Lateral Global Buckling of Submarine Pipelines Based on the Model of Nonlinear Pipe–Soil Interaction

With the increasing development and utilization of offshore oil and gas resources, global buckling failures of pipelines subjected to high temperature and high pressure are becoming increasingly important. For unburied or semi-buried submarine pipelines, lateral global buckling represents the main form of global buckling. The pipe–soil interaction determines the deformation and stress distribution of buckling pipelines. In this paper, the nonlinear pipe–soil interaction model is introduced into the analysis of pipeline lateral global buckling, a coupling method of PSI elements and the modified RIKS algorithm is proposed to study the lateral global buckling of a pipeline, and the buckling characteristics of submarine pipeline with a single arch symmetric initial imperfection under different pipe–soil interaction models are studied. Research shows that, compared with the ideal elastic–plastic pipe–soil interaction model, when the DNV-RP-F109 model is adopted to simulate the lateral pipe–soil interactions in the lateral global buckling of a pipeline, the buckling amplitude increases, however, the critical buckling force and the initial buckling temperature difference decreases. In the DNV-RP-F109 pipe–soil interaction model, the maximum soil resistance, the residual soil resistance, and the displacement to reach the maximum soil resistance have significant effects on the analysis results of pipeline global buckling.


Introduction
Oceanic hydrocarbon reservoirs have become the main source of future economic oil and gas reserves. Submarine pipelines served as high efficiency and continuous conveying means that are widely used in the development of ocean resources (Liu et al., 2011). Under the influences of both temperature stress and the Poisson effect, the considerable additional stress is produced on the pipeline. Owing to the constraints of foundation soil, it is difficult to release additional stress of the pipeline through free deformation. Thus, when the additional stress accumulates to a certain level, the global buckling, which is similar to compressive bar instability, will occur. The pipe-soil interaction has a great influence on the global buckling of pipelines. During the process of the pipeline lateral global buckling, the variation of soil resistance with pipeline displacement affects the buckling trajectory of a pipeline. Meanwhile, the pipe-soil interaction determines the deformation and stress distribution of the buckling pipelines. From the previous studies (Lyons, 1973;Wagner et al., 1989;Bruton et al., 2006;Wang and Liu, 2016), it shows that many factors, including soil characteristics, loading modes, self-weight of pipelines and initial embedment depths, affect the soil resistance to the pipeline, thus, the pipe-soil interaction is nonlinear and complicated. Thus, it is of great practical significance to study the global buckling for pipelines under the nonlinear pipe-soil interaction model.
In recent years, the application of numerical methods has made advances for the researches on the pipe-soil interaction. Bruton et al. (2007) used ABAQUS to study the effects of the axial and lateral soil resistance on the pipeline buckling morphology, the axial force of a pipeline, and the initial buckling conditions. However, the soil resistance to a pipeline was simplified as the Coulomb friction in the analysis, which could not truly simulate the variation of the soil resistance with the pipeline displacement. Wang et al. (2015) built the finite element models to investigate the lateral buckling response of the SP (sandwich pipes) with nonlinear spring elements used to simulate the dynamic soil resistance. However, the soil resistance was related to the dis-tribution density of spring elements in the analysis, which easily caused the stress concentrations at the spring nodes with a large load and a sparse spring distribution.  developed the user subroutine VFRIC in ABAQUS to simulate the dynamical friction in the pipe-soil interactions for the accurate pipeline global buckling analysis. In VFRIC, the soil resistance was defined by using the basic incremental method. However, there exists a sudden change with its displacement of the pipeline during the global buckling process. When the displacement between the two adjacent incremental steps was large, the results of the soil resistance obtained using VFRIC were not accurate. At present, there is still a lack of effective simulation of the dynamic soil resistance, and the accurate simulation of nonlinear pipe-soil interactions is an important premise to study the pipeline lateral global buckling.
