LIU Run, HAO Xintong, LI Chengfeng, LI Qingxin, YU Zheng, and ZHAO Dang

Theoretical and Numerical Studies on the Coupling Deformation of Global Lateral Buckling and Walking of Submarine Pipeline

LIU Run, HAO Xintong, LI Chengfeng*, LI Qingxin, YU Zheng, and ZHAO Dang

State Key Laboratory of Hydraulic Engineering Intelligent Construction and Operation, Tianjin University, Tianjin 300072, China

Buckling initiation devices/techniques, including sleepers, distributed buoyancy, snake lay, and residual curvature method (RCM), have recently been widely applied in engineering. These initiated buckles may induce a long pipeline to transform into mul- tiple short pipeline segments, which promote the occurrence of pipeline walking. Thus, a pipeline, which is designed to buckle later- ally, may laterally and axially displace over time when subjected to repeated heating and cooling cycles. This study aims to reveal the coupling mechanism of pipeline walking and global lateral buckling. First, an analytic solution is proposed to estimate the walking of pipeline segments between two adjacent buckles. Then, the sensitivity of this method to heating and cooling cycles is analyzed. Re- sults show the applicability of the proposed walking analytical solution of buckling pipelines. Subsequently, an influence analysis of walking on global buckling, including the capacity of buckling initiation, buckling amplitude, buckling mode, and failure assessment ofthe buckling pipeline, is performed. The results reveal that the effect of walking on the buckling axial force is negligible. However, pipeline walking will aggravate the asymmetry of the pipeline buckling and the failure parameters of the pipeline during the post-buckling.

submarine pipeline; global lateral buckling; pipeline walking; coupling deformation; analytic solution

1 Introduction

Global buckling always causes stress-strain accumula- tion in the pipeline with the increase in temperature, which may jeopardize the structural integrity and cause pipeline failure (DNV, 2012). Therefore, global buckling is a key aspect that is frequently considered in subsea pipeline de- signs (Wang and Tang, 2020). The options for pipeline buckling control and mitigation can be divided into two categories. The first category prohibits the occurrence of pipeline global buckling with design schemes, including reducing the design temperature and pressure on pipelines and increasing the flexural stiffness of the pipeline and soil resistance to the pipeline. The second category initi- ates a controlled buckling response in preset locations of pipelines by design schemes, including introducing artifi- cial imperfections and reducing local soil resistance in the pipeline during installation (Bruton, 2003).

Thedesignschemeof‘prohibitbuckling’isalwaysadopt- ed for pipelines in the shallow sea. By contrast, ‘initiatecontrolled buckling’ was recommended for pipelines in thedeep sea by SAFEBUCK projects (Bruton and Carr, 2005) because a remarkably sophisticated and cost-effective solu- tion is to work withinstead of working againstthe pipeline by controlling the formation of lateral buckles along thepipeline. Buckling initiation techniques, including sleep- ers (Wang and Jukes, 2008; Sinclair, 2009), distrib- uted buoyancy (Thompson, 2009; Li, 2016; Wang, 2018), snake lay (Matheson, 2004; Peek, 2004; Luo, 2013), and residual curvature meth- od(RCM) (Teigen and Ibrahim, 2015, 2017), have been widely applied in engineering and studied by scholars in the last 20 years. The configurations of these typical buck- ling initiation devices and methods are shown in Fig.1.

As shown in Fig.1, the spacing of trigger devices/con- figurations (snake lay bend and permanent imperfections by reel-lay) is typically 2–5km. The pipeline segment between two adjacent buckles possibly becomes a short pipeline with the successful initiation of global buckling in most trigger devices/configurations (Carr, 2008). Pipeline walking may occur for these short pipeline seg- ments when a pipeline is subjected to tension from the steel catenary riser (SCR), global seabed slope along the pipeline length, or thermal gradient along the pipeline (Bru- ton, 2003, 2010; Rong, 2009). Thus, a pipeline, which is designed to buckle laterally, may laterally and axi- ally displace over time when subjected to repeated heat- ing and cooling cycles.

