Numerical studies on global buckling of subsea pipelines

Run Liu a,n, Hao Xiong a, Xinli Wu b, Shuwang Yan a

a State Key Laboratory of Hydraulic Engineering Simulation and Safety, Tianjin University, Tianjin 300072, Chinab School of Engineering Design, Technology, and Professional Programs, Pennsylvania State University, University Park, PA 16802, USA

a r t i c l e i n f o

Article history:Received 4 June 2013Accepted 29 December 2013Available online 23 January 2014

Keywords:Subsea pipelinePipeline lateral bucklingFEMInitial imperfection

a b s t r a c t

Subsea pipelines buckle globally because of their movement relative to surrounding soil. Global bucklingis often triggered by high operational temperature of the oil in pipelines, initial imperfections in thepipeline, and/or a combination of both. Pipeline global buckling is a failure mode that must be consideredin the design and in-service assessment of submarine pipelines because it can jeopardize the structuralintegrity of the pipelines. Global buckling is increasingly difficult to control as temperature and pressureincrease. Therefore, location prediction and buckling control are critical to pipeline design. Finite elementanalysis (FEA) is often used to analyze the behavior of pipelines subject to extreme pressures andtemperatures. Four numerical simulation methods based on the finite element method (FEM) programABAQUS, i.e., the 2D implicit, 2D explicit, 3D implicit, and 3D explicit methods, are used to simulatepipeline global buckling under different temperatures. The analysis results of the four typical methodswere then compared with classical analytical solutions. The comparison indicates that the resultsobtained using the 2D implicit and 2D explicit methods are similar and the results obtained using the 2Dimplicit method are closer to those obtained using traditional analytical solutions. The analysis showsthat the results of the 3D implicit and 3D explicit methods are similar, but the results obtained using the3D methods are significantly different from those obtained using the analytical solution. A novel methodto introduce initial pipeline imperfections into the FEA model in global buckling analysis is alsopresented in this paper.

& 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Pipeline design faces numerous engineering challenges as oiland gas resources are being obtained from deep waters. One ofthese challenges is pipeline global buckling. Pipelines are beingrequired to operate at increasing temperatures and pressures indeep water. Thermal lateral buckling is a typical global bucklingmode of deep-water pipelines because pipelines are typically laiddirectly on the seabed rather than being trenched and buried.In-service hydrocarbons must be transported at a high temperatureand pressure to ease the flow. Thus, the thermal stress induced bythe difference between the operational and ambient temperaturescoupled with the Poisson effect causes a pipeline to expand longi-tudinally. However, the pipeline cannot expand freely because thesurrounding soil restricts it. Axial compressive stress builds up onthe wall of a pipeline approximately one kilometer long, andsudden deformation occurs when the compressive load reaches orexceeds the soil foundation constraint to release the internal stressaccumulated on the pipe wall. Uncontrolled buckling can haveserious effects on pipeline integrity.

Studies on thermal buckling in pipelines can be traced back tothe early 1970s. Hobbs (1981, 1984) derived analytical solutions tothe buckling and post-buckling behavior of a heated pipeline byassuming a pipeline buckling curve. He established the relation-ship between buckling temperature and buckling length in con-sideration of axial pipe–soil interaction. Taylor and Gan (1986)derived an analytical solution to the global buckling of an initiallyimperfect pipeline based on the analytical solution obtained byHobbs (1984) to lateral buckling in an ideal pipeline. Theyassumed that the shape of a deformed pipeline is symmetricaland similar to that of an initially imperfect. The seabed trenchbottom deformation are neglected and the soil resistance force isfully mobilized per unit length acting against the lateral bucklingmechanism. Solving the total potential energy equation withspecial boundary conditions, Taylor and Gan (1986) obtained theanalytical solution of buckling force, buckling amplitude, andmaximum compressive stress. A sophisticated finite elementmethod (FEM) that considers all of the pertinent pipeline opera-tion data has been applied to pipeline buckling analysis with thedevelopment of modern computers. FEM studies on subsea-pipeline global buckling can be classified into two categories.One focuses on the interaction between a pipeline and its subsoilbecause a reliable pipe–soil resistance assessment plays a signifi-cant role in pipeline global buckling analysis. Several researchers

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Ocean Engineering

0029-8018/$ - see front matter & 2014 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.oceaneng.2013.12.018

n Corresponding author.E-mail address: liurun@tju.edu.cn (R. Liu).

