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Static and dynamic analysis on upheaval buckling of unburied subsea pipelines
Ocean Engineering 104 (2015) 249–256
Contents lists available at ScienceDirect
Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng
Static and dynamic analysis on upheaval buckling of unburied subsea pipelines Zhe Wang a, Zhi Huachen a,b,n, Hongbo Liu a, Yidu Bu c a
Department of Civil Engineering, Tianjin University, Tianjin 300072, China Key Laboratory of Coast Civil Structure and Safety, Ministry of Education (Tianjin University), Tianjin 300072, China c Department of Civil and Environmental Engineering, Imperial College London, London SW7 2AZ, United Kingdom b
art ic l e i nf o
a b s t r a c t
Article history: Received 20 August 2014 Accepted 15 May 2015
Upheaval buckling is one of the most common problems threatening the safe operation of subsea pipelines, which is trigged by the increasing of temperature and inner pressure. In order to predict the critical buckling temperature and post buckling path of upheaval buckling, ABAQUS is used to build four kinds of numerical models, and they are static and dynamic models both in 2D and 3D. Two analysis procedures which combine the static and dynamic processes are applied to aforementioned models. The results show good agreement with existing test data. For snap upheaval buckling, pipelines have two different buckling modes. Such buckling modes are not found in experiments. In addition, only 3D dynamic model can catch such buckling modes. For bifurcation upheaval buckling, predicted buckling temperatures of those models are all acceptable with an error of 5%. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Subsea pipeline Upheaval buckling Snap buckling Finite element (FE)
1. Introduction Subsea pipelines are one of the most important parts of the marine oil industry and the design of such pipelines creates and has many engineering challenges and potential risks. Global buckling is a common phenomenon in pipelines which may cause failure such as excessive bending, fatigue and buckling propagation (Karampour and Albermani, 2014). With increasing mining depth of subsea oil, higher transport temperature and pressure are required to satisfy production requirements for crude oil. As a result, the global buckling problem is more likely to occur in deeper subsea pipelines. Generally, there are two types of global buckling modes for unburied subsea pipelines, and they are lateral buckling and upheaval buckling. Unburied subsea pipelines tend to buckle laterally because the critical buckling temperature of lateral buckling is lower than upheaval buckling (Karampour et al., 2013). However if the seabed holding pipelines has upheaval imperfections, upheaval buckling may occur and do more harm. In addition, upheaval buckling may go through dynamic response and transform to lateral buckling (Nystrom et al., 1997). Hobbs (1984) derived the analytical solution of upheaval buckling of ideal pipelines based on the research of Kerr (1976), who studied lateral buckling of railway tracks. Bringing in three types of seabed
n
Corresponding author. E-mail address: [email protected] (Z. Huachen).
http://dx.doi.org/10.1016/j.oceaneng.2015.05.019 0029-8018/& 2015 Elsevier Ltd. All rights reserved.
imperfections into the govern differential equation of pipelines, Ju and Kyriakides (1988) obtained the analytical solution of upheaval buckling force for imperfect pipelines. Maltby and Calladine (1995a, 1995b) considered the nonlinear pipe–soil interaction and presented a simple formula for the axial load at which upheaval buckling occurs. In order to verify the solution, they carried out a small-scale upheaval buckling experiment which used a screw arrangement and interior oil pressure to trigger buckling. Taylor and Tran (1996) studied upheaval buckling of unburied pipelines by analytical and experimental methods. They discussed the snap upheaval buckling of pipelines. Aforesaid researches covered some basic factors which affect buckling of pipelines, including initial imperfections, pipe–soil interactions and critical temperatures (or axial loads). However those solutions cannot fully explain buckling transformation and hardly catch the dynamic post-buckling path. Traditional analytical method assumes that pipelines go through small and linear deformation; such assumption is invalid when considering post-buckling behavior of upheaval buckling. Considering geometric and material nonlinear factors, some researchers studied global buckling of pipelines by Finite Element Method. Jukes et al. (2009) reported a highly nonlinear FE program based on ABAQUS, which can be used to analysis complex pipeline responses including upheaval buckling. Liu et al. (2014a) studied dynamic lateral buckling of pipeline by 3D explicit method without taking loading time into consideration. In this study, two analysis procedures are proposed which simulate the embedded process and upheaval buckling of pipelines. A static method is used in the first procedure and dynamic
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method is used in the second procedure uses. Four finite element models of pipelines are established by ABAQUS including 2D static model, 2D dynamic model, 3D static model and 3D dynamic model. First, the analysis results obtained using the 2D models are compared with experimental data given by Taylor and Tran (Maltby and Calladine, 1995a). Then two typical upheaval buckling phenomena (snap buckling and bifurcation buckling) of imperfect pipelines are discussed in details.