Studies on the coupling interaction of pipeline buckling and walking are limited. In 2003, Bruton first conducted a study on the coupling response of global buckling and walking by pipeline with free ends measuring 12in. in diameter and 5km in length. The study claimed that pipe- line walking would cause asymmetrical development of its global buckling configuration. Depending on changes in local seabed slopes, a combination of pipeline walking drivers and global buckling was investigated by Cumming(2009). Through response analysis along the pipeline route, Cumming divided the effects of axial movement on pipeline buckling during heating and cooling cycles into the following three categories: experiencing decreasing axial feed-in (givers), maintaining the buckle shape with minimal changes in axial feed-in (stayers), and accumulat- ing the axial feed-in (collectors). Based on the nonlinear finite element simulation results, the lateral buckling and pressure circulation in the pipeline will promote the axial movement of pipelines, as described by Zhou(2010).A strong walking tendency due to the buckle has been identified by Solano(2014) for long pipelines, whichhave a low tendency to walk. The effects of the number of buckling along a short pipeline on walking rate were eval- uated as a controlled walking strategy by Seyfipour(2019, 2021). The study shows that setting more buckling than traditional along a short pipeline with a relatively small VAS can alleviate pipeline walking.

Research on the coupling deformation of pipeline buckl- ing and walking is not only limited in number till now but also in guidance capability to pipeline safety design in en- gineering. According to the classical research and engi- neering application (Carneiro, 2009), the design prin- ciple of the buckling mitigation method, which aims to in-itiate controlled buckling, can be summarized as two points. First, global buckling at most preset pipeline loca- tions, wherein the critical axial force required for buckling initiation must be sufficiently low, should be successfully triggered. Second, the maximum stress and strain of the pipeline in post-buckling should be controlled in the allow- able range; that is, the limit state of the pipeline buckling should conform to the design specification. Owing to the two principles, the influence analysis of walking on pipe- line buckling should focus on the effect of walking on the capacity of buckling initiation (critical axial force required for buckling) and the influence on the integrity and failure probability of the pipeline post-buckling. However, the above-mentioned existing studies mostly focused on the influence of pipeline walking on the buckling deforma- tion mode. Investigations on the effect of walking on the capacity of buckling initiation and the integrity and fail- ure probability of the pipeline post-buckling are unavail- able.

The monitoring data of submarine pipelines in some pro- jects show that pipeline walking per kilometer can reach several centimeters in one opening and closing cycle, and the total cumulative displacement is several meters for a typical design life (T?rnes, 2000). The existing stud- ies on pipeline walking are mostly focused on short pipe- lines (Liu, 2020; Hong, 2021). However, mul- tiple global buckles along a long pipeline will facilitate di- vision into multiple short pipeline segments. The walking of each short pipeline during heating and cooling cycles will cause cumulative displacement along the long pipe- line.The accumulation of axial displacement may induce the failure of submarine pipelines in the form of excessive structural stress and loss of prestress in SCRs, which se- riously threatens the safety of the end connection of pipe- lines (Perinet and Simon, 2011). Therefore, the key to the influence analysis of global buckling on pipeline walking is to establish a calculation method for the walking of a short pipeline segment. However, existing research on the coupling deformation of pipeline buckling and walking only indicates that global buckling may induce pipeline walking, with no research aimed at estimating the walk- ing of pipeline segments between two adjacent buckles.

Submarine pipelines to explore and exploit oil and gas resources in the deep sea are increasingly required to sat- isfy the ‘deep-shallow-land’ development model, which in- evitably lays considerable long-distance pipelines on the seabed with a definite slope. Thus, a calculation method of the axial movement of a buckling pipeline is proposed witha pipeline on a sloping seabed. Subsequently, the couplingmechanism of lateral buckling and pipeline walking is studied with the influence law of axial movement on global lateral buckling.