ABAQUS provides the pipe-soil interaction (PSI) elements for modeling the interaction between a pipeline and the surrounding soil, which was used to study the nonlinear pipe-soil interaction by many scholars. Zhang et al. (2010) presented a kind of closed-form analytical solution to the interaction between a pipeline and the soil induced by nearby excavation processes. Then the analytical results were compared with the FEM solutions based on the PSI elements provided by ABAQUS. Sun et al. (2011) created the fullscale FE models to study the upheaval buckling of partially buried pipelines. The PSI elements were utilized to model the relationship between the soil resistance and the pipe displacement for the buried sections. They investigated the effects of soil cover height, vertical prop size, and soil resistance on the upheaval and lateral buckling response of a partially buried pipeline. Zeng et al. (2013) utilized the FEM to analyze the upheaval buckling behavior of an actual PIP system, among which the nonlinear pipe-soil interaction was simulated by the PSI elements. Compared with the nonlinear spring elements which are closely related to the spring distribution density, the constitutive behavior for the PSI elements are defined by the force per unit length. PSI elements can be infilled in the severe buckling deformation section of the pipeline without causing the stress concentration at the nodes. Meanwhile, PSI elements are also applicable to the sudden deformation of a pipeline during the global buckling process and will not produce the kind of error problem that the subroutine VFRIC makes. Therefore, more accurate results of the deformation and stress for the pipeline lateral global buckling can be obtained from using the PSI elements to simulate the nonlinear pipe-soil interaction. From the above studies, it shows that PSI elements are currently only used to simulate the pipe-soil interaction in ground deformation and pipeline upheaval global buckling, and they are less commonly applied to study the lateral global buckling of a pipeline.
In this paper, the nonlinear pipe-soil interaction models were established to simulate the dynamic soil resistance dur-ing the process of the pipeline lateral global buckling. A coupling method of the PSI elements and the modified RIKS algorithm was proposed to study the lateral global buckling of a pipeline for the first time. The buckling characteristics of a submarine pipeline with a single arch symmetric initial imperfection under different pipe-soil interaction models were studied.

Coupling method of PSI elements and the modified RIKS algorithm
2.1 PSI element ABAQUS provides two-dimensional (PSI24 and PSI26) and three-dimensional (PSI34 and PSI36) pipe-soil interaction elements to model the interaction between a pipeline and the surrounding soil (2011). One side or edge of the PSI element shares nodes with the elements that model the pipeline (Fig. 1). The nodes on the other edge represent a far-field surface, such as the ground surface, and are used to prescribe the far-field ground motion via the boundary conditions together with the amplitude references as needed. It is important to note that PSI elements do not discretize the actual domain of the surrounding soil; the extent of the soil domain is reflected through the stiffness of the elements.
PSI elements have only the displacement degrees of freedom at their nodes; the deformation of a PSI element is characterized by the relative displacements between the two edges of the element. When the element is "strained" by the relative displacements, forces are applied to the pipeline nodes. Positive "strains" are defined by where, are the relative displacements between the two edges, ( are the far-field displacements, and are the pipeline displacements); are local directions, and the index (=1, 2, 3) refers to the three local directions. For three-dimensional elements, all three strain components , , and exist.
The constitutive behavior for a pipe-soil interaction is defined by the force per unit length, or "stress, " at each point along the pipeline, , caused by relative displacement or "strain, " , between that point and the point on the far- Fig. 1. Pipe-soil interaction model. LIU Run, WANG Xiu-yan China Ocean Eng., 2018, Vol. 32, No. 3, P. 312-322 313 field surface: where denotes the state variables (such as plastic strains), and represents the temperatures and/or field variables.

Modified RIKS algorithm with the PSI elements
The modified RIKS method has become the main method used in the analysis of non-linear structure stability problems. The modified RIKS method uses the load magnitude as an additional unknown; it solves simultaneously for loads and displacements. It is a generalized displacement control method that can be used to speed up the convergence of illconditioned or snap-through problems that do not exhibit instability. The modified RIKS algorithm provided by the ABAQUS software has become one of the most important methods used in the analysis of non-linear structure stability problems.