Ocean Engineering 78 (2014) 62–72

over the past 30 years have paid particular attention to pipe–soilinteraction in the analysis of on-bottom pipeline strength andstability behavior. Numerous excellent research studies have beenconducted, and several useful achievements have been adopted forpractical use, such as those of Lyons (1973), Friedmann (1986),Schaminee et al. (1990), Palmer et al. (1990), Hesar (2004),Merifield, White, and Randolph (2007, 2009), Bruton et al.(2008), Wang et al. (2013) and Bruton et al. (2011). The othercategory simulates pipeline global buckling under high-pressureand high-temperature conditions. Numerical pipeline global buck-ling analysis tools, such as PIPLIN-III (Structural SoftwareDevelopment Inc., 1981), PlusOne (Palmer and Associates, 1995),PIPSOL (Nixon, 1994), ABP, and UPBUCK (Klever et al., 1990) havebeen used in different situations over the past 30 years. Shaw andBomba (1994) have developed a finite element (FE) analysismethod that considers both nonlinear geometry and materialeffects to examine pipeline response to upheaval buckling. Casestudies show that the temperature difference corresponding topipeline buckling with nonlinear material behavior is smaller thanthat in the elastic model. Andreuzzi and Perrone (2001) present amathematical model that considers soil resistance to beam lateraldeflections by introducing linear spring resistance to beam lateraldisplacement, and report that FE and finite differences maygenerate errors in the results because of the discretization relatedto the modeling of the various axial compressive forces in theelements. A formula to analyze initially imperfect undergroundpipelines has been developed, and an issue regarding a 2D, initiallyimperfect, buried pipeline has been analyzed by Villarraga et al.(2004). All of these programs are based on pipe beam elementsand elastic–plastic soil springs. Using simplified analytical modelshas been a standard approach to analyze upheaval buckling ofhigh-temperature and high-pressure pipelines.

However, simplified approaches may be excessively conserva-tive because they may fail to identify vulnerable features andunderlying upheaval buckling risks that can result in severeeconomic consequences (Zhang and Tuohy, 2002). Therefore,understanding pipeline response to various loading conditions iscritical in increasing pipeline design efficiency. Zhang and Tuohy(2002) conduct a global-buckling analysis case study on a trenchedbut unburied 6.0-inch production flowline using the commercialFEM program ANSYS (Kohnke and Peter, 1999). The results showthat FE technology can be adopted as an effective tool to evaluatepotential offshore flowline buckling behavior. ABAQUS (Hibbittet al., 2000) also incorporates pipeline beam elements, soil–pipeinteraction, and large displacements to model considerable pipe-line length and predict overall structural behavior under differentload conditions. Jukes et al. (2009) report the advantage of the FEanalysis tools, which can be used in the design and simulation ofsubsea pipelines and their components. SIMULATOR, a highly non-linear FE program, has been developed using ABAQUS as the FEengine. Case studies show that the developed program can beapplied to complex pipeline design cases, such as global analysis,local modeling, and pipeline route selection. The case study resultsalso imply that the advanced numerical tools are suitable forpipeline design and simulation, particularly of deep-water pipe-lines. These tools are also suited to extreme conditions becausethey can simulate highly non-linear cases quickly and efficiently.Wang et al. (2009) and Jukes et al. (2009) developed the FE tool asa SIMULATOR component. The in-house pipeline analysis packagedesigned by Kenny, which has been developed using the ABAQUSplatform, can simulate global buckling with different pipelineconfigurations under various conditions. Global pipeline FE ana-lyses have been widely used to investigate complex practicalproblems associated with lateral pipeline buckling and the walk-ing pipeline phenomenon (Jukes et al., 2008; Jukes et al., 2009;Sinclair et al., 2009; Cumming et al., 2009; Cumming and

Rathbone, 2010; Jin et al., 2010; Bruton et al., 2011; Sun et al.,2011). Literature reviews have shown that the existing 3D,thermal-pipeline-buckling finite element analysis (FEA) method,which collectively considers temperature field and stress fields,initial imperfections in a pipe, and soil/pipe interaction, is ineffi-cient, although the FEA of offshore pipeline upheaval buckling hasprogressed rapidly in recent years.