2 FE Model to Simulate the Coupling Deformation of Pipeline Buckling and Walking

2.1 FE Model

Liu(2014)proposed a 3D explicit method based on ABAQUS to simulate the global buckling of pipelines.This method was applied to pipeline buckling analysis in many concepts by Liu, including the calculation of the perfect VAS length (Liu, 2016), lateral buckling based on the model of nonlinear pipe-soil interaction (Liu and Wang, 2018), and critical length calculation of buck- ling pipelines (Liu and Li, 2018). These applications fur- ther increase the applicability of the 3D explicit method in the simulation of global buckling in pipelines. Li and Liu (2020) recently improved the method and applied it to the multiple buckling simulation of a pipeline with more than one initial imperfection, which detailed the method to introduce double imperfections into the FEA model. The coupling deformation simulation of pipeline buckling and walking is also related to submarine pipelines involv- ing multiple imperfections. Thus, the improved 3D explic- it method is used in this study.

The improved 3D explicit method can be briefly de- scribed as follows. The beam element with element-type PIPE 31 is used to model the pipeline. The constitutive model of the pipeline is the linear elastic model, where the material properties include the pipeline density, outside diameter and wall thickness of the pipeline, elastic modu- lus, Poisson ratio, and thermal expansion coefficient. The boundary conditions of the pipeline are free on both ends. The average element size of the pipeline is 1m. The ini- tial imperfections of pipelines are introduced into the pipe- line model by the ‘import sketch’ step in ABAQUS. The form of a single initial imperfection among these multiple pipeline imperfections is calculated by the ‘buckle’ step in ABAQUS. The C3D8R solid element is used to model the seabed. The constitutive model of the soil is the Mohr-Coulomb model, where the material properties of the soil include the effective density, elastic modulus, Poisson ratio, cohesion, and internal friction angle of soil. The element size of the soil is 1m. The boundary conditions around the seabed are laterally and axially constrained, and the om- nidirectional freedom is constrained at the bottom of the seabed. Hard contact between pipeline and soil in the ver- tical direction is applied to simulate the interaction betweenpipeline and seabed, whereas the lateral interaction is simu- lated by the penalty function. This condition implies that the lateral resistance is governed by the friction coefficient and submerged weight of the pipeline.

A total of 11 steps are considered to simulate the cou- pling deformation of pipeline buckling and walking. The first step is to apply the gravity field to ensure a hard con- tact between the pipeline and the soil. The second step is to increase the temperature of the pipeline to simulate a high-temperature load until the pipeline exhibits global buckling and walking. The third step is to cool the pipe from the design temperature to the ambient temperature. The second step combined with the third step is a heating and cooling cycle. The second and third steps are repeatedin steps 4–11 to allow four additional heating and cooling cycles in the pipeline. Moreover, the step must be appro- priately defined during the dynamic explicit calculation to limit the ratio of the kinematic energy to the internal en- ergy, which results in the required quasistatic calculation by this study. Considering the calculation cost and accu- racy, the period of each step is 300s in this study, and its rationality can be verified in Fig.2.

2.2 Validation of the FE Model

The 3D explicit method has been applied and verified in the analysis of independent buckling of pipelines and joint buckling of pipelines with multiple imperfections but has not been adopted for the pipeline walking prob- lem. Therefore, the walking simulation using the 3D ex- plicit method is validated by the walking induced by the seabed slope of a pipeline in Bohai Bay (the design pa- rameters of the pipeline and soil are shown in Table 1). The design temperature of the pipeline is 95℃ with an ambient temperature of 10℃.

Table 1 Design parameters (Li and Liu, 2020)

According to Carr(2008), pipeline walking per heating and cooling cycles caused by the seabed slope can be calculated by Eq. (1).

whereXis the walking per heating and cooling cycles at the midpoint of the pipeline,is the elastic modulus,is the cross-sectional area of the pipe,is the friction coef- ficient between pipe and seabed,is the weight of the pipe per unit length,is the pipeline length,is the sea- bed slope, andis the fully constrained effective axial force of the pipeline.

whereis the thermal expansion coefficient of the mate- rial,is the operating temperature, and0is the ambient temperature.