With the obvious advantages of the PSI elements and the modified RIKS method, they are combined to study the influence of the nonlinear pipe-soil interaction model on the lateral global buckling of pipelines in this paper. Three-dimensional four-node pipe-soil interaction elements (PSI34) are adopted to simulate the nonlinear pipe-soil interaction during the process of the pipeline lateral global buckling. PIPE31 elements are adopted to model the pipeline. The PSI elements are defined so that one edge of the element shares nodes with the pipeline elements, and the nodes on the other edge represent a far-field surface where the ground motion is prescribed. All degrees of freedom on the remaining far-field nodes are fully fixed.
The calculation process consists of three analysis steps: (1) An initial imperfection is introduced into the pipeline model based on the results of the modal analysis method. The models in the buckle analysis method are analyzed according to the geometric parameters of the pipeline to obtain the most possible geometric imperfection and then output the nodes displacement of the pipeline. (2) A new model, for the coupling method of the PSI elements and the modified RIKS algorithm, is then established, and both models must have identical node numbers. The node displacement is introduced into the new calculation model in the form of a scale factor. (3) Temperature load has been applied to simulate the high temperature loading process until the design temperature difference is reached.

Reliability verification
To verify the reliability of the proposed calculation method, the numerical results are compared with the model test results by Poiate et al. (2004). In the test, the deformation displacements of a single-layer pipe model under the temperature load were measured. The total length of the pipe model was 16 m, the diameter was 76.2 mm, and the wall thickness was 2 mm. The experimental apparatus was capable of simulating the lateral reaction of the soil. The fi-nite element model is established by using the PSI elements to simulate the pipe-soil interaction. The relationship between the lateral soil resistance per unit length and lateral displacement in the test and FEM are illustrated by the solid line and the dotted line, respectively in Fig. 2. It can be seen that the soil resistance in the FE model is consistent with that in the test. Hence, PSI elements can successfully simulate the nonlinear pipe-soil interaction in the test. The test and numerical results of the pipeline deformation are illustrated in Fig. 3.
It can be seen from Fig. 3 that the buckling amplitude at the midpoint of the pipeline obtained from the proposed numerical method is basically the same as the test results. The coupling method can successfully simulate the three bending segments in the model test, and the buckling lengths of the three bending segments are almost consistent with the test value. These are the most important targets of the buckling morphology of a pipeline. And the buckling displacements on both sides of the midpoint are slightly smaller than the test results, but the difference is small. Therefore, it is reliable to use this coupling method to simulate the lateral global buckling of a pipeline.

Case
The coupling method of the PSI elements and the modified RIKS algorithm is adopted to study the lateral global  buckling behavior of a pipeline; therefore, nonlinear pipe-soil interaction models are required to be established in the analysis. The pipeline considered has an outer diameter of 323.9 mm, a wall thickness of 12.7 mm, and the material specification of the pipeline is API 5L X65. The designed internal pressure and temperature difference are 4.65 MPa and 88°C, respectively. To facilitate the analysis of the pipeline buckling under the temperature load, the internal pressure is converted into the temperature difference. According to the research results of Hobbs (1984), the equivalent temperature difference is 5°C when the internal pressure is 4.65 MPa. Thus, the total temperature difference is 93°C. The parameters of the pipeline and foundation soil are presented in Table 1.
A finite element model is established based on Table 1, as shown in Fig. 4. To simulate the process of the global buckling more accurately under a real condition, the length for the calculated model is 20000 m, and both the ends of the pipeline are completely free. The pipeline model is modeled with 100000 PIPE31 elements. PIPE31 elements which belong to the Timoshenko beams can be subjected to large axial strains and the axial strains due to the torsion are assumed to be small. A single arch symmetric initial imperfection is introduced at the midpoint of the pipeline based on the modal analysis. Changing the initial geometric imperfection will change the buckling morphology of a pipeline, but will not affect the influence characteristics of the parameters in the nonlinear pipe-soil interaction model on the global buckling. Therefore, an initial imperfection with an amplitude v 0 =0.8 m and a wavelength L 0 =30 m is selected for calculation. PSI elements are adopted to simulate the nonlinear pipe-soil interaction; hence, the stiffness of the PSI elements should be set in the lateral, vertical and axial directions (k L , k V and k A ).