In this study, four typical FEA methods, namely, the 2D implicit,2D explicit, 3D implicit, and 3D explicit methods, are used toanalyze subsea-pipeline global buckling under high temperatureand high pressure conditions. Initial pipeline imperfection as aresult of fabrication or installation and seabed undulation is alsoconsidered in the FE analyses using a novel method. The analysisresults obtained using the four typical FEA methods are thencompared with analytical solutions derived by Taylor and Gan(1986).

2. Global buckling analysis methods

2.1. Establishment of 2D and 3D FEA models

A global buckling analysis model is characterized by the longaxial direction of the pipeline and the relatively small pipelinecross-section. Therefore, the beam element is highly suitable forpipeline structure simulation. The moment distribution alongthe pipeline can be obtained easily. The 2D FEA model can beadopted to simplify the study on one-direction global buckl-ing under thermal stress conditions. The 2D FEA model offers atrustworthy solution to exposed pipelines on an even seabed,and it allows the pipeline to move both axially and laterally.The 2D FEA model not only can assess thermal expansion andlongitudinal thermal loading, but also can investigate lateraland upheaval buckling. The horizontal 2D FEA model can bebuilt as shown in Fig. 1.

The 2D FEA model employed in this study uses ABAQUS as theunderlying FE engine. PIPE32H beam elements and dimensionsrepresent the pipeline in this model. The expansion coefficient ofthe pipeline material is also defined to determine the role ofthermal stress induced by temperature. The pipeline is modeledusing linear elastic material, and the seabed is assumed to be arigid foundation. Contact elements are created between the pipe-line and the rigid foundation, and these contact elements arepositioned in two areas: one is perpendicular to the plane of thefoundation soil, which displays normal contact behavior and has a“hard” contact parameter, and the other is parallel to the founda-tion soil surface/plane, which displays tangential contact behaviorand has a penalty function parameter. This form of contact cannotdetermine pipeline settlement deformation as a result of self-weight, but it can simulate the increasing resistance of thefoundation soil to the increasing weight of the submerged pipe-line. The friction length effect cannot be extended to both ends ofthe pipeline because the pipeline is long (Z500 m). Both ends ofthe pipeline are completely fixed, and the boundary conditionsof the soil resist both laterally and axially. The bottom boundary ofthe soil is completely fixed as well. The following three steps havebeen adapted in the simulation: (1) introduction of the initialcurved pipeline section to simulate the initial imperfection gen-erated in the manufacturing process and placement of the pipeline

Fig. 1. 2D FEA model.

R. Liu et al. / Ocean Engineering 78 (2014) 62–72 63

on the seabed, (2) application of a gravitational field to modelcontact behavior between the pipeline and the rigid foundationsoil, and (3) increase in pipeline temperature to simulate the high-temperature loading process until global buckling occurs.

A 3D model is considered to simulate the pipeline globalbuckling process, pipeline self-penetration procedure, and seabedbehavior. C3D8R solid elements are used to model the pipeline andfoundation soil in the 3D FE model. Pipeline mechanical behavioris assumed to be linear-elastic. The Mohr–Coulomb model is usedas the soil foundation constitutive model to simulate the stress–strain relationship with respect to soil properties. Tables 1 and 2provide additional details on these parameters. Soil dilatancy isnot considered, and the contact between pipeline and soil occursin three directions. The first direction runs along the deadweightdirection of the pipeline that displays normal contact behavior andhas a contact parameter set to “hard contact.” The second directionruns along the longitudinal direction of the pipeline that exhibitstangential contact behavior and with contact parameter set to“penalty” function. The third direction runs along the radialdirection of the pipeline with tangential contact behavior andwith contact parameter set to “penalty” function. The 3D modelboundary is similar to that of the 2D model and the specificsimulation process involves three steps. First, the initial curvedsection of the pipeline is introduced to simulate the initialimperfection generated in the manufacturing process and theplacement of the pipeline on the seabed. Second, a gravitationalfield is applied to model the contact behavior between the pipe-line and the rigid foundation soil. The pipeline sinks into the soilbecause of its own weight. Subsequent computation iterationbegins when the deformation between the pipeline and soil isstabilized. Third, pipeline temperature is increased to simulate thehigh-temperature loading process until global buckling occurs.A significant difference between the 3D and 2D models is thataside from the lateral and axial friction effects along the pipeline,strong lateral earth pressure from the deformed soil affects thepipeline during buckling because the pipeline partially sinks intothe foundation soil. This sinking results in foundation soil defor-mation. The 3D analysis model is presented in Fig. 2.