With the 3D explicit method, pipeline walking during five heating (from ambient temperature to the design tem- perature) and cooling (from the design temperature to am- bient temperature) cycles are calculated for a straight pipe- line with a length of 2000m, which is a short pipeline ac- cording to the definition of Carr(2008):/is less than 1. The slope of the seabed varies from 3? to 11?, and other parameters of the pipeline and seabed are shown in Table 1. The FEA results of the axial displacement per cycle at the midpoint of the pipeline are determined by the average walking during cycles 2 to 5. The results of pipeline walking per cycle calculated by the FEA method and the analytical solution are shown in Fig.2, where the FE results of pipeline walking calculated by static meth- ods are also provided for comparison.

Fig.2 Pipeline walking per heating and cooling cycles caused by the seabed slope.

To simulate the coupling deformation of pipeline buck- ling and walking, on the one hand, the dynamic explicit method has a significant advantage in the convergence of pipeline buckling analysis (Liu and Li, 2018), especially for the global buckling caused by several heating and cool- ing cycles, wherein the improved static method risk is no longer applicable. Therefore, the dynamic method becomes an option. On the other hand, Fig.2 shows that the magni- tude of errors is acceptable despite errors between analyti- cal and FEA results using the 3D explicit method. More-over, the simulation results show that the ratio of dynamic and potential energy during the pipeline heating and cool- ing process is low, as shown in Fig.3. This finding re- flects the good convergence of dynamic analysis. There- fore, the dynamic explicit method is adopted to simulate the coupling deformation of pipeline walking and buckling in this study.

Fig.3 Ratio of the dynamic and potential energy.

3 Analysis of Walking in a Pipeline with Global Buckling

3.1 Walking Phenomenon of a Buckling Pipeline

A double-imperfection pipeline on the inclined seabed is considered to study the walking phenomenon of a buckl- ing pipeline. As shown in Fig.1, the spacing between two adjacent imperfections caused by buckling initiation tech- niques in pipeline engineering is usually 2 to 3km (Sinclair, 2009; Wang, 2018). In this study, the pipeline length is set as 6000m, and its two ends are free. The two imperfections (I and H) divide the 6000m pipeline into three pipeline segments with lengths of 2000m, where A, B, and C are the midpoints of each pipeline segment, as shown in Fig.4. The wavelength0mof the two imperfec- tions is 40m, and the imperfection amplitudes0mIand0mHare 0.5m. Other parameters of the pipeline are shown in Table 1.

The seabed slopein Fig.4 is modified to demonstrate the walking phenomenon of a buckling pipeline. The ax- ial displacements of points A, B, and C during five heat- ing and cooling cycles are shown in Fig.5. The positive displacement in the figures refers to the axial movement toward the lower end of the seabed slope.

Fig.4 Pipeline with double imperfections.

Fig.5 Axial displacements of points A, B, and C in five heating and cooling cycles.

Fig.5(a) shows the absence of a cumulative axial move- ment at midpoint B of pipeline segment IH with increas- ing cycle numbers when the seabed slope is 0?. When the seabed is 5? and 9?, all axial displacements at points A, B, and C accumulate with cycle numbers, and the trend of axial displacement accumulation increases with the sea- bed slope. Thus, the walking phenomenon of a buckling pipeline on the inclined seabed is directly demonstrated.

For pipelines with multiple imperfections, the walking of one pipeline segment between two adjacent imperfect- tions is identical to that of pipeline segment IH in Fig.4. Thus, the walking behavior of a pipeline with multiple im- perfections can be studied on the basis of pipeline segment IH. The axial displacement at the midpoint of pipeline seg- ment IH (2000m) and that at the midpoint of a short straight pipeline with a length of 2000mare shown in Fig.6.