Vertical pipe-soil interaction model
There is no uniform vertical pipe-soil interaction model in the existing international specifications. In this paper, the vertical pipe-soil interaction model is established by using the recommendation formula of the ASCE specification (ASCE, 2005). The relationship between the vertical soil resistance per unit length and vertical displacement of a pipeline is shown in Fig. 5. According to the calculation, the maximum vertical downward soil resistance F vd = 2.732× 10 4 N/m, and the corresponding vertical displacement V v1 = 0.2D = 0.06478 m. The computational method in the ASCE specification is applied when initial embedment depth of a F vu = 100 N/m pipeline >0.5D; however, the unburied pipeline has only a small initial burial depth due to the pipeline self-penetration. Therefore, to reasonably simulate the soil resistance in the actual project, the maximum vertical upward soil resistance should be set to a small value , and the corresponding vertical displacement V v2 = 0.2D = 0.06478 m. The above relationship between the vertical soil resistance and vertical displacement is taken as the vertical pipe-soil interaction model.

Axial pipe-soil interaction model
There is no uniform axial pipe-soil interaction model in the existing international specifications. White et al. (2011) carried out a series of the axial pipe-soil interaction model tests to investigate the axial movement behavior of a pipeline. In this paper, the axial pipe-soil interaction model is established using the model tests results of White et al. (2011). In the tests, the length of the pipe section is 8 m and the diameter is 90 mm. The relationship between the equivalent axial friction factor (axial soil resistance/submerged weight of pipeline per unit length) and axial displacement is illustrated by the solid line in Fig. 6. The dotted line is obtained by the linear fitting to the solid line (as shown in Fig. 6). Corresponding to the fitted curve, the relationship between the soil resistance and axial displacement is adopted to establish the axial pipe-soil interaction model (as shown in Fig. 7). According to calculation, F a1 = 595.2 N/m, the corresponding axial displacement V a1 = 0.004 m,  , and the corresponding axial displacement ; this curve is symmetric at the origin.

Lateral pipe-soil interaction model
There is no uniform lateral pipe-soil interaction model in the existing international specifications. In DNV-RP-F109, the soil resistance consists of two parts in general: a pure Coulomb friction part; and a passive resistance due to the buildup of the soil penetration as the pipe moves laterally (see the DNV line in Fig. 8).
F f (1) For the Coulomb friction , the coefficient of the friction for a concrete coated pipe can normally be taken as 0.6 for sand, 0.2 for clay and 0.6 for rock. F R (2) For the passive resistance , a typical model for the passive soil resistance consists of four distinct regions: In the elastic region, where the lateral displacement is typically smaller than 2% of the pipe diameter, the stiffness can be taken as 50-100 N/m for sand and 20-40 N/m for clay.
The maximum is recorded as . In the region where the displacement is up to the half of the pipe diameter, the pipesoil interaction creates work, which again increases the penetration and thus the passive resistance. The maximum soil resistance is recorded as . This stage is called the upward stage. In the region where the displacement is up to one pipe diameter, the resistance and penetration decrease. This stage is called the downward stage. When the displacement exceeds one pipe diameter, the passive resistance and penetration may be assumed to be constant. The constant soil resistance is recorded as . This stage is called the residual soil resistance stage.
Passive resistance on clay can be taken as: where is the vertical contact force between pipe and soil; is the un-drained clay shear strength; D is the pipe outer diameter; is the dry unit soil weight and can be taken as 18000 N/m 3 for clay.