2.2. Analysis type definition

The non-linear static buckling problem is a part of pipeline globalbuckling under thermal stress conditions. The structure must releasestrain energy to maintain equilibrium when negative stiffness occursin response to load displacement. Two current FEA methods simulatethis behavior. One is the displacement control method in which adampler controls the stable structure displacement and addresses

the static equilibrium of the unstable response segment in implicitanalysis. This method is known as the modified Riks method (Hibbittet al., 2000). The other method is the acceleration control method inwhich inertia controls acceleration and addresses the steady struc-ture displacement in explicit analysis, thereby simulating the snap-ping of the pipeline. This method is known as the explicit dynamicmethod (Hibbitt et al., 2000).

2.2.1. Implicit arc-length iterative method (modified Riks method)The Riks method, originally proposed by Riks (1972, 1979) and

Wempner (1971), tracks the nonlinear structural equilibrium path.This method has been modified and developed further by Crisfield(1981) and Ramm (1981) and has become the main method used in

Table 1Physical and mechanical parameters of pipeline.

Wall thickness External diameter Young0s modulus Poisson ratio Bulk density Oil density seawater density Thermal coefficient yield stressT (mm) D (mm) E (N/mm2) ν γs (kg/m3) γo (kg/m3) ρ (kg/m3) α (1C) sy (MPa)

0.0127 0.3239 206000 0.3 7850 800 1120 0.000011 448

Table 2Physical and mechanical parameters of subsoil (from Bohai Gulf).

Soil layer Thicknessof soil (m)

Moisturecontent (w%)

Effective bulk density(kN/m3)

Voidratio e

Plasticityindex IP

Cohesion(kPa)

Internal frictionangle (1)

Compression modulusEs1–2 (MPa)

Shearstrength (kPa)

Mucky silty clay 0–2 38.8 7.8 1.05 14.3 18 18.6 3.26 4.0–7.5Silt clay 2–3 45.8 9.4 1.27 21.5 10 15.9 2.58 7.5–16.0

Fig. 2. 3D FEA model.

Fig. 3. Load–displacement response curve of the unstable equilibrium problem.

R. Liu et al. / Ocean Engineering 78 (2014) 62–7264

the analysis of non-linear structure stability problems. The relation-ship between load and displacement in the unstable equilibriumproblem corresponding to the load-displacement response curve isnot monotonic. The stress or strain release phenomenon occursduring the response, as shown in Fig. 3. The modified Riks method isan effective solution algorithm in such cases.

The Riks algorithm assumes that the loading process is applied inproportion to the overall response of the structure and that theprocess is smooth. All of the equilibrium solutions are viewed asnodal displacement variables during the development of instability,and a single equilibrium path in space is defined by the loading pro-portion coefficient. All size increments must be artificially restrictedwhen Newton0s method is used in the iterative solution to ensureconvergence. The size increment of this equilibrium path is the mov-ing distance from the equilibrium point along its tangential directionto the next solving point in the modified Riks algorithm. Distance isdetermined by the automatic increase in convergence speed in thisalgorithm and does not have to be artificially restricted in the com-putation process. The computation principle is described as follows.

(1) A reference load is defined, and the proportionality factor λ ofthis load to the ultimate load corresponding to pipeline failureis calculated. ABAQUS/Standard uses the “arc length (l)” alongthe static equilibrium path in the load–displacement space tomeasure the progress of the solution and determines therelationships among load, arc length, and displacement.

(2) The load is proportionally applied. The load magnitude (Ptotal)is defined as

Ptotal ¼ P0þλðPref �P0Þ ð1Þwhere P0 is the initial load; Pref is the reference load vector; and λis the load proportionality factor, which is considered as a part ofthe solution. ABAQUS/Standard determines the current value ofthe load proportionality factor at each increment.