2. Numerical models 2.1. Create FE models First, the element type is selected. The subsea pipeline is an ultra-slender structure which means the length of one direction (pipeline axial direction) is larger than the length of another two directions (pipeline cross-section). Therefore, the beam element is suitable for simulation of pipeline structures. The seabed is assumed to be a rigid surface, which is a common assumption in the traditional analytical method (Liu et al., 2014b). Thus rigid surface element is selected to simulate the seabed. Second, the length of the pipeline is determined. The global buckling of pipelines is characterized by localization (Ju and Kyriakides, 1988), which means buckling occurs only in a certain range of pipeline length and does not influence the whole pipeline. Therefore, instead of establishing the entire pipeline, only the part affected by buckling needs to be built. In this study, the length of the pipeline is as same as that in Taylor and Tran (1996). Third, the end constrains of the pipeline are determined. Both ends of the pipeline are pinned (Liu et al., 2014a) because the pipeline is long enough to avoid effect of constrains. Fourth, the triggering method of buckling is selected. The increase of temperature and inner pressure are the main factors triggering upheaval buckling. However, the triggering mechanism of inner pressure is more complicated than temperature (Knut Vedelda et al., 2014). The increase of temperature is selected to be the loading method, which simplifies the problem. Fifth, the initial imperfection of the pipeline is induced. It is assumed that the initial imperfection of the pipeline is only induced by imperfect profile of the seabed. An isolated imperfection is assumed here, which is a kind of symmetry initial imperfection (Hobbs, 1984). Based on these principles, four FE models are established (see Table 1) by ABAQUS. Models 1 and 2 are two dimensional models and Model 3 and 4 are three dimensional models. For model 1 and 2, the pipeline is simulated by B21 element, a 2-node 2D linear beam element. R2D2 element, a 2-node 2D line rigid element, is used to simulate the seabed. For models 3 and 4, the pipeline is simulated by B31 element, a 3-node 3D linear beam element. R3D4 element, a 4-node 3D surface rigid element, is used to simulate the seabed. Pipeline mechanical behavior is assumed to be linear-elastic. The friction model is the Coulomb friction model. For 2D models (Model 1 and 2), only axial friction coefficient is considered. For 3D models (Model 3 and 4), both axial and lateral friction coefficients are considered. The initial imperfection of the pipeline is applied according to the analysis procedure discussed later. 2.2. Analysis procedure Before unfolding the details of the analysis procedure, it is appropriate to make a simple description of two method types of solving nonlinear problems. The first type is static method, such as arc-length method (modified Riks method in ABAQUS) and Newton–Raphson method. Arc-length method is more suitable for
highly nonlinear buckling problems because it is easy to deal with the negative stiffness which may occur in some buckling analyses (e.g. snap buckling). The second type is dynamic method such as implicit dynamic and explicit dynamic method. Implicit and explicit dynamic methods are all adaptive to catch dynamic effect of buckling problems. Implicit dynamic analysis is able to link with a static analysis, which explicit analysis cannot do. The computational expense of explicit analysis is higher than implicit analysis (Hibbitt et al., 2000). Both analysis procedures (Procedure 1 and 2) are divided into two steps, the in place step and the upheaval buckling step, as shown in Fig. 1. The seabed is established with vertical imperfection while the pipeline is ideal at the beginning of the analysis procedure (see Fig. 2). The in place step for Procedure 1 is as same as Procedure 2, which is moving the pipeline downward to the seabed and applying gravity (vertical load). To make sure that the ends of the pipeline touch the seabed. At the end of in place step, reliable contact between the pipeline and the seabed is built and initial imperfection of the pipeline occurs due to the imperfection of the seabed (see Fig. 3). Newton–Raphson method is used in the in place step because it is an efficient way to calculate moderate nonlinear behaviors. When it comes to Upheaval buckling step, Procedure 1 and 2 are different. For Procedure 1, a static analysis is carried out by the arc-length method, which can calculate the post-buckling path of pipelines. For Procedure 2, a dynamic analysis is put into use by the implicit dynamic method, which can catch both post-buckling path and dynamic effect of pipelines. The damping factor for the implicit dynamic analysis is 0.05 (Hibbitt et al., 2000; Kyriakides and Netto, 2000). In other words, Procedure 1 is a combination of two static analyses while a static analysis and a dynamic analysis comprise Procedure 2. In following studies, Procedure 1 is applied to Model 1 and 3 and Procedure 2 is applied to Model 2 and 4 (see Table 1). 2.3. Verification The computation results obtained using Model 1 in this study is compared with classic test results (Taylor and Tran, 1996) to verify the accuracy of the proposed methods. Table 2 shows basic parameters of the test. Test 1–6 and Test 25–30 are used as prototypes of Model 1. The amplitude of initial vertical imperfections (v0) for Test 1–6 and Test 25–30 are 30 mm and 2 mm, respectively. The reason to choose these two tests is that experiment phenomena of these two tests represent two typical Table 1 Four numerical models. Model 1 Element type Seabed type Friction type Analysis procedure
In place step of pipelines
Model 2
B21 B21 Rigid line Coulomb friction model Procedure 1 Procedure 2
Thermal increase
Static analysis Newton-Raphson method
Model 3
Model 4
B31 B31 Rigid surface Procedure 1
Procedure 2
Upheaval buckling step of pipelines
Static analysis Arc-length method
Procedure 1 (Model1, 3)
Transient analysis Implicit dynamic method
Procedure 2 (Model 2, 4)
Fig. 1. Analysis procedure.
Z. Wang et al. / Ocean Engineering 104 (2015) 249–256
upheaval buckling types, say snap buckling and bifurcation buckling. Based on the steps in Section 3.1 and Table 1, the pipeline is simulated by 60 B21 elements and R2D2 element is used to build the seabed, as shown in Fig. 2. The coefficient of friction between pipelines and the seabed is 0.2. First, the in place step of Procedure 1 is applied to Model 1. The pipeline moves downward to the seabed and gravity is applied. At the end of this step, initial imperfection of pipelines is established (both 2 mm and 30 mm imperfections), as shown in Fig. 3a and b. Fig. 3a illustrates that the end of the pipeline is pinned and the pipeline is symmetrical. Fig. 3b shows a zooming picture of 2 mm imperfection because the imperfection amplitude of Test 25–30 is very small. Second, the temperature load is applied to the pipeline and arc-length method is used to analyze the buckling and postbuckling response. Two key points here: 1. Pipelines with relative large imperfection amplitude, say 30 mm, have a stable buckling path. Therefore, Newton–Raphson method is also suitable for upheaval buckling
Fig. 2. 2D FE models (at the beginning of in place step).
step. 2. The peak point of the seabed should be modeled by arcs rather than straight lines (see Fig. 3b) to avoid excessive angle between rigid elements which may cause convergence problems. Fig. 4 shows simulated and experimental buckling path of Test 4 and Test 27. As shown in Fig. 4a, when the amplitude of initial imperfection is large, bifurcation buckling occurs. The critical buckling temperatures of the tests and the simulation are 5.9 1C and 6.03 1C, respectively. After the bifurcation point, the buckling amplitude increases with the growth of temperature. Although the appearance of both curves is similar to each other, the buckling amplitude of Model 1 grows slowly than test values. Fig. 4b reflects a snap buckling path which has a descending branch. The critical buckling temperatures of the tests and the simulation are 9.14 1C and 9.19 1C, respectively. In the snap process, the buckle amplitude grows rapidly. However, it is difficult to monitor the snap process in the experiment, so the curve representing test data is discontinuous. The curve representing data of Model 1 shows good agreement with the test data and describes the snap process. Nevertheless, the buckling path calculated by Model 1 is not a real path because the snap process is in fact a dynamic process which is not appropriate to be explained by static methods as we will discuss later in this paper. Comparing to test data, results obtained from finite element models shows good accuracy. As a result, Model 1 can be used as the basis of other models in the following analysis. 2D and 3D snap buckling are discussed separately since different buckling modes occur in 2D and 3D models. Then Models 1–4 for bifurcation buckling are studied in the same section because there is no essential distinction between them.
Fig. 3. 2D FE models (at the end of in place step) (b) 2 mm imperfection (Zoom in the imperfection) (a) 30 mm imperfection.
Table 2 Test parameters of pipelines. Diameter (mm) Thickness (mm) Young’s modulus (MPa) Yield stress (MPa) Length of pipelines (mm) Coefficient of linear expansion (1C 1) Poisson’s ratio Coefficient of friction (Axial) Coefficient of friction (Lateral) only for 3D model
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9.53 1.6 195,000 117 6000 1.10E 05 0.3 0.2 0.2 Fig. 5. The influence of loading time on the critical buckling temperature.