Total penetration can be taken as the sum of initial penetration and penetration due to the pipe movement : z pi Initial penetration on clay can be taken as: The stiffness can be taken as 40 N/m for clay. According to the research of Wang et al. (2017), when the lateral displacement reaches half and one pipe diameter, the penetration depth is 15% and 7.5% of the pipe diameter, respectively. According to the calculation, , F e = 232 N/m, , and . Based on the research of Maltbya and Calladineb (1995) and White et al. (2011), with the increase of the lateral displacement, the soil resistance does not increase significantly after reaching the maximum soil resistance. The pipeline tends to experience a simple lateral movement. This soil resistance model can be called the ideal elastic-plastic model (see the "ideal elastic-plastic model" line in Fig. 8).
During the process of the lateral global buckling, the lateral soil resistance to the pipeline plays a critical role in the deformation and buckling characteristics of a pipeline. The following research focuses on the effect of the lateral pipe-soil interaction model on the lateral global buckling of a pipeline.   The coupling method of PSI elements and the modified RIKS algorithm is applied to study the lateral global buckling characteristics of a submarine pipeline using both the DNV model and ideal elastic-plastic model. The vertical pipe-soil interaction model and axial pipe-soil interaction model in Section 3.2 and Section 3.3 are adopted to simulate the nonlinear vertical and axial pipe-soil interactions. Under the combined effect of high pressure and high temperature, the pipeline morphologies of finial deformation for the lateral buckling under two lateral pipe-soil interaction models are obtained, as shown in Fig. 9.
In Fig. 9, when the temperature difference reaches the designed value of 93°C, the buckling amplitude (the lateral amplitude of the pipeline midpoint) and total buckling deformation length under the DNV model are 6.35 m and 235.8 m, respectively; those values under the ideal elastic-plastic model are 6.01 m and 211.8 m, respectively. This is mainly because by the DNV model, after the lateral displacement of the pipeline midpoint exceeding the half of the pipe diameter, the soil resistance decreases significantly and it is difficult to suppress the buckling deformation of a pipeline with a smaller soil resistance. Therefore, compared with those of the ideal elastic-plastic model, the buckling amplitude and total buckling deformation length of a pipeline obtained with the DNV model are larger. Fig. 10 details the relationships between the buckling amplitude, temperature difference and buckling force using both the DNV model and ideal elastic-plastic model.
Under the combined effect of high pressure and high temperature, the thermal stress together with the Poisson effect will cause the pipeline to expand longitudinally. Internal additional stress has continued to accumulate until it exceeds the soil resistance. Sudden deformation occurs to release the internal stress. During the process of the global buckling, the buckling force increases to the maximum value (the critical buckling force), and then decreases. In Fig. 10a, the critical buckling force of the pipeline by the ideal elastic-plastic model is 408.8 kN, and that by the DNV model is 358.2 kN. In Fig. 10b, the initial buckling temperature difference by the ideal elastic-plastic model is 31.7°C, and that by the DNV model is 28.8°C. When the lateral displacement of the pipeline midpoint is smaller than one pipe diameter, the soil resistances by the ideal elastic-plastic model and DNV model are similar; therefore the buckling force curves by these two models are relatively consistent. When the lateral displacement of the pipeline midpoint exceeds one pipe diameter, the soil resistance to the pipeline by the DNV model begins to decrease. Therefore, the buckling force and initial buckling temperature difference of the pipeline by the DNV model are smaller than those by the ideal elastic-plastic model.

Effect of the maximum soil resistance in the pipe-soil interaction model
To study the effect of the maximum soil resistance in the pipe-soil interaction model on the lateral global buckling of a pipeline, it assumes that the DNV models have different maximum soil resistances of 700, 1050, 1400, and 1750 N/m, while the other parameters in the model remain unchanged, and four calculation models (Peak 1-Peak 4) are established. Under the combined effect of high pressure and high temperature, the pipeline morphologies of finial deformation for tye lateral buckling by these four models are obtained, as shown in Fig. 11.
In Fig. 11, the model with higher maximum soil resistance has a larger buckling amplitude and a smaller maxim-  LIU Run, WANG Xiu-yan China Ocean Eng., 2018, Vol. 32, No. 3, P. 312-322 317 um negative deformation. The model with lower maximum soil resistance has a smaller buckling amplitude and a larger maximum negative deformation.