Fig.6 shows that for pipelines on a seabed of the same slope, the walking of the short straight pipeline is always larger than that of pipeline IH and the gap increases with the number of cycles. To study the cause of this gap, the axial force curves of pipeline segment IH and the short straight pipeline during five cycles are shown in Figs.7 and 8, respectively, where letters M and N correspondingly represent the anchors during heating and cooling.

Fig.6 Axial displacements of buckling pipeline segment IH and a short straight pipeline.

Fig.7 Axial force of buckling pipeline segment IH during five cycles.

Fig.8 Axial force of the short straight pipeline during five cycles.

The pipeline gravity component along the seabed slope combined with soil resistance on the pipeline causes the asymmetry in axial force along the pipeline. Therefore, virtual anchor M during the heating process does not co- incide with virtual anchor N in the cooling process. The pipeline movement direction on both sides of one virtual anchor is indicated by the arrows in Fig.7. The movement direction of pipeline segment MN points to the lower end of the seabed in heating and cooling processes. Therefore, the pipeline walks downward in the heating and cooling cycles.

The comparison between Figs.7 and 8 shows the differ- ence in axial force distribution at the ends of the short straight pipeline and pipeline segment IH. Unlike the com- plete free condition of the short straight pipeline, the end conditions of pipeline segment IH are determined by the buckling of the pipeline. This phenomenon is one of the reasons for the difference in walking between the buck- ling pipeline and the short straight pipeline. Another alter- native reason lies in the difference in axial force along the short straight pipeline and pipeline segment IH, especially during the cooling process. Axial force curves of cooling 1–5 of the short straight pipeline almost overlap, whereas the residual axial force of pipeline IH increases with the rising cycles, which will eventually increase the distance between virtual anchor points M and N of the pipeline.

3.2 Analytical Solution for Walking of the Buckling Pipeline

3.2.1 Walking analytical solution of the short straight pipeline

The axial force curves of a short straight pipeline dur- ing a heating and cooling cycle, which is placed on the sea- bed inclined along the length direction of the pipeline, are shown in Fig.9.

Fig.9 Axial force curves of a short straight pipeline (2000m) during a heating and cooling cycle.

Carr(2008) showed that the pipeline walking per cycle induced by the seabed inclination is equal to the ex- pansion between two virtual anchors during one heating and cooling cycle.

Fig.9 shows that the axial force of pipeline section MN changes as Δduring the heating and cooling processes:

The expansion amount per heating and cooling cycle of pipeline segment MNXis:

According to the geometric relationship in Fig.9, the spacing between virtual anchors M and N (MN) is:

Thus, the walking per heating and cooling cycle at the midpoint of pipelineXis:

The meanings of the letters in Eqs. (3)–(6) are identical to the above.

3.2.2 Analytical solution of walking for the buckling pipeline

The FEA results of pipeline walking have been relativelystable since the second cycle. Thus, the analytical solutionof buckling pipeline axial movement is derived on the basis of the axial force curve of the second cycle in Fig.7(a).

Fig.10 shows that the change in axial force Δof pipe- line section MN during the heating and cooling cycle is:

1and2can be determined by the pipeline buckling analysis. Expansion amountXper heating and cooling cycle of pipeline segment MN is:

The geometric relationship in Fig.10 shows that:

1and2are approximately equal to4and3, respec- tively, as shown in Fig.10. Thus, Eq. (9) is simplified to:

Therefore, by comparing Figs.10 and 7(a), the effect of the number of cycles on the analytical formula of walking is analyzed as follows. From cycles 1 to 5, axial forces2and3hardly change while1and4both accumulate with the cycle numbers. This phenomenon gradually in- creases the difference between1and4in Eq. (9); there- fore,MNincreases accordingly. In addition, the growth in1will increase Δin Eq. (7).

Take cycles 2 and 3 as examples to analyze the effect of changes in1and4on the walking analytical formula.