Fig. 4. Comparison of tests and Model 1 (a) Test 4 (v0 ¼ 30 mm, bifurcation buckling). (b) Test 27 (v0 ¼ 2 mm, snap buckling).
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3. 2D snap buckling 3.1. The influence of loading time Loading time is a significant factor in dynamic analysis especially when dynamic effect is prominent (Liu et al., 2012). As a result, the influence of loading time on snap buckling is discussed first. The same pipeline prototype (Taylor and Tran, 1996) is used in this Section. A 2D dynamic model (Model 2) is established and Procedure 2 is applied to Model 2. The loading time for implicit dynamic method is set as 1 s–1200 s.
Fig. 5 shows the relationship of loading time and the buckling temperature. As it can be seen in Fig. 5, the critical buckling temperature decreases with the growth in loading time. When loading time is longer than 40 s (the heating speed is 0.5 1C=s), the critical buckling temperature is convergent at 10.03 1C. The influence of loading time on the critical buckling path is shown in Fig. 6. The oscillation decay fast with the increase of loading time. It is noticeable that when the loading period is too short, 1 s for example; the pipeline experiences dynamic responses in the whole period of loading time.
Fig. 6. The influence of loading time on the buckling path.
Fig. 9. Axial force distribution in upheaval buckling (Model 1 and 2) (Total energy ¼Strain energy þ Kinetic energy External work)(Hibbitt et al., 2000).
Fig. 7. The buckling path of Model 1 and 2.
Fig. 10. Axial compressive stress distribution in upheaval buckling (Model 1 and 2).
Fig. 8. Energy change in upheaval Buckling (Model 2).
Z. Wang et al. / Ocean Engineering 104 (2015) 249–256
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3.2. Results of 2D static and dynamic models
Fig. 11. 3D FE models.
Fig. 12. The influence of loading time on the buckling temperature.
Based on the analysis in Section 3.1, a typical loading time for Model 2 is set as 60 s which results in a steady buckling temperature. Fig. 7 shows the buckling path of Model 1 and 2. As shown in this figure, the critical buckling temperature of Model 2 (dynamic model) is slightly larger than Model 1 (static model), which are 10.03 1C and 9.19 1C, respectively. However, the buckling paths are quite different after the bifurcation point. For Model 1, the buckling amplitude increases rapidly from 2 mm to about 27 mm. In this process, the temperature falls down at the beginning of the buckling, then gain rise again. For Model 2, the buckling amplitude also increases rapidly after the bifurcation point, and then goes through a fluctuant growth, ending with a steady increase. The max fluctuant amplitude, however, is about 38 mm, which is 31% larger than that of Model 1. In addition, the temperature keeps stable when buckling occurs which is more realistic than Model 1. At the end of snap buckling process, both buckling paths show the same tendency. In order to understand the snap buckling process of pipelines, energy change in snap upheaval buckling process is analyzed as shown in Fig. 8. It can be seen that snap buckling process is a releasing process of strain energy. Strain energy of the system accelerates in the form of a quadratic function before the critical buckling temperature. In this period, total energy of the system is comprised of strain energy and a bit of positive external work. When the temperature reaches the critical point, strain energy of the system transforms into kinetic energy which means the pipeline moves upward rapidly. As a result, the gravity does negative work and total energy of the system decreases. Finally, the energy distribution tends to be stable after a short period of fluctuation. Fig. 9 shows the axial force distribution against the temperature. This figure describes how the two Models release axial force.
Fig. 13. The influence of loading time on the critical buckling path.
Fig. 14. Different buckling modes of Model 4 (a) Buckling Mode 1. (1) Upheaval buckling (2) Lateral buckling. (b) Buckling Mode 1.
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For the reason of differences in critical temperatures, the peak axial force of Model 1 ( 778 N) is smaller than Model 2 ( 850 N). Model 1 and 2 have very similar post-buckling axial force. For Model 1, however, the temperature decreases in the releasing process of axial force, which seems to be unrealistic. This may because that in Model 1, a static model, there is no way to transform strain energy into kinetic energy. Fig. 10 shows the axial compressive stress of integration point 1 during the buckling process. As can be seen in Fig. 10, the max stress of Model 2 is 53.6 MPa, which is approximately 56% larger than that of Model 1. The stresses of the two models are similar to each other in postbuckling process.
4. 3D snap buckling 4.1. The influence of loading rate Similar to Section 3, 3D static and dynamic models (Model 3 and 4) are established in this section as shown in Fig. 11. Procedure 1 and 2 are applied to Model 3 and 4, respectively. For Model 4, the time period for implicit dynamic method is also set as 1 s–1200 s with the max temperature 20 1C to study the influence of loading period on buckling response of the pipeline. As show in Fig. 12, the critical buckling temperature decreases with the growth in loading time and the steady critical buckling temperature is 10.01 1C.
Fig. 15. The buckling path of Model 3 and 4.
Fig. 13 illustrates the influence of loading time on the buckling path. There are two buckling modes with the increase of loading time. Buckling mode 1: For loading time which is less than 7 s (the heating speed is larger than 2.85 1C/s and larger than 75 s (the heating speed is less than 0.25 1C/s), the pipeline only buckles vertically as shown in Fig. 14a. Buckling mode 2: For other loading period, the pipeline first buckles vertically (see Fig. 14b1) then falls back to the seabed and buckles laterally (Fig. 14b2). When the loading period is less than 7 s, 1 s for example; the pipeline experiences a great dynamic response in vertical direction during the whole loading period. When the loading period is between 7 s and 75 s, 10 s for example; buckling mode 1 occurs. The buckling amplitude increases rapidly when the temperature touches the critical value. After a short period it falls back to the seabed moving laterally at the same time. It is obvious that 2D models cannot catch the buckling transformation because there is no lateral degree of freedom in these models. When the loading period is larger than 75 s, the buckling path of the pipelines is similar to 2D dynamic models. The oscillation decays faster with the increase in loading period. The amplitude of oscillation with different loading period shows similarity. 4.2. Results of 3D static and dynamic models The typical loading time in this section is 600 s because it leads to a steady critical buckling temperature on the basis of results in Section 4.1. Fig. 15 shows the buckling path of model 3 and 4. The critical buckling temperature of Model 3 is 8.28 1C, which is lower
Fig. 17. Axial force distribution in upheaval buckling (Model 3 and 4).
Fig. 16. Energy change in upheaval buckling (Model 4).