To study this phenomenon, the maximum positive and negative deformation points of Peak 1 are recorded as A1 and B1, respectively, and the maximum positive and negative deformation points of Peak 4 are recorded as A4 and B4, respectively. The relationships between the lateral displacement and temperature difference of these four points are illustrated in Fig. 12 (the lateral deformation displacements of B1 and B4 are converted to positive values).
As shown in Fig. 12, the lateral deformation process of these four points consists of three stages.
(1) 0°C <temperature difference <35.2°C When the lateral displacements of A1 and A4 are smaller than one pipe diameter (1.0D), the soil resistance of Peak 4 is always larger than that of Peak 1, thus, the lateral displacement of A1 is larger than that of A4. At this stage, there is almost no lateral deformation of B1 or B4, which has no effect on A1 and A4. When the lateral displacements of A1 and A4 exceed 1.0D, the soil resistances of A1 and A4 enter the residual soil resistance stage and deformation velocities of the two points are almost the same. With an increase in the temperature difference, the lateral displacement of B1 occurs, and the soil resistance of B1 increases, B4 still has no deformation. The negative deformation of B1 suppresses the positive deformation of A1, which causes a decrease in the deformation velocity of A1. The deforma-tions of A1 and B1 are almost the same at the temperature difference of 35.2°C.
(2) 35.2°C <temperature difference <43.5°C With an increase in the temperature, the lateral negative deformation of B4 occurs; however, at this stage, the deformation value is small. The soil resistance of B1 enters the residual soil resistance stage. The deformation of B1 increases further, continually suppressing the positive deformation of A1. The deformation velocity of A1 is smaller than that of A4.
(3) Temperature difference> 43.5°C At this stage, the soil resistance of B4 enters the residual soil resistance stage; the deformation velocity of B4 is close to that of B1. The inhibitory effects of the negative deformation of B1 and B4 on the positive deformation of A1 and A4 are almost identical. Therefore, the final lateral displacement of A4 is larger than that of A1, and the final lateral displacement of B1 is larger than that of B4. Fig. 13 details the relationships between the buckling amplitude, temperature difference and buckling force using these four models.
In Fig. 13a, the critical buckling forces of the pipelines in Peak 1-Peak 4 are 338.5, 357.0, 371.5 and 385.6 kN, respectively. The calculation model, with higher maximum soil resistance, has larger critical buckling force of a pipeline. When the same deformation occurs at the midpoint of the pipeline, the calculation model, with higher maximum soil resistance, has a larger axial force accumulated in the pipeline. In Fig. 13b, the initial buckling temperature differences of the pipelines in four calculation models are in the range of 28°C to 28.8°C. In each model, the lateral displacement of the pipeline midpoint has exceeded 1.0D when the global buckling occurs; that is, the soil resistance of the pipeline midpoint is in the residual soil resistance stage, thus, the change of the maximum soil resistance has little effect on the initial buckling temperature difference.

Effect of the residual soil resistance in the pipe-soil interaction model
To study the effect of the residual soil resistance in the pipe-soil interaction model on lateral global buckling of a pipeline, assuming that the DNV models have different residual soil resistances of 300, 500, 700, and 900 N/m, while the other parameters in the model remaining unchanged, four calculation models (Res 1-Res 4) are established. Under the combined effect of high pressure and high temperature, the pipeline morphologies of finial deformation for the lateral buckling in these four models are obtained, as shown in Fig. 14. In Fig. 14, the model with higher residual soil resistance has a larger buckling amplitude, a smaller maximum negative deformation and a shorter total buckling deformation length. The model with lower residual soil resistance has a smaller buckling amplitude, a larger maximum negative deformation and a longer total buckling deformation length. Similar to Fig. 11, this is because, under the combined effect of high pressure and high temperature, the points on the pipeline are located at different stages of the pipe-soil interaction model. Fig. 15 details the relationships between the buckling amplitude, temperature difference and buckling force under the four models.
