. 2023, 13, x FOR PEER REVIEW 11 of 16 Figure 7. The pressure–displacement curves of FIPP liner with different DR under (a) a/R = 0.99; (b) of a/R on critical buckling pressure reduced as the value of DR increased. a/R = 0.98. Figure 8. Comparing the influence of increasing DR for critical buckling pressure under different a/R. 3.2.2. Effect of Ovality The pressure versus displacement curves when the value of q is changed are shown in Figure 9. Similar to the effect caused by changing the value of DR on the buckling be- havior, as the value of q increases, the critical buckling pressure of the liner decreases and the displacement of buckling lobe increases. Unlike the effect caused by the changing value of DR, the curves of Group C-6 in Figure 9a and Group D-6 in Figure 9b have distinct peaks compared to the other curves and the increasing rate of pressure in the pre-buckling equilibrium path and the decreasing rate of pressure in the post-buckling equilibrium path are significantly greater than the other groups. As the value of q increases, the curve peak is gradually less obvious and the effect of changing the liner cross-sectional shape from circular to elliptical on the buckling equilibrium path is greater than the effect of Figure 8. Comparing the influence of increasing DR for critical buckling pressure under different a/R. continuing to increase the value of q for elliptical cross-sections. The decreasing of the 3.2.2. Effect of Ovality value of a/R has the greatest effect on the case where the liner cross-section is circular (q = The pressure versus displacement curves when the value of q is changed are shown 0); the critical buckling pressure decreases and the displacement of the buckling lobe in- in Figure 9. Similar to the effect caused by changing the value of DR on the buckling creases, the increasing rate of pressure in the pre-buckling equilibrium path and the de- behavior, as the value of q increases, the critical buckling pressure of the liner decreases creasing rate of pressure in the post- buckling equilibrium path decrease, but there are and the displacement of buckling lobe increases. Unlike the effect caused by the changing still obvious peaks in the curves. value of DR, the curves of Group C-6 in Figure 9a and Group D-6 in Figure 9b have distinct The comparison of the critical buckling pressure between the numerical result and peaks compared to the other curves and the increasing rate of pressure in the pre-buckling calculation results when the value of q is changed under a different value of a/R are shown equilibrium path and the decreasing rate of pressure in the post-buckling equilibrium path in Figure 10. The critical buckling pressure of the simulation results are significantly are significantly greater than the other groups. As the value of q increases, the curve peak is greater than the calculation results. The greatest discrepancies are 22.92 kPa and 7.68 kPa, gradually less obvious and the effect of changing the liner cross-sectional shape from circular which occur at q = 0 when the a/R = 0.99 and a/R = 0.98. The reduction of the value of a/R to elliptical on the buckling equilibrium path is greater than the effect of continuing to increase diminishes the reduction effect caused by the changed value of q on the critical buckling the value of q for elliptical cross-sections. The decreasing of the value of a/R has the greatest pressure; the decreased rate of critical buckling pressure gradually decreases as the value effect on the case where the liner cross-section is circular (q = 0); the critical buckling pressure of q is increasing. The critical buckling pressure of Group C-7 decreased 51.45% compared decreases and the displacement of the buckling lobe increases, the increasing rate of pressure to Group C-6, while Group D-7 decreased 33.72% compared to Group D-6; the decreasing in the pre-buckling equilibrium path and the decreasing rate of pressure in the post- buckling of the value of a/R will cause the reduction in the decreasing rate of critical buckling pres- equilibrium path decrease, but there are still obvious peaks in the curves. sure under the same increment of the value of q. Figure 9. The pressure–displacement curves of FIPP liner with different q under (a) a/R = 0.99; (b) Figure 9. The pressure–displacement curves of FIPP liner with different q under (a) a/R = 0.99; a/R = 0.98. (b) a/R = 0.98. Appl. Sci. 2023, 13, 679 11 of 15 The comparison of the critical buckling pressure between the numerical result and calculation results when the value of q is changed under a different value of a/R are shown in Figure 10. The critical buckling pressure of the simulation results are significantly greater than the calculation results. The greatest discrepancies are 22.92 kPa and 7.68 kPa, which occur at q = 0 when the a/R = 0.99 and a/R = 0.98. The reduction of the value of a/R diminishes the reduction effect caused by the changed value of q on the critical buckling pressure; the decreased rate of critical buckling pressure gradually decreases as the value of q is increasing. The critical buckling