(3) Newton0s method is used to obtain the nonlinear equilibriumequations. The modified Riks procedure extrapolates only a 1%strain increment. The ABAQUS program provides an initial arc-length increment ðΔlinÞ along the static equilibrium path in theRiks step definition. The initial load proportionality factorðΔλinÞ is computed as

Δλin ¼Δlinlperiod

ð2Þ

where lperiod is a user-specified total arc-length scale factor(typically equal to 1). This Δtstable value is used during the firstiteration of a Riks step. The λ value is computed automaticallyaccording to Eq. (2) in subsequent iterations and increments.

(4) Compared with the Riks method, the modified Riks methodaddresses the unstable collapse and conducts post-bucklinganalysis more effectively (see Fig. 4). The initial load propor-tionality factor can be determined using this method as shownin the following paragraphs.

The solution is assumed to have developed to point A0 ¼ ðuN0 ; λ0Þ.

The tangent stiffness ðKNM0 Þ has been formed and KNM

0 vM0 ¼ PN has

been determined. The increment size (A0 to A1 in Fig. 4) is selectedbased on a specified path length ðΔlinÞ in the solution space. Therefore,

Δλ20ðνN0 ;1Þ : ðνN0 ;1Þ ¼Δlin2 ð3Þ

Thus,

Δλ0 ¼7Δlin

ðνN0 νN0 þ1Þ1=2ð4Þ

where λ0 is the initial load magnitude parameter in the modified Riksmethod; N andM denote the degrees of freedom of the model; uN and

vN0 are the displacements; νN0 is the normalized tangential displace-ment vector at the initial iterative step, where νN0 is vN0 scaled by u; andu is the maximum absolute value of all displacement variables.

The value Δlin is initially suggested by the user and is adjustedusing the ABAQUS/Standard load increment method in staticproblems based on the convergence rate.

2.2.2. Explicit dynamic methodThe explicit integration algorithm in the ABAQUS/explicit

model adopts the central interpolation method, i.e., the nodalforce (external force P minus internal force I) can be calculated inthe following manner by multiplying the mass matrix M with theacceleration matrix at time t if the dynamic equilibrium conditionsare satisfied:

M €u¼ P� I ð5ÞThe acceleration at initial time increment t can be calculated as

€ujðtÞ ¼ ðMÞ�1ðP� IÞjðtÞ ð6ÞThe acceleration can be obtained directly and the entire system

of equations in the explicit integration algorithm need not be usedbecause the lumped mass matrix is typically adopted. Accelerationat a certain node can be determined simply through node massand nodal force, thereby significantly reducing calculation time.The velocity increment can be calculated using the central inter-polation method based on the assumption that the accelerationremains constant at a certain time increment. The velocity at themidpoint of the current time increment can be obtained by addingthe velocity increment to the velocity at the midpoint of theprevious time increment.

_ujðtþð1=2ÞΔtÞ ¼ _ujðt�ð1=2ÞΔtÞ þðΔtjðtþð1=2ÞΔtÞ þΔtjðtÞÞ

2€ujðtÞ ð7Þ

Deformation at the end of the time increment can be calculatedby integrating velocity as follows:

ujðtþΔtÞ ¼ ujðtÞ þΔtujðtþð1=2ÞΔtÞ ð8Þ

Explicit procedure stability is conditional, i.e., the time incre-ment cannot exceed the limitationΔtstable. This limitation can beestimated by

Δtstable ¼2

ωmaxð

ffiffiffiffiffiffiffiffiffiffiffiffi1þξ2

q�ξÞ ð9Þ

where ωmax is the supreme natural frequency and ξ is thecorresponding critical damping.

As shown in preceding paragraphs, initial acceleration can bedetermined if the dynamic equilibrium conditions are satisfied atthe initial time increment. Then, velocity and deformation can becalculated using the explicit procedure. The selected time incre-ment must be adequately small to obtain precise solutions; thus,the assumption that acceleration remains constant through thetime increment is reasonable. The explicit procedure therefore

Fig. 4. Modified Riks method.