In Fig. 15a, the critical buckling forces of the pipelines in Res 1-Res 4 are 309.7, 335.7, 365.7 and 391.9 kN, respectively. The calculation model with higher residual soil resistance has a larger critical buckling force of the pipeline. In Fig. 15b, the initial buckling temperature differences in Res 1-Res 4 are 19.9°C, 26.8°C, 29.0°C and 30.6°C, re-spectively. In each model, the lateral displacement of the pipeline midpoint has exceeded 1.0D when the global buckling occurs; that is, the soil resistance of the pipeline midpoint is in the residual soil resistance stage. Unlike the results of changing maximum soil resistance in Section 4.2, the change of residual soil resistance has a great influence on the initial buckling temperature difference. It can be seen from Fig. 15 that compared with the model with a larger residual soil resistance, the critical buckling force and the initial buckling temperature difference of the pipeline with a lower residual soil resistance decrease significantly, which threatens the safe operation of the pipeline.
4.4 Effect of the displacement to reach the maximum soil resistance in the pipe-soil interaction model To study the effect of the displacement to reach the maximum soil resistance in the pipe-soil interaction model on the lateral global buckling of a pipeline, assuming that the DNV models have different displacements of 0.1D, 0.4D, 0.7D and 0.9D to reach the maximum soil resistance, while the other parameters in the model remain unchanged, four calculation models (Rise 1-Rise 4) are established. Under the combined effect of high pressure and high temperature, the pipeline morphologies of finial deformation for the lateral buckling in the four models are obtained, as shown in Fig. 16.
In Fig. 16, the model with larger displacement to reach the maximum soil resistance has a smaller buckling amp-  LIU Run, WANG Xiu-yan China Ocean Eng., 2018, Vol. 32, No. 3, P. 312-322 319 litude and a larger maximum negative deformation, but the differences of the results are rather small. Additionally, there are no significant differences in the total buckling deformation length of the four models. Fig. 17 details the relationships between the buckling amplitude, temperature difference and buckling force under the four models.
In Fig. 17a, the critical buckling forces of the pipelines in Rise 1-Rise 4 are 364.5, 360.1, 353.2 and 347.4 kN, respectively. The calculation model with larger displacement to reach the maximum soil resistance has a smaller critical buckling force of the pipeline. The critical buckling force is reduced by only 17.1 kN when the displacement increases from 0.1D to 0.9D. In Fig. 17b, the initial buckling temperature differences in Rise 1-Rise 4 are in the range of 27.8°C to 29.6°C. In each model, the lateral displacement of the pipeline midpoint has exceeded 1.0D when the global buckling occurs; that is, the soil resistance of the pipeline midpoint is in the residual soil resistance stage, thus, the change of the displacement to reach the maximum soil resistance has little effect on the critical buckling force and initial buckling temperature difference. Furthermore, in the four models, the displacement to reach the maximum soil resistance is smaller than 1.0D. The displacement changes little; therefore, it has no significant influence on the buckling characteristics of the pipeline.
4.5 Effect of the displacement to reach the residual soil resistance in the pipe-soil interaction model To study the effect of the displacement to reach the residual soil resistance in the pipe-soil interaction model on the lateral global buckling of the pipeline, assuming that the DNV models have different displacements reach the residual soil resistance (Y 3 ) of 0.6D, 0.8D, 1.0D, ..., 3.6D, 3.8D and 4.0D, while the other parameters in the model remain unchanged, eighteen calculation models (Down 1-Down 18) are established.
According to calculation, due to the change of Y 3 in the pipe-soil interaction model, when the global buckling occurs, the soil resistances of the pipeline midpoints in the Down 1-Down 7 models and Down 8-Down 18 models are in the residual soil resistance stage and downward stage, respectively; and thus present different buckling characteristics. When the global buckling occurs, based on different stages in the pipe-soil interaction model where the pipeline midpoint is located, Down 1-Down 18 are divided into two groups.