pressure of Group C-7 decreased 51.45% compared to Group C-6, while Group D-7 decreased 33.72% compared to Group D-6; the decreasing of Appl. Sci. 2023, 13, x FOR PEER REVI the EW value of a/R will cause the reduction in the decreasing rate of critical buckling pr12 essur of 16 e under the same increment of the value of q. Figure 10. Comparing the influence of increasing q for critical buckling pressure under different a/R. Figure 10. Comparing the influence of increasing q for critical buckling pressure under different a/R. 3.2.3. Effect of the Ratio of Liner Major Axis to Host Pipe Radius 3.2.3. Effect of the Ratio of Liner Major Axis to Host Pipe Radius The pressure versus displacement curves when the value of a/R is changed are shown The pressure versus displacement curves when the value of a/R is changed are shown in Figure 11. As the value of a/R increases, the critical buckling pressure of the liner in Figure 11. As the value of a/R increases, the critical buckling pressure of the liner de- decreases and the displacement of the buckling lobe increases. Comparing the curves in creases and the displacement of the buckling lobe increases. Comparing the curves in Fig- Figure 11a,b, it can be found that the critical buckling pressure of the circular liner (q = 0) is ure 11a,b, it can be found that the critical buckling pressure of the circular liner (q = 0) is significantly higher than the elliptical liner (q = 2.5). The critical buckling pressure in Group significantly higher than the elliptical liner (q = 2.5). The critical buckling pressure in C-6 decreased by 46.21% compared to Group B-2 and the displacement of the buckling lobe Group C-6 decreased by 46.21% compared to Group B-2 and the displacement of the buck- increased by 100.52%, while the critical buckling pressure in Group C-1 decreased by 10.64% ling lobe increased by 100.52%, while the critical buckling pressure in Group C-1 de- compared to Group B-1 and the displacement of the buckling lobe increased by 21.77%. creased by 10.64% compared to Group B-1 and the displacement of the buckling lobe in- Apparently, the increase rate and decrease rate caused by the decrease of a/R during the creased by 21.77%. Apparently, the increase rate and decrease rate caused by the decrease buckling equilibrium path is greater when the liner cross-section is circular (q = 0) than of a/R during the buckling equilibrium path is greater when the liner cross-section is cir- when the liner cross-section is elliptical (q = 5). The deformation of the buckling lobe when cular (q = 0) than when the liner cross-section is elliptical (q = 5). The deformation of the Group B-2 and Group C-6 reach the critical buckling pressure is shown in Figure 12 and buckling lobe when Group B-2 and Group C-6 reach the critical buckling pressure is the stress in the tensile part of the liner does not change significantly but the range of the shown in Figure 12 and the stress in the tensile part of the liner does not change signifi- detached portion increases when the value of a/R decreases, which leads to the decrease in cantly but the range of the detached portion increases when the value of a/R decreases, critical buckling pressure and increase in the displacement of the buckling lobe. which leads to the decrease in critical buckling pressure and increase in the displacement The comparison of the critical buckling pressure between numerical results and calcula- of the buckling lobe. tion results when the value of a/R is changed under different values of q is shown in Figure 13; The comparison of the critical buckling pressure between numerical results and cal- the effect of the ratio of liner major axis to host pipe radius (a/R) in Equation (1) is ignored. culation results when the value of a/R is changed under different values of q is shown in However, the critical buckling pressure of the simulation results reduce by 33.95 kPa and Figure 13; the effect of the ratio of liner major axis to host pipe radius (a/R) in Equation (1) 1.79 kPa under q = 0 and q = 5 when a/R changed from 0.999 to 0.99. The critical buckling is ignored. However, the critical buckling pressure of the simulation results reduce by pressure of the simulation results reduce by 3.77 kPa and 0.89 kPa under q = 0 and q = 5 when 33.95 kPa and 1.79 kPa under q = 0 and q = 5 when a/R changed from 0.999 to 0.99. The a/R changed from 0.99 to 0.98. The simulation results show that the critical buckling pressure critical buckling pressure of the simulation results reduce by 3.77 kPa and 0.89 kPa under of the liner decreases as the value of a/R is decreasing, while the reduction amount of the q = 0 and q = 5 when a/R changed from 0.99 to 0.98. The simulation results show that the critical buckling pressure of the liner decreases due to the value of q increasing. critical buckling pressure of the liner decreases as the value of a/R is decreasing, while the reduction amount of the critical buckling pressure of the liner decreases due to the value of q increasing. Appl. Sci. 2023, 13, x FOR PEER REVIEW 13 of 16 Appl. Sci. 2023, 13, x FOR PEER REVIEW 13 of 16 Appl. Sci. 2023, 13, x FOR PEER REVIEW 13 of 16 Appl. Sci. 2023, 13, 679 12 of 15 Figure 11. The pressure–displacement curves of FIPP liner with different a/R under (a) q = 0; (b) q = Figure 11. The pressure–displacement curves of FIPP liner with different a/R under (a) q = 0; (b) q = Figure 11. The pressure–displacement curves of FIPP liner with different a/R under (a) q = 0; (b) q = 5. Figure 11. The pressure–displacement curves of FIPP liner with different a/R under (a) q = 0; (b) q = Figure 12. The stress distribution of (a) Sample B-2; (b) Sample C-6 under the critical buckling pres- Figure 12. The stress distribution of (a) Sample B-2; (b) Sample C-6 under the critical buckling pres- Figure sure. 12. The stress distribution of (a) Sample B-2; (b) Sample C-6 under the critical buckling pressure. sure. Figure 12. The stress distribution of (a) Sample B-2; (b) Sample C-6 under the critical buckling pres- sure. Figure Figure 13. 13. Comparing Comparingthe the influence influence of of incr inceasing reasing a/ aR /R for for critical criticabuckling l buckling pr pr essur essur e under e unde dif r d fer iffent eren q.t Figure 13. Comparing the influence of increasing a/R for critical buckling pressure under different q. 3.3. Quantification of Enhancement Factor q. 3.3. Q The uantifica parameters tion of E innh Table ancem 1 ar ent e br Fact ought or into Equation (1) to find that the critical buckling Figure 13. Comparing the influence of increasing a/R for critical buckling pressure under different 3.3. Quantification of Enhancement Factor pressure value deviates significantly. The sensitivity of different parameters to the critical q. The parameters in Table 1 are brought into Equation (1) to find that the critical buck- buckling pressure of the liner was analyzed in Section 3.2. Figures 8, 10 and 13 show the The parameters in Table 1 are brought into Equation (1) to find that the critical buck- ling pressure value deviates significantly. The sensitivity of different parameters to the deviates 3.3. Quantifica of critical tion o buckling f Enhancem pressur ent Fact e values or between the calculation results and simulation ling pressure value deviates significantly. The sensitivity of different parameters to the results, which indicate that the enhancement factor K is significantly relevant with the The parameters in Table 1 are brought into Equation (1) to find that the critical buck- values of DR, a/R and q. DR, a/R and q are used as variables and a polynomial fit is ling pressure value deviates significantly. The sensitivity of different parameters to the Appl. Sci. 2023, 13, x FOR PEER REVIEW 14 of 16 critical buckling pressure of the liner was analyzed in Section 3.2. Figures 8, 10 and 13 Appl. Sci. 2023, 13, 679 13 of 15 show the deviates of critical buckling pressure values between the calculation results and simulation results, which indicate that the enhancement factor K is significantly relevant with the values of DR, a/R and q. DR, a/R and q are used as variables and a polynomial fit performed based on the numerical results to obtain a prediction model for enhancement is performed based on the numerical results to obtain a prediction model for enhancement factor K as in Equation (2). For simplification of Equation (2), a/R is replaced by D, as factor K as in Equation (2). For simplification of Equation (2), a/R is replaced by Δ, as shown in Equation (3). shown in Equation (3). 1/2 3/2 1/2 1/2 2 2 2 1/2 1/2 KΔ = 37.94DR Δ − 170.9Δ q − 0.000069DR + 11561Δ + 0.0964q + 0.1068q − 36.522DR − 44831Δ + 1/2 3/2 1/2 1/2 2 2 2 K = 37.94DR D 170.9D q 0.000069DR + 11561D + 0.0964q + 0.1068q (2) 1/2 (2) 162.57q + 33287 1/2 1/2 1/2 36.522DR 44831D + 162.57q + 33287 Δ = a/R (3) D = a/R (3) The The comparison comparison between between th the e re results sults ca calculated lculated by byEEquation quation (1(1) ) an and d ththe e rer sul esult t cacal- lcu- culated after using the enhancement factor K is shown in Figure 14 and after revising lated after using the enhancement factor KΔ is shown in Figure 14 and after revising the the enhancement factor K, the calculation results are more accurate and fit better with the enhancement factor K, the calculation results are more accurate and fit better with the numerical results and the average difference rate between predicted results and numerical numerical results and the average difference rate between predicted results and numerical results was reduced from 23.43% to 5.52%. results was reduced from 23.43% to 5.52%. Figure 14. Comparing the critical buckling pressure between the simulation result and result calcu- Figure 14. Comparing the critical buckling pressure between the simulation result and result calcu- lated by the prediction model. lated by the prediction model. 4. Conclusions 4. Conclusions In this paper, the buckling behavior of a loosely fitted FIPP liner under external In this paper, the buckling behavior of a loosely fitted FIPP liner under external pres- pressure was investigated experimentally; the sensitivity of the critical buckling pressure of sure was investigated experimentally; the sensitivity of the critical buckling pressure of the loosely fitted FIPP liner to different parameters was investigated by 2D finite element the loosely fitted FIPP liner to different parameters was investigated by 2D finite element model and a prediction method of the critical buckling pressure was proposed. Based on model and a prediction method of the critical buckling pressure was proposed. Based on the results, the following conclusions can be obtained. the results, the following conclusions can be obtained. (1) A loosely fitted FIPP liner will float under the action of buoyancy. While rotation (1) A loosely fitted FIPP liner will float under the action of buoyancy. While rotation may occur, the buckling lobe of the initial elliptical liner is likely to appear in the direction may occur, the buckling lobe of the initial elliptical liner is likely to appear in the direction of the short axis of the liner cross-section when the pressure reaches the critical buckling of the short axis of the liner cross-section when the pressure reaches the critical buckling pressure, the buckling lobe will move in the axial direction, the buckling part has not had pressure, the buckling lobe will move in the axial direction, the buckling part has not had plastic deformation and the liner will rebound after the movement of the buckling lobe. plastic deformation and the liner will rebound after the movement of the buckling lobe. (2) Through test and simulation, results indicate that the value of critical buckling (2) Through test and simulation, results indicate that the value of critical buckling pressure calculated by the equation from ASTM 1216 is significantly lower than the results pressure calculated by the equation from ASTM 1216 is significantly lower than the results under working conditions. The difference in the critical buckling pressure between the under working conditions. The difference in the critical buckling pressure between the calculated result and actual result will increase when the installation condition of the liner calculated result and actual result will increase when the installation condition of the liner is closed to the tightly fitted status. is closed to the tightly fitted status. (3) The decrease in the value of a/R causes a change in the contact conditions between (3) The decrease in the value of a/R causes a change in the contact conditions between the liner and the host pipe under the loosely fitted condition, the increase in the gap between the liner and the host pipe under the loosely fitted condition, the increase in the gap be- the liner and the host pipe will lead to the movement of the contact point between the tween the liner and the host pipe will lead to the movement of the contact point between liner and the original pipe and the range of the buckling portion increases under external pressure conditions, resulting in a reduction in the critical buckling pressure. (4) The effects of changes in the dimensional parameters of the liner on the critical buckling pressure are not independent of each other. With the determined value of DR, a Appl. Sci. 2023, 13, 679 14 of 15 decrease in the value of a/R will diminish the effect of DR and q on the critical buckling pressure, while an increase in the value of q will diminish the effect of a/R on the critical buckling pressure with the determined value of DR. Considering the interaction between the dimensional parameters, a polynomial fit is used to propose the equation for the enhancement factor K and the average difference rate was reduced from 23.43% to 5.52%. The present study only considers 2D plane conditions; any possible deformation in the axial direction is ignored, while the possible rotation of the liner under the water pressure is also ignored and the results are idealized. Future research can focus on the buckling behavior of the FIPP liner which is suspended in the host pipe under water pressure and not in contact with the inner surface of the host pipe. Author Contributions: Conceptualization, Z.Z., X.Y. and S.T.A.; methodology, Z.Z. and Y.Z.; soft- ware, Z.Z. and Y.Z.; validation, X.Y. and W.X.; formal analysis, Z.Z.; investigation, X.Y. and S.T.A.; resources, X.Y. and S.T.A.; data curation, W.X. and Z.Z.; writing—original draft preparation, Z.Z.; writing—review and editing, S.T.A., Y.Z. and X.Y.; visualization, Z.Z.; supervision, X.Y.; project administration, X.Y.; funding acquisition, X.Y. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by the National Natural Science Foundation of China under grant (No. 42002284) and Zhejiang Provincial Natural Science Foundation of China under grant (No. LGG20E080005). Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: Data are contained within the article. Conflicts of Interest: The authors declare no conflict of interest. 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Journal

Applied Sciences – Multidisciplinary Digital Publishing Institute

Published: Jan 4, 2023

Keywords: FIPP; buckling; ovality; FEM

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