The geometric relationships in Fig.11 show that

Thus,

The comparison of Eqs. (13) and (11) shows that the increase in1weakens axial movementXand the incre- ment in the difference between4and1strengthensX. Thus,the two factors have opposite effects.According to the buckling mechanism, the changes in1and4in each cycle are relatively small to. In addition, if the effect of the changes in1and4on the walking is considered, then the walking of the buckling pipeline must be calculated cycle by cycle, which significantly increases the calcula- tion cost. This study recommends simply adopting Eq. (11) to estimate the walking of buckling pipelines based on the aforementioned reasons.

Fig.10 Axial force curves of pipeline IH (2000m) during a heating and cooling cycle.

Fig.11 Axial force of pipeline section IH in cycles 2 and 3 (seabed slope=5?).

3.2.3 Validation of the analytic solution for the walking of a buckling pipeline

The analytical results of the walking of pipeline segment IH in a double-imperfection pipeline on the seabed of 5?, 7?, and 9? were determined by Eq. (11) and are plotted in Fig.12 with the FEA results of axial displacement at the midpoint of pipeline segment IH during cycles 2–5. The analytical results determined by Carr(2008) are also provided for comparison in Fig.12.

Fig.12 Analytical results vs. FEA results of each cycle.

Fig.12 shows that the analytical results determined by Eq. (11) and the FEA results have consistent trends with each other, but the analytical results determined by Carr(2008) markedly differ from the numerical results. This difference is due to the analytical solution of Carr, which is derived from short straight pipelines and cannot be directly applied to the walking calculation of buckling pipelines. Thus, considering the influence of pipeline buckl- ing force is necessary when calculating the walking of a buckling pipeline, as shown in Eq. (11).

Fig.12 also reveals that the analytical results determinedby Eq. (11) and the FEA results have consistent trends with each other, but some differences in specific values with anerror of approximately 1% to 7% are found. The FEA re- sults show that the walking error between cycles is within 10%. Two conclusions can be drawn from this analysis. First, the walking difference between cycles is limited, andthe computational cost of going through each cycle is high. Therefore, the difference between cycles should be ig- nored to simplify the calculation. Second, when the cyclic effect on walking is ignored, the analytical results using Eq. (11) are consistent with the FEA results, which validates the proposed walking analytical solution of the buckling pipeline.

The comparison of the analytical and FEA results for pipelines with different outside diameters and operation temperatures is shown in Figs.13 and 14 to further verify the applicability of the analytical solution for the walking of a buckling pipeline (Eq. (11)) for varying parameters, such as pipe size and environmental conditions.

Figs.13 and 14 show that for buckling pipelines with different outside diameters and operating temperatures, the FEA results of walking are consistent with the analytical results proposed in this paper. This finding verifies the ap- plicability of the analytical solution for the walking of a buckling pipeline.

Fig.13 Analytical results vs. FEA results for pipelines with different outside diameters.

Fig.14 Analytical results vs. FEA results for pipelines with different temperatures.

4 Influence Analysis of Walking on Global Buckling

4.1 Effect of Walking on the Capacity of the Buckling Initiation

The capacity of buckling initiation is one of the most important factors in pipeline buckling design. Thus, the influence analysis of walking on the critical axial force required for buckling is conducted through a group of double-imperfection pipelines (Fig.4) on the seabed with a slope of 0?, 5?, and 9?.

Fig.15 shows that when the seabed slope is 0?, the axial force at imperfections I and H completely coincide in five cycles, which indicates that the buckling axial forces dur- ing heating are consistent at imperfections I and H, and the same is true for the residual axial force during cooling.A slight difference in buckling axial force at imperfectionsI and H is observed for pipelines on the seabed with slopes of 5? and 9? when the cycle numbers increase. The effect of walking on the buckling axial force can be neglected because the difference is remarkably small. However, the residual axial force of pipelines on the seabed with slopes of 5? and 9? evidently varies with the number of cycles. The residual axial force at imperfection H is larger than that at imperfection I, and the difference accumulates with the cycle numbers.