R. Liu et al. / Ocean Engineering 78 (2014) 62–72 65

typically requires numerous time increments. The computationtime required for each time increment solution is quite smallbecause the entire system of equations does not have to be solved.However, internal element computation is time consuming andincludes calculating strain with deformation, computing stressaccording to constitutive relations, and finally determining theinternal force at the nodes.

2.3. Introduction of initial imperfection into the FEA model

Initial imperfection is considered in FEA because of pipelineout-of-straightness as a result of fabrication or installation andseabed undulation. An approach to introduce initial pipelineimperfection based on modal analysis results is presented in thispaper. Modal analysis is often used to study the probability ofdifferent structure-buckling shapes under unit load. Buckling loaddecreases if structural deformation results in the generation ofpositive work by external loads because this deformation cannotresist external loads, according to the work–energy principle.Conversely, buckling load increases if structural deformationcauses external loads to produce negative work because energyis absorbed. Thus, the deformation is advantageous to the struc-tural ability to resist external forces. The proposed method isbased on the work–energy principle and focuses on the first-orderbuckling mode of the pipeline structure. Pipeline deformation isrecorded to update the pipeline geometry when external loads

generate positive work. The implementation method used in thenumerical analysis is described as follows. First, pipeline bucklingmodes are calculated and a geometric model identical to the globalbuckling analysis model is generated. Both models must haveidentical node numbers. The calculation subspace of the pipelinemodal analysis is set to 2 and the nNODEFILE keywords areinputted into the INP file. The pipeline displacement modal fileis outputted as FIL, which is used as the basis for the bucklinganalysis model. The model adds an nIMPERFECTION scale factorstatement to introduce pipeline imperfection, as shown in Fig. 5.

In this study, investigations and analyses of an in-serviceoperational pipeline subject to high temperature are conductedusing the aforementioned numerical analysis methods, i.e., the 2Dimplicit, 2D explicit, 3D implicit, and 3D explicit methods.

3. Analysis results using different FEA methods

3.1. A subsea pipeline case

A pipeline engineering project is constructed in Bohai Gulf. Theunburied subsea pipeline has an outside diameter of 323.9 mm andwall thickness of 12.7 mm. The designated temperature difference is60 1C. The designed pipeline and foundation soil parameters arereported in Table 1 and 2, respectively. The four numerical analysismethods mentioned are used in this study to analyze pipeline global

Fig. 5. Introduction of initial imperfection into the pipeline model. (a) Straight pipeline and (b) pipeline with initial imperfection.

Table 3Characteristics of different FEA models.

FEA Model Load step 2D-implicit 2D-explicit 3D-implicit 3D-explicitRiks Dynamic explicit Riks Dynamic explicit

Subsoil Material behavior Rigid Rigid M–C M–CElement type ——— ——— C3D8R C3D8RProperty

γ0 (kN/m3) ——— ——— 7.8 7.8E (MPa) ——— ——— 3.26 3.26c (kPa) ——— ——— 18 18ϕ (1) ——— ——— 18.6 18.6

Pipeline Material behavior Elastic Elastic Elastic ElasticElement type PIPE32H PIPE32H C3D8R C3D8RProperty

ρ (kg/m3) 7850 7850 7850 7850E (MPa) 206000 206000 206000 206000α (1C) 0.000011 0.000011 0.000011 0.000011

Contact type between pipeline and seabed type Node to surface Node to surface Surface to surface Surface to surfaceCoefficient 0.4 0.4 0.4 0.4

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buckling under thermal stress. The results are computed using thevarious computational methods and subsequently compared.

3.2. Parameters of different FEA models

The 2D implicit, 2D explicit, 3D implicit, and 3D explicitmethods are employed in this study to examine the differencesin the results obtained using the various numerical analysis/simulation methods in terms of pipeline global buckling andestablish the FEA model. The computation parameters of thedifferent analysis models are shown in Table 3. Initial imperfec-tions are introduced using the modal analysis method in thisstudy. The initial imperfection amplitude is 150 mm and thewavelength is 23 m. The 2D and 3D computation models arepresented in Fig. 4 and Fig. 5, respectively.