Z. Wang et al. / Ocean Engineering 104 (2015) 249–256
than that of Model 4, 10.01 1C. The max fluctuant amplitude of Model 4 is about 40 mm, which is 43% larger than that of Model 3, 28 mm. Energy change in snap upheaval buckling process of Model 4 is similar to 2D dynamic models as shown in Fig. 16. Fig. 17 shows the axial force distribution of 3D models against temperature. The trend of axial force distribution of 3D models looks similar to that of 2D models’. The peak axial force of Model 3 ( 706 N) is smaller than Model 4 ( 851 N), and Model 3 and 4 have very similar postbuckling axial force. Fig. 18 shows the axial compressive stress of integration point 1 during the buckling process. As shown in
Fig. 18. Axial compressive stress distribution in upheaval buckling (Model 3 and 4).
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Fig. 18, when upheaval buckling occurs in Model 4, the stress of integral point 1 grows rapidly to about 53 MPa which is 57% larger than that of Model 3.
5. Bifurcation buckling For pipelines with larger initial imperfection (for example v0 ¼ 30 mm), bifurcation upheaval buckling occurs. Model 1–4 are analyzed based on the pipeline model mentioned in Section 2. The loading time for Model 2 and 4 is 30 s. Unlike models for pipelines with small imperfections, different models for pipelines with large imperfections show similar structure response as illustrated in Fig. 19. The critical buckling temperatures of Model 1–4 are 6.03 1C, 6.33 1C, 6.2 1C and 6.55 1C, respectively. The critical buckling temperature of 3D models is larger than 2D models and the critical buckling temperature of dynamic models is larger than static models. The axial compressive stress of the pipeline shows similar pattern with buckling path as shown in Fig. 19b. The energy change in bifurcation buckling is shown in Fig. 20. As shown in this figure, there is no oscillation and kinetic energy in the buckling process which means the dynamic effect in bifurcation buckling is very small. Strain energy of the system accelerates in the form of a quadratic function before the critical buckling temperature. Then the gravity does negative work which indicates that the pipeline moves upwards.
Fig. 19. Structural response of bifurcation buckling (a) Buckling path. (b) Axial compressive stress.
Fig. 20. Energy change in bifurcation upheaval buckling (Model 2 and 4) (a) Model 2. (b) Model 4.
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6. Conclusions
References
This study aims to study the static and dynamic responses of upheaval buckling. Two analysis procedures are proposed, which put in place step and upheaval buckling step into a single procedure. Procedure 1 links Newton–Raphson and arc-length method. Procedure 2 links static and dynamic nonlinear analysis seamlessly which effectively solves the convergence problem when dealing with snap buckling. In addition, Procedure 2 can capture the dynamic process of snap buckling. Procedures 1 and 2 are applied to 4 finite element models, 2D/3D static/dynamic models (Model 1–4). Notable conclusions are summarized as follows:
Hibbitt, H.D., Karlsson, B.I., Sorensen, P., 2000. ABAQUS theoretical manual. Version 4, 5. Hobbs, R.E., 1984. In-service buckling of heated pipelines. . J. Transp. Eng. 110 (2), 175–189. Ju, G.T., Kyriakides, S., 1988. Thermal buckling of offshore pipelines. J. Offshore Mech. Arct. Eng. 110 (4), 355–364. Jukes, P., Eltaher, A., Sun, J. (2009). The latest developments in the design and simulation of deepwater subsea oil and gas pipelines using FEA. In: The Proceedings of The Third (2009) ISOPE International Deep-Ocean Technology Symposium: Deepwater Challenge, IDOT-2009, International Society of Offshore and Polar Engineers, United States, pp. 70–82. Karampour, H., Albermani, F., Gross, J., 2013. On lateral and upheaval buckling of subsea pipelines. Eng. Struct. 52, 317–330. Karampour, Hassan, Albermani, Faris, 2014. . Experimental and numerical investigations of buckle interaction in subsea pipelines. Eng. Struct. 66, 81–88. Kerr A.D. On Thermal Buckling of Straight Railroad Tracks and the Effect of Track Length on the Track Response. 1976. Knut Vedelda, Havar A. Sollunda, Helleslanda, Jostein, et al., 2014. Effective axial forces in offshore lined and clad pipes. . Eng. Struct. 66, 66–80. Kyriakides, S., Netto, T.A., 2000. On the dynamics of propagating buckles in pipelines. Int. J. Solids Struct. 37 (46), 6843–6867. Liu, R., Wang, W.G., Yan, S.W., Wu, X.L., 2012. Engineering measures for preventing upheaval buckling of buried submarine pipelines. Appl. Math. Mech. 33, 781–796. Liu, R.u.n., Hao, Xiong, Xilin, W.u., et al., 2014a. Numerical studies on global buckling of subsea pipelines. Ocean Eng. 78, 62–72. Liu, Run, Liu, Wenbin, Xinli, W.u., Shuwang, Y.a.n., 2014b. Global lateral buckling analysis of idealized subsea pipelines. J. Cent. South Univ. Technol. 21 (1), 416–427. Maltby, Timothy Chrichton, Calladine, Christopher Reuben, 1995a. An investigation into upheaval buckling of buried pipelines—II. Theory and analysis of experimental observations. Int. J. Mech. Sci. 37 (9), 965–983. Maltby, Timothy Chrichton, Calladine, Christopher Reuben, 1995b. An investigation into upheaval buckling of buried pipelines—I. Experimental apparatus and some observations. Int. J. Mech. Sci. 37 (9), 943–963. Nystrom, Per R., et al., 1997. 3-D dynamic buckling and cyclic behavior of HP/HT flowlines // ISOPE. In: The Seventh International Offshore and Polar Engineering Conference. International Society of Offshore and Polar Engineers (ISOPE). International Society Offshore & Polar Engineers Cupertino, Honolulu, pp. 299–307. Taylor, Neil, Tran, Vinh, 1996. Experimental and theoretical studies in subsea pipeline buckling. Mar. Struct. 9 (2), 211–257.
(1) Newton–Raphson method is suitable for simulating the in place step of the pipeline and at the end of this step the imperfection is applied to the pipeline. When it comes to snap buckling step, however, arc-length method or dynamic method is more effective. In addition, for the buckling process which shows apparent dynamic phenomenon, say snap buckling, dynamic method is able to describe the real buckling process. (2) In 3D dynamic analysis, pipelines with small imperfections may turn into 2 different buckling modes depending on the loading time. It is recommended to apply a long loading period or practical loading period to obtain a steady buckling temperature. (3) By studying the energy distribution of the pipeline by dynamic method, the snap buckling process can be understood as transforming strain energy into kinetic energy and other energy. (4) For snap buckling, the critical buckling temperature obtained by dynamic method is larger than by static method and the critical buckling temperature obtained by 3D models is smaller than by 2D models. The max axial compressive stress of dynamic analysis is about 50% larger than that of static analysis. For bifurcation buckling, the critical buckling temperature obtained by 3D models is larger than by 2D models. Axial compressive stress of dynamic analysis is slightly larger than that of static analysis.