Fig.15 Axial force at imperfections I and H.

4.2 Effect of Walking on the Buckling Mode of the Pipeline

The lateral displacement curves along the pipeline on the seabed with slopes of 0?, 5?, and 9? are plotted as shown in Fig.16 to analyze the effect of walking on the buckling mode.

Fig.16 shows that the buckling of the pipeline at two imperfections is completely symmetrical for the case where the seabed slope is 0?. However, buckling amplitude and mode at the two imperfections are no longer symmetrical when the seabed slope increases to 5? and 9?. Take the case where the seabed slope is equal to 5? as an example to ana- lyze the asymmetry phenomenon. The lateral displacement curves of each cycle are separately extracted to analyze the change process of the buckling mode during the cycles, as shown in Fig.17.

For pipeline imperfection H at the low end of the seabed slope, Fig.17 shows that the maximum buckling amplitudedecreases while the secondary buckling amplitude increas- es with the cycle numbers, facilitating the changes in buckl- ing mode from 3 to 2 as described by Hobbs (1984). How- ever, the modes remain unchanged for pipeline imperfec- tion I at the high end of the seabed slope despite the limited variation of the pipe buckling amplitude with the number of cycles. This condition explains the gradual asymmetry of buckles at the two imperfections. In addition, this analy- sis shows that the effect of walking produces asymmetri- cal buckling at one imperfection; that is, the amplitudes of the secondary buckling on the left and right sides of the primary buckling are not equal, which is consistent with the study of Bruton(2003).

Fig.16 Lateral displacement curves of pipelines on the seabed with different slopes.

Fig.17 Lateral displacement curves of pipelines on a seabed with slope=5?.

The lateral displacement at the midpoint of imperfec- tions I and H during heating and cooling cycles is shown in Fig.18.

Fig.18 shows that the buckling displacement at the mid- point of the upslope imperfection I gradually increases from cycles 1 to 4 but decreases in the fifth cycle. Mean- while, the downslope imperfection H shows an early and severely weakened trend of the buckling amplitude. Thus, for pipelines with multiple imperfections, further studying this walking effect, which weakens the maximum buck- ling amplitude of the pipeline, and its accumulation with imperfection numbers, is reasonable.

Fig.18 Lateral displacement at the midpoint of imperfections I and H (seabed slope=5?).

4.3 Effect of Walking on the Failure Assessment of a Buckling Pipeline

The visual expression of the walking effect on pipeline buckling lies in the changes in buckling amplitude and mode. These changes are accompanied by variations in bending moment, stress, and strain. These factors may jeopardize the structural integrity and cause pipeline failure. There- fore, the influence analysis of walking on the failure as- sessment of global lateral buckling in pipelines is neces- sary.

DNV-OS-F101 (2012) suggests two design criteria for the failure assessment: displacement- and load-controlled criteria. A pipeline checked for displacement-controlled cri- teria will typically have tensile strains above 0.4%. Frac- ture assessment is required if tensile strains exceed 0.4%. For the load-controlled criterion, pipe members subjected to axial force and bending moment will be designed to satisfy the following criterion at all cross-sections.

wheremis the material resistance factor,SCis the safety class resistance factor,Sdis the design moment,cis the flow stress parameter,pis the plastic capacity of the mo- ment,Sdis the design effective axial force,pis the plas- tic capacity of axial force, andis the failure parameter; when≤1, the pipeline efficiently works under the cur- rent load combination. A large value of, which corre- sponds to a high load, indicates a high failure probability.

The buckling results of the pipeline during five load cy- cles when the seabed slope is 0? are shown in Table 2. All integration points of each pipe element are checked, but only the worst point is presented in Table 2.

Table 2 Buckling results of the pipeline on a seabed with a slope of 0?

Table 2 shows that failure parameters and total strain of the pipelines reach the peak in the first cycle. This finding is attributed to the weakened subsequent buckling defor- mation due to residual stress and strain in the pipeline generated at the end of the first cycle. Without walking (seabed slope=0?), the values of failure parameters at the two imperfections of the pipeline are close but only vary in the fifth cycle.

The pipeline laid on the seabed with a slope of 9? istaken as a case to analyze the effect of walking on the fail- ure assessment of the buckling pipeline. The failure as- sessment of global buckling in pipelines with walking dur- ing five heating and cooling cycles is described in Table 3.

Similar to the pipelines in Table 2, failure parameters and total strain of the pipelines are maximal in the first cycle in Table 3. However, the failure parameters at the two imperfections differ after the first cycle, as well as the total strain for the pipeline with walking (seabed slope=9?). This finding indicates that pipeline walking affects the failure assessment of global lateral buckling in pipelines.

Table 3 Buckling results of the pipeline on a seabed with a slope of 9?

Fig.19 shows the comparison of the failure parameters of the pipelines with seabed slopes of 0? and 9? in Tables 1 and 3, respectively.

Fig.19 shows that the failure parameters of the pipe- lines with/without walking are approximately identical to each other in the first and second cycles. With the increase in the cycles, the failure parameter of the pipeline with walking (seabed slope=9?) is gradually larger than that of the pipeline without walking (seabed slope=0?). This finding indicates that pipeline walking is unfavorable to the integrity of a pipeline that exhibits global buckling.

Fig.19 Failure parameters of the pipelines with/without walking.

5 Conclusions

Global lateral buckling is mainly used to release the ax- ial force of submarine pipelines subjected to high tem- peratures and pressures. However, the buckling at multi- ple locations of a long pipeline may divide the originally fully constrained pipeline into multiple short pipeline seg- ments, which may increase the risk of pipeline walking. Thus, a pipeline, which is designed to buckle laterally, may laterally and axially displace over time when subjected to repeated heating and cooling cycles. Hence, the coupling deformation of global lateral buckling and walking of sub- marine pipelines is studied with theoretical and FEA meth- ods. The conclusions are as follows.

1) A walking phenomenon in the pipeline segment be- tween two adjacent buckles is observed for a buckled pipeline laid on an inclined seabed. This buckling pipeline segment walks differently from a short straight pipeline for two reasons. The end conditions of the pipeline seg- ment determined by the buckling are different from the complete free condition of the short straight pipeline. The residual axial force of the pipeline segment and distance between two anchors MN change with the increase in cy- cles, while those of the short straight pipeline remain un- changed.

2) Compared with the walking of the short straight pipe-line, the axial movement mechanism of the long pipeline with global buckling is analyzed. An analytical solution for the walking of buckling pipelines is proposed on the basis of the aforementioned analysis. Subsequently, the sensi- tivityof this method to heating and cooling cycles is ana- lyzed, which validates the proposed walking analytical solution of the buckling pipeline.

3) The effect of walking on the buckling axial force is negligible. The effect of pipeline walking on post-buckling can be described as the gradually asymmetric buckling at various imperfections with heating and cooling cycles. In addition, the buckling at one imperfection is no longer sym- metrical.

4) The influence analysis of walking on the failure as- sessment of buckling pipelines shows that the pipelines with/without walking have identical failure parameters in the first and second cycles. With the increase in cycle num-bers of heating and cooling, the failure parameters of the pipeline with walking (seabed slope=9?) are gradually larger than those of the pipeline without walking (seabed slope=0?), which indicates that the pipeline walking is un-favorable for pipeline safety.

Acknowledgements

The study is supported by the China National Postdoc- toral Program for Innovative Talents (No. BX2021213), and the Natural Science Foundation for Distinguished Young Scholars of China (No. 51825904).

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(March 19, 2022;

May 9, 2022;

July 11, 2022)

? Ocean University of China, Science Press and Springer-Verlag GmbH Germany 2023

. E-mail: licf@tju.edu.cn

(Edited by Xie Jun)