Under the combined effect of high pressure and high temperature, the pipeline morphologies of finial deformation for lateral buckling in the 18 models are obtained. Since there are too many models, only the results of models Down 1, Down 6, Down 12 and Down 18 are illustrated in Fig.  18a. The relationships between the buckling amplitude (black line), the maximum negative deformation of the pipeline (grey line) and Y 3 /D (the displacement to reach the residual soil resistance in the pipe-soil interaction model/one pipe diameter) are illustrated in Fig. 18b. The relationships between the critical buckling force and Y 3 /D are illustrated in Fig. 18c; the relationships between the initial buckling temperature difference and Y 3 /D are illustrated in Fig. 18d.
In Fig. 18a and Fig. 18b, the model with larger Y 3 has a larger buckling amplitude, a smaller maximum negative deformation and a shorter total buckling deformation length. Similar to Fig. 11 and Fig. 14, this is because, under the combined effect of high pressure and high temperature, the points on the pipeline are located at different stages of the  pipe-soil interaction model.
In Fig. 18c, the critical buckling force of the pipeline increases with Y 3 , regardless of the stage in the pipe-soil interaction model where the pipeline midpoint is located. When the displacement of the pipeline midpoint is smaller than 0.5D, the same deformation of the pipeline occurs in each model. After the pipeline midpoint entering the downward stage of the pipe-soil interaction model, the model, with smaller Y 3 , has a decline in the soil resistance and a significant increase in the lateral displacement of the pipeline midpoint, which is equivalent to the increase of the imperfection, and thus the critical buckling force decreases.
In Fig. 18d, when the global buckling occurs, the pipeline midpoints in the Down 1-Down 7 models are located at the residual soil resistance stage, and the initial buckling temperature differences in Down 1-Down 7 are in the range of 28.3°C to 29.0°C. There is no significant increase or decrease in the initial buckling temperature difference of the pipeline with the increase of Y 3 . Y 3 has no obvious effect on the initial buckling temperature difference of the pipeline.
When the global buckling occurs, the pipeline midpoints of Down 8-Down 18 models are in the downward stage of the pipe-soil interaction model, and the initial buckling temperature difference of the pipeline increases with Y 3 . When the soil resistance of the pipeline midpoint reaches the maximum soil resistance, the same axial force is accumulated in the pipeline of each model. When the pipeline midpoint enters the downward stage of the pipe-soil interaction model, the model with smaller Y 3 has a significant decline in the soil resistance, which makes the global buckling occurrence more prone.

Conclusions
The nonlinear pipe-soil interaction models were established to simulate the variation of the soil resistance with the pipeline displacement in the pipeline global buckling, and a coupling method of PSI elements and the modified RIKS algorithm was proposed to study the lateral global buckling of a pipeline for the first time. The reliability of this coupling method is verified by comparing with the classical model test (Poiate et al., 2004), and the effect of nonlinear pipe-soil interaction model on the lateral global buckling of a pipeline with a single arch symmetric initial imperfection is analyzed. The main conclusions are as follows.
(1) Compared with the ideal elastic-plastic soil resistance model which is commonly used in the pipeline lateral global buckling, when the pipe-soil interaction model in the DNV specification is adopted, the critical buckling force and initial buckling temperature difference decrease and the buckling amplitude increases.
(2) The pipe-soil interaction determines the deformation and initial occurrence condition of the pipeline global buckling directly. The maximum soil resistance in the pipe-soil interaction model affects the buckling amplitude, total buckling deformation length and critical buckling force of a pipeline to some degree. The influence of the residual soil resistance on the pipeline global buckling is very significant, which affects not only the buckling amplitude, total buckling deformation length and critical buckling force of a pipeline but also the initial occurrence condition of the (3) The displacement to reach different stages in the pipe-soil interaction models also affects the deformation and initial occurrence condition of a pipeline global buckling. Among these, the displacement to reach the maximum soil resistance has little effect on the lateral global buckling of a pipeline. However, the buckling amplitude, total buckling deformation length, critical buckling force and initial occurrence condition of the pipeline global buckling are affected by the displacement to reach the residual soil resistance. When the global buckling occurs, this effect is more significant when the pipeline midpoint is in the downward stage of the pipe-soil interaction model.