Acknowledgment The authors are grateful for the support provided by the National Basic Research Program of China (no. 2014CB046801).
Contents lists available at ScienceDirect
Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng
Static and dynamic analysis on upheaval buckling of unburied subsea pipelines Zhe Wang a, Zhi Huachen a,b,n, Hongbo Liu a, Yidu Bu c a
Department of Civil Engineering, Tianjin University, Tianjin 300072, China Key Laboratory of Coast Civil Structure and Safety, Ministry of Education (Tianjin University), Tianjin 300072, China c Department of Civil and Environmental Engineering, Imperial College London, London SW7 2AZ, United Kingdom b
art ic l e i nf o
a b s t r a c t
Article history: Received 20 August 2014 Accepted 15 May 2015
Upheaval buckling is one of the most common problems threatening the safe operation of subsea pipelines, which is trigged by the increasing of temperature and inner pressure. In order to predict the critical buckling temperature and post buckling path of upheaval buckling, ABAQUS is used to build four kinds of numerical models, and they are static and dynamic models both in 2D and 3D. Two analysis procedures which combine the static and dynamic processes are applied to aforementioned models. The results show good agreement with existing test data. For snap upheaval buckling, pipelines have two different buckling modes. Such buckling modes are not found in experiments. In addition, only 3D dynamic model can catch such buckling modes. For bifurcation upheaval buckling, predicted buckling temperatures of those models are all acceptable with an error of 5%. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Subsea pipeline Upheaval buckling Snap buckling Finite element (FE)
1. Introduction Subsea pipelines are one of the most important parts of the marine oil industry and the design of such pipelines creates and has many engineering challenges and potential risks. Global buckling is a common phenomenon in pipelines which may cause failure such as excessive bending, fatigue and buckling propagation (Karampour and Albermani, 2014). With increasing mining depth of subsea oil, higher transport temperature and pressure are required to satisfy production requirements for crude oil. As a result, the global buckling problem is more likely to occur in deeper subsea pipelines. Generally, there are two types of global buckling modes for unburied subsea pipelines, and they are lateral buckling and upheaval buckling. Unburied subsea pipelines tend to buckle laterally because the critical buckling temperature of lateral buckling is lower than upheaval buckling (Karampour et al., 2013). However if the seabed holding pipelines has upheaval imperfections, upheaval buckling may occur and do more harm. In addition, upheaval buckling may go through dynamic response and transform to lateral buckling (Nystrom et al., 1997). Hobbs (1984) derived the analytical solution of upheaval buckling of ideal pipelines based on the research of Kerr (1976), who studied lateral buckling of railway tracks. Bringing in three types of seabed
n
Corresponding author. E-mail address: [email protected] (Z. Huachen).
http://dx.doi.org/10.1016/j.oceaneng.2015.05.019 0029-8018/& 2015 Elsevier Ltd. All rights reserved.
imperfections into the govern differential equation of pipelines, Ju and Kyriakides (1988) obtained the analytical solution of upheaval buckling force for imperfect pipelines. Maltby and Calladine (1995a, 1995b) considered the nonlinear pipe–soil interaction and presented a simple formula for the axial load at which upheaval buckling occurs. In order to verify the solution, they carried out a small-scale upheaval buckling experiment which used a screw arrangement and interior oil pressure to trigger buckling. Taylor and Tran (1996) studied upheaval buckling of unburied pipelines by analytical and experimental methods. They discussed the snap upheaval buckling of pipelines. Aforesaid researches covered some basic factors which affect buckling of pipelines, including initial imperfections, pipe–soil interactions and critical temperatures (or axial loads). However those solutions cannot fully explain buckling transformation and hardly catch the dynamic post-buckling path. Traditional analytical method assumes that pipelines go through small and linear deformation; such assumption is invalid when considering post-buckling behavior of upheaval buckling. Considering geometric and material nonlinear factors, some researchers studied global buckling of pipelines by Finite Element Method. Jukes et al. (2009) reported a highly nonlinear FE program based on ABAQUS, which can be used to analysis complex pipeline responses including upheaval buckling. Liu et al. (2014a) studied dynamic lateral buckling of pipeline by 3D explicit method without taking loading time into consideration. In this study, two analysis procedures are proposed which simulate the embedded process and upheaval buckling of pipelines. A static method is used in the first procedure and dynamic
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method is used in the second procedure uses. Four finite element models of pipelines are established by ABAQUS including 2D static model, 2D dynamic model, 3D static model and 3D dynamic model. First, the analysis results obtained using the 2D models are compared with experimental data given by Taylor and Tran (Maltby and Calladine, 1995a). Then two typical upheaval buckling phenomena (snap buckling and bifurcation buckling) of imperfect pipelines are discussed in details.
2. Numerical models 2.1. Create FE models First, the element type is selected. The subsea pipeline is an ultra-slender structure which means the length of one direction (pipeline axial direction) is larger than the length of another two directions (pipeline cross-section). Therefore, the beam element is suitable for simulation of pipeline structures. The seabed is assumed to be a rigid surface, which is a common assumption in the traditional analytical method (Liu et al., 2014b). Thus rigid surface element is selected to simulate the seabed. Second, the length of the pipeline is determined. The global buckling of pipelines is characterized by localization (Ju and Kyriakides, 1988), which means buckling occurs only in a certain range of pipeline length and does not influence the whole pipeline. Therefore, instead of establishing the entire pipeline, only the part affected by buckling needs to be built. In this study, the length of the pipeline is as same as that in Taylor and Tran (1996). Third, the end constrains of the pipeline are determined. Both ends of the pipeline are pinned (Liu et al., 2014a) because the pipeline is long enough to avoid effect of constrains. Fourth, the triggering method of buckling is selected. The increase of temperature and inner pressure are the main factors triggering upheaval buckling. However, the triggering mechanism of inner pressure is more complicated than temperature (Knut Vedelda et al., 2014). The increase of temperature is selected to be the loading method, which simplifies the problem. Fifth, the initial imperfection of the pipeline is induced. It is assumed that the initial imperfection of the pipeline is only induced by imperfect profile of the seabed. An isolated imperfection is assumed here, which is a kind of symmetry initial imperfection (Hobbs, 1984). Based on these principles, four FE models are established (see Table 1) by ABAQUS. Models 1 and 2 are two dimensional models and Model 3 and 4 are three dimensional models. For model 1 and 2, the pipeline is simulated by B21 element, a 2-node 2D linear beam element. R2D2 element, a 2-node 2D line rigid element, is used to simulate the seabed. For models 3 and 4, the pipeline is simulated by B31 element, a 3-node 3D linear beam element. R3D4 element, a 4-node 3D surface rigid element, is used to simulate the seabed. Pipeline mechanical behavior is assumed to be linear-elastic. The friction model is the Coulomb friction model. For 2D models (Model 1 and 2), only axial friction coefficient is considered. For 3D models (Model 3 and 4), both axial and lateral friction coefficients are considered. The initial imperfection of the pipeline is applied according to the analysis procedure discussed later. 2.2. Analysis procedure Before unfolding the details of the analysis procedure, it is appropriate to make a simple description of two method types of solving nonlinear problems. The first type is static method, such as arc-length method (modified Riks method in ABAQUS) and Newton–Raphson method. Arc-length method is more suitable for
highly nonlinear buckling problems because it is easy to deal with the negative stiffness which may occur in some buckling analyses (e.g. snap buckling). The second type is dynamic method such as implicit dynamic and explicit dynamic method. Implicit and explicit dynamic methods are all adaptive to catch dynamic effect of buckling problems. Implicit dynamic analysis is able to link with a static analysis, which explicit analysis cannot do. The computational expense of explicit analysis is higher than implicit analysis (Hibbitt et al., 2000). Both analysis procedures (Procedure 1 and 2) are divided into two steps, the in place step and the upheaval buckling step, as shown in Fig. 1. The seabed is established with vertical imperfection while the pipeline is ideal at the beginning of the analysis procedure (see Fig. 2). The in place step for Procedure 1 is as same as Procedure 2, which is moving the pipeline downward to the seabed and applying gravity (vertical load). To make sure that the ends of the pipeline touch the seabed. At the end of in place step, reliable contact between the pipeline and the seabed is built and initial imperfection of the pipeline occurs due to the imperfection of the seabed (see Fig. 3). Newton–Raphson method is used in the in place step because it is an efficient way to calculate moderate nonlinear behaviors. When it comes to Upheaval buckling step, Procedure 1 and 2 are different. For Procedure 1, a static analysis is carried out by the arc-length method, which can calculate the post-buckling path of pipelines. For Procedure 2, a dynamic analysis is put into use by the implicit dynamic method, which can catch both post-buckling path and dynamic effect of pipelines. The damping factor for the implicit dynamic analysis is 0.05 (Hibbitt et al., 2000; Kyriakides and Netto, 2000). In other words, Procedure 1 is a combination of two static analyses while a static analysis and a dynamic analysis comprise Procedure 2. In following studies, Procedure 1 is applied to Model 1 and 3 and Procedure 2 is applied to Model 2 and 4 (see Table 1). 2.3. Verification The computation results obtained using Model 1 in this study is compared with classic test results (Taylor and Tran, 1996) to verify the accuracy of the proposed methods. Table 2 shows basic parameters of the test. Test 1–6 and Test 25–30 are used as prototypes of Model 1. The amplitude of initial vertical imperfections (v0) for Test 1–6 and Test 25–30 are 30 mm and 2 mm, respectively. The reason to choose these two tests is that experiment phenomena of these two tests represent two typical Table 1 Four numerical models. Model 1 Element type Seabed type Friction type Analysis procedure
In place step of pipelines
Model 2
B21 B21 Rigid line Coulomb friction model Procedure 1 Procedure 2
Thermal increase
Static analysis Newton-Raphson method
Model 3
Model 4
B31 B31 Rigid surface Procedure 1
Procedure 2
Upheaval buckling step of pipelines
Static analysis Arc-length method
Procedure 1 (Model1, 3)
Transient analysis Implicit dynamic method
Procedure 2 (Model 2, 4)
Fig. 1. Analysis procedure.
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upheaval buckling types, say snap buckling and bifurcation buckling. Based on the steps in Section 3.1 and Table 1, the pipeline is simulated by 60 B21 elements and R2D2 element is used to build the seabed, as shown in Fig. 2. The coefficient of friction between pipelines and the seabed is 0.2. First, the in place step of Procedure 1 is applied to Model 1. The pipeline moves downward to the seabed and gravity is applied. At the end of this step, initial imperfection of pipelines is established (both 2 mm and 30 mm imperfections), as shown in Fig. 3a and b. Fig. 3a illustrates that the end of the pipeline is pinned and the pipeline is symmetrical. Fig. 3b shows a zooming picture of 2 mm imperfection because the imperfection amplitude of Test 25–30 is very small. Second, the temperature load is applied to the pipeline and arc-length method is used to analyze the buckling and postbuckling response. Two key points here: 1. Pipelines with relative large imperfection amplitude, say 30 mm, have a stable buckling path. Therefore, Newton–Raphson method is also suitable for upheaval buckling
Fig. 2. 2D FE models (at the beginning of in place step).
step. 2. The peak point of the seabed should be modeled by arcs rather than straight lines (see Fig. 3b) to avoid excessive angle between rigid elements which may cause convergence problems. Fig. 4 shows simulated and experimental buckling path of Test 4 and Test 27. As shown in Fig. 4a, when the amplitude of initial imperfection is large, bifurcation buckling occurs. The critical buckling temperatures of the tests and the simulation are 5.9 1C and 6.03 1C, respectively. After the bifurcation point, the buckling amplitude increases with the growth of temperature. Although the appearance of both curves is similar to each other, the buckling amplitude of Model 1 grows slowly than test values. Fig. 4b reflects a snap buckling path which has a descending branch. The critical buckling temperatures of the tests and the simulation are 9.14 1C and 9.19 1C, respectively. In the snap process, the buckle amplitude grows rapidly. However, it is difficult to monitor the snap process in the experiment, so the curve representing test data is discontinuous. The curve representing data of Model 1 shows good agreement with the test data and describes the snap process. Nevertheless, the buckling path calculated by Model 1 is not a real path because the snap process is in fact a dynamic process which is not appropriate to be explained by static methods as we will discuss later in this paper. Comparing to test data, results obtained from finite element models shows good accuracy. As a result, Model 1 can be used as the basis of other models in the following analysis. 2D and 3D snap buckling are discussed separately since different buckling modes occur in 2D and 3D models. Then Models 1–4 for bifurcation buckling are studied in the same section because there is no essential distinction between them.
Fig. 3. 2D FE models (at the end of in place step) (b) 2 mm imperfection (Zoom in the imperfection) (a) 30 mm imperfection.
Table 2 Test parameters of pipelines. Diameter (mm) Thickness (mm) Young’s modulus (MPa) Yield stress (MPa) Length of pipelines (mm) Coefficient of linear expansion (1C 1) Poisson’s ratio Coefficient of friction (Axial) Coefficient of friction (Lateral) only for 3D model
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9.53 1.6 195,000 117 6000 1.10E 05 0.3 0.2 0.2 Fig. 5. The influence of loading time on the critical buckling temperature.
Fig. 4. Comparison of tests and Model 1 (a) Test 4 (v0 ¼ 30 mm, bifurcation buckling). (b) Test 27 (v0 ¼ 2 mm, snap buckling).
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3. 2D snap buckling 3.1. The influence of loading time Loading time is a significant factor in dynamic analysis especially when dynamic effect is prominent (Liu et al., 2012). As a result, the influence of loading time on snap buckling is discussed first. The same pipeline prototype (Taylor and Tran, 1996) is used in this Section. A 2D dynamic model (Model 2) is established and Procedure 2 is applied to Model 2. The loading time for implicit dynamic method is set as 1 s–1200 s.
Fig. 5 shows the relationship of loading time and the buckling temperature. As it can be seen in Fig. 5, the critical buckling temperature decreases with the growth in loading time. When loading time is longer than 40 s (the heating speed is 0.5 1C=s), the critical buckling temperature is convergent at 10.03 1C. The influence of loading time on the critical buckling path is shown in Fig. 6. The oscillation decay fast with the increase of loading time. It is noticeable that when the loading period is too short, 1 s for example; the pipeline experiences dynamic responses in the whole period of loading time.
Fig. 6. The influence of loading time on the buckling path.
Fig. 9. Axial force distribution in upheaval buckling (Model 1 and 2) (Total energy ¼Strain energy þ Kinetic energy External work)(Hibbitt et al., 2000).
Fig. 7. The buckling path of Model 1 and 2.
Fig. 10. Axial compressive stress distribution in upheaval buckling (Model 1 and 2).
Fig. 8. Energy change in upheaval Buckling (Model 2).
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3.2. Results of 2D static and dynamic models
Fig. 11. 3D FE models.
Fig. 12. The influence of loading time on the buckling temperature.
Based on the analysis in Section 3.1, a typical loading time for Model 2 is set as 60 s which results in a steady buckling temperature. Fig. 7 shows the buckling path of Model 1 and 2. As shown in this figure, the critical buckling temperature of Model 2 (dynamic model) is slightly larger than Model 1 (static model), which are 10.03 1C and 9.19 1C, respectively. However, the buckling paths are quite different after the bifurcation point. For Model 1, the buckling amplitude increases rapidly from 2 mm to about 27 mm. In this process, the temperature falls down at the beginning of the buckling, then gain rise again. For Model 2, the buckling amplitude also increases rapidly after the bifurcation point, and then goes through a fluctuant growth, ending with a steady increase. The max fluctuant amplitude, however, is about 38 mm, which is 31% larger than that of Model 1. In addition, the temperature keeps stable when buckling occurs which is more realistic than Model 1. At the end of snap buckling process, both buckling paths show the same tendency. In order to understand the snap buckling process of pipelines, energy change in snap upheaval buckling process is analyzed as shown in Fig. 8. It can be seen that snap buckling process is a releasing process of strain energy. Strain energy of the system accelerates in the form of a quadratic function before the critical buckling temperature. In this period, total energy of the system is comprised of strain energy and a bit of positive external work. When the temperature reaches the critical point, strain energy of the system transforms into kinetic energy which means the pipeline moves upward rapidly. As a result, the gravity does negative work and total energy of the system decreases. Finally, the energy distribution tends to be stable after a short period of fluctuation. Fig. 9 shows the axial force distribution against the temperature. This figure describes how the two Models release axial force.
Fig. 13. The influence of loading time on the critical buckling path.
Fig. 14. Different buckling modes of Model 4 (a) Buckling Mode 1. (1) Upheaval buckling (2) Lateral buckling. (b) Buckling Mode 1.
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For the reason of differences in critical temperatures, the peak axial force of Model 1 ( 778 N) is smaller than Model 2 ( 850 N). Model 1 and 2 have very similar post-buckling axial force. For Model 1, however, the temperature decreases in the releasing process of axial force, which seems to be unrealistic. This may because that in Model 1, a static model, there is no way to transform strain energy into kinetic energy. Fig. 10 shows the axial compressive stress of integration point 1 during the buckling process. As can be seen in Fig. 10, the max stress of Model 2 is 53.6 MPa, which is approximately 56% larger than that of Model 1. The stresses of the two models are similar to each other in postbuckling process.
4. 3D snap buckling 4.1. The influence of loading rate Similar to Section 3, 3D static and dynamic models (Model 3 and 4) are established in this section as shown in Fig. 11. Procedure 1 and 2 are applied to Model 3 and 4, respectively. For Model 4, the time period for implicit dynamic method is also set as 1 s–1200 s with the max temperature 20 1C to study the influence of loading period on buckling response of the pipeline. As show in Fig. 12, the critical buckling temperature decreases with the growth in loading time and the steady critical buckling temperature is 10.01 1C.
Fig. 15. The buckling path of Model 3 and 4.
Fig. 13 illustrates the influence of loading time on the buckling path. There are two buckling modes with the increase of loading time. Buckling mode 1: For loading time which is less than 7 s (the heating speed is larger than 2.85 1C/s and larger than 75 s (the heating speed is less than 0.25 1C/s), the pipeline only buckles vertically as shown in Fig. 14a. Buckling mode 2: For other loading period, the pipeline first buckles vertically (see Fig. 14b1) then falls back to the seabed and buckles laterally (Fig. 14b2). When the loading period is less than 7 s, 1 s for example; the pipeline experiences a great dynamic response in vertical direction during the whole loading period. When the loading period is between 7 s and 75 s, 10 s for example; buckling mode 1 occurs. The buckling amplitude increases rapidly when the temperature touches the critical value. After a short period it falls back to the seabed moving laterally at the same time. It is obvious that 2D models cannot catch the buckling transformation because there is no lateral degree of freedom in these models. When the loading period is larger than 75 s, the buckling path of the pipelines is similar to 2D dynamic models. The oscillation decays faster with the increase in loading period. The amplitude of oscillation with different loading period shows similarity. 4.2. Results of 3D static and dynamic models The typical loading time in this section is 600 s because it leads to a steady critical buckling temperature on the basis of results in Section 4.1. Fig. 15 shows the buckling path of model 3 and 4. The critical buckling temperature of Model 3 is 8.28 1C, which is lower
Fig. 17. Axial force distribution in upheaval buckling (Model 3 and 4).
Fig. 16. Energy change in upheaval buckling (Model 4).
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than that of Model 4, 10.01 1C. The max fluctuant amplitude of Model 4 is about 40 mm, which is 43% larger than that of Model 3, 28 mm. Energy change in snap upheaval buckling process of Model 4 is similar to 2D dynamic models as shown in Fig. 16. Fig. 17 shows the axial force distribution of 3D models against temperature. The trend of axial force distribution of 3D models looks similar to that of 2D models’. The peak axial force of Model 3 ( 706 N) is smaller than Model 4 ( 851 N), and Model 3 and 4 have very similar postbuckling axial force. Fig. 18 shows the axial compressive stress of integration point 1 during the buckling process. As shown in
Fig. 18. Axial compressive stress distribution in upheaval buckling (Model 3 and 4).
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Fig. 18, when upheaval buckling occurs in Model 4, the stress of integral point 1 grows rapidly to about 53 MPa which is 57% larger than that of Model 3.
5. Bifurcation buckling For pipelines with larger initial imperfection (for example v0 ¼ 30 mm), bifurcation upheaval buckling occurs. Model 1–4 are analyzed based on the pipeline model mentioned in Section 2. The loading time for Model 2 and 4 is 30 s. Unlike models for pipelines with small imperfections, different models for pipelines with large imperfections show similar structure response as illustrated in Fig. 19. The critical buckling temperatures of Model 1–4 are 6.03 1C, 6.33 1C, 6.2 1C and 6.55 1C, respectively. The critical buckling temperature of 3D models is larger than 2D models and the critical buckling temperature of dynamic models is larger than static models. The axial compressive stress of the pipeline shows similar pattern with buckling path as shown in Fig. 19b. The energy change in bifurcation buckling is shown in Fig. 20. As shown in this figure, there is no oscillation and kinetic energy in the buckling process which means the dynamic effect in bifurcation buckling is very small. Strain energy of the system accelerates in the form of a quadratic function before the critical buckling temperature. Then the gravity does negative work which indicates that the pipeline moves upwards.
Fig. 19. Structural response of bifurcation buckling (a) Buckling path. (b) Axial compressive stress.
Fig. 20. Energy change in bifurcation upheaval buckling (Model 2 and 4) (a) Model 2. (b) Model 4.
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6. Conclusions
References
This study aims to study the static and dynamic responses of upheaval buckling. Two analysis procedures are proposed, which put in place step and upheaval buckling step into a single procedure. Procedure 1 links Newton–Raphson and arc-length method. Procedure 2 links static and dynamic nonlinear analysis seamlessly which effectively solves the convergence problem when dealing with snap buckling. In addition, Procedure 2 can capture the dynamic process of snap buckling. Procedures 1 and 2 are applied to 4 finite element models, 2D/3D static/dynamic models (Model 1–4). Notable conclusions are summarized as follows:
Hibbitt, H.D., Karlsson, B.I., Sorensen, P., 2000. ABAQUS theoretical manual. Version 4, 5. Hobbs, R.E., 1984. In-service buckling of heated pipelines. . J. Transp. Eng. 110 (2), 175–189. Ju, G.T., Kyriakides, S., 1988. Thermal buckling of offshore pipelines. J. Offshore Mech. Arct. Eng. 110 (4), 355–364. Jukes, P., Eltaher, A., Sun, J. (2009). The latest developments in the design and simulation of deepwater subsea oil and gas pipelines using FEA. In: The Proceedings of The Third (2009) ISOPE International Deep-Ocean Technology Symposium: Deepwater Challenge, IDOT-2009, International Society of Offshore and Polar Engineers, United States, pp. 70–82. Karampour, H., Albermani, F., Gross, J., 2013. On lateral and upheaval buckling of subsea pipelines. Eng. Struct. 52, 317–330. Karampour, Hassan, Albermani, Faris, 2014. . Experimental and numerical investigations of buckle interaction in subsea pipelines. Eng. Struct. 66, 81–88. Kerr A.D. On Thermal Buckling of Straight Railroad Tracks and the Effect of Track Length on the Track Response. 1976. Knut Vedelda, Havar A. Sollunda, Helleslanda, Jostein, et al., 2014. Effective axial forces in offshore lined and clad pipes. . Eng. Struct. 66, 66–80. Kyriakides, S., Netto, T.A., 2000. On the dynamics of propagating buckles in pipelines. Int. J. Solids Struct. 37 (46), 6843–6867. Liu, R., Wang, W.G., Yan, S.W., Wu, X.L., 2012. Engineering measures for preventing upheaval buckling of buried submarine pipelines. Appl. Math. Mech. 33, 781–796. Liu, R.u.n., Hao, Xiong, Xilin, W.u., et al., 2014a. Numerical studies on global buckling of subsea pipelines. Ocean Eng. 78, 62–72. Liu, Run, Liu, Wenbin, Xinli, W.u., Shuwang, Y.a.n., 2014b. Global lateral buckling analysis of idealized subsea pipelines. J. Cent. South Univ. Technol. 21 (1), 416–427. Maltby, Timothy Chrichton, Calladine, Christopher Reuben, 1995a. An investigation into upheaval buckling of buried pipelines—II. Theory and analysis of experimental observations. Int. J. Mech. Sci. 37 (9), 965–983. Maltby, Timothy Chrichton, Calladine, Christopher Reuben, 1995b. An investigation into upheaval buckling of buried pipelines—I. Experimental apparatus and some observations. Int. J. Mech. Sci. 37 (9), 943–963. Nystrom, Per R., et al., 1997. 3-D dynamic buckling and cyclic behavior of HP/HT flowlines // ISOPE. In: The Seventh International Offshore and Polar Engineering Conference. International Society of Offshore and Polar Engineers (ISOPE). International Society Offshore & Polar Engineers Cupertino, Honolulu, pp. 299–307. Taylor, Neil, Tran, Vinh, 1996. Experimental and theoretical studies in subsea pipeline buckling. Mar. Struct. 9 (2), 211–257.
(1) Newton–Raphson method is suitable for simulating the in place step of the pipeline and at the end of this step the imperfection is applied to the pipeline. When it comes to snap buckling step, however, arc-length method or dynamic method is more effective. In addition, for the buckling process which shows apparent dynamic phenomenon, say snap buckling, dynamic method is able to describe the real buckling process. (2) In 3D dynamic analysis, pipelines with small imperfections may turn into 2 different buckling modes depending on the loading time. It is recommended to apply a long loading period or practical loading period to obtain a steady buckling temperature. (3) By studying the energy distribution of the pipeline by dynamic method, the snap buckling process can be understood as transforming strain energy into kinetic energy and other energy. (4) For snap buckling, the critical buckling temperature obtained by dynamic method is larger than by static method and the critical buckling temperature obtained by 3D models is smaller than by 2D models. The max axial compressive stress of dynamic analysis is about 50% larger than that of static analysis. For bifurcation buckling, the critical buckling temperature obtained by 3D models is larger than by 2D models. Axial compressive stress of dynamic analysis is slightly larger than that of static analysis.
Acknowledgment The authors are grateful for the support provided by the National Basic Research Program of China (no. 2014CB046801).