3.3. Analysis results

3.3.1. Comparison of pipeline global buckling shapesThe aforementioned four types of numerical analysis/simula-

tion methods can simulate the lateral pipeline global bucklingprocess that occurs at the initially imperfect area under tempera-ture loads. The lateral pipeline global buckling shape develop-ments with the increase in temperature obtained using the foursimulation methods are shown in Fig. 6. This figure presents onlythe local variation in the pipeline deformation segments. Thedeformation segment length is shown along the x-axis, and thelateral deformation amplitude is shown along the y-axis.

Deformation first occurs at the initially imperfect area ofthe pipeline with the increase in temperature and its amplitudegradually increases from 150 mm. A reverse bending deformationoccurs on both sides of the initially imperfect pipeline wavelength

when this amplitude increases to a certain extent. Bucklingamplitude and wavelength increase gradually in both positiveand reverse directions in the pipeline, and the global bucklingscope is enlarged.

The pipeline global buckling deformation process in the 2Dmodel is significantly different from that in the 3D model. Themaximum pipeline buckling amplitudes differ under similar tem-perature variations. Buckling amplitude in the 2D model is greaterthan 1.3 m when the temperature difference is 60 1C, whereas themaximum buckling amplitude in the 3D model is less than 1.2 m.The mutational temperatures in pipeline global buckling also differwith the increase in temperature variation. Pipeline bucklingamplitude increases by over three times when the temperaturedifference is increased from 20 1C to 30 1C, as shown in the 2Dmodel. However, mutational increase in the buckling amplitude inthe 3D model occurs when the temperature difference is increasedfrom 40 1C to 60 1C. The pipeline buckling shapes are different aswell. Reverse pipeline buckling amplitude and wavelength in the2D model undergo significant changes as observed in the devel-opment of reverse pipeline buckling amplitude. However, theincrease in the reverse pipeline amplitude is significant and thewavelength change in the 3D model is substantial only if thetemperature is higher than 40 1C. Buckling amplitude develop-ment in the 2D model under 40 1C is greater than 50% of theamplitude under 60 1C. This finding indicates that the develop-ment speed of buckling amplitude in the 2D model is faster thanthat in the 3D model. The differences mentioned are mainlycaused by the partial embedding of the pipeline in the 3D modelin the foundation soil; thus, soil resistance to lateral pipelinedeformation increases significantly.

Maximum pipeline buckling amplitude in the 2D modelreaches 1.49 m when the temperature difference is 60 1C and the

Fig. 6. Thermal buckling modes of different FEA models. (a) 2D-Implicit FEA model, (b) 2D-Explicit FEA model, (c) 3D-Implicit FEA model and (d) 3D-Explicit FEA model.

R. Liu et al. / Ocean Engineering 78 (2014) 62–72 67

implicit method is used. This amplitude is greater than the 1.35 mobtained using the explicit dynamic method, as shown in thecomparison of the results obtained using both implicit and explicitmethods. Maximum reverse-pipeline buckling amplitude reaches0.52 m, which is less than the 0.57 m obtained via the explicitdynamic method. A similar trend appears in the results obtainedin the 3D model, as shown in Fig. 6(c) and (d). The deformationregularities obtained using both the implicit and dynamic explicitmethods are highly similar in the 2D model, whereas a significantdifference between the 3D model results obtained using theimplicit and dynamic explicit methods is observed when thecalculation results of both the 2D and 3D models are compared.The results of the two 3D models differ because the pipelinedeformation rates are different.

3.3.2. Comparison of maximum stresses along the pipelineThe maximum axial stress on the compressed side along the

pipeline longitudinal direction changes with the increase intemperature when different simulation methods are used, asshown in Fig. 7. The figure shows only one side of the axial stresswithin the deformed segment experienced by the symmetricalpipeline axis in consideration of the symmetrical pipeline defor-mation. The horizontal axis is notably distant from the deformedsymmetrical pipeline axis, whereas the stress on the compressedside of the pipeline is located on the vertical axis. Compressivestress is denoted as negative in the figure.

Internal pipeline stress continues to accumulate under tem-perature variation, and the entire pipeline is compressed by thefoundation soil constraints, as shown in Fig. 7. The compressivestress in the pipeline continuously increases with the increase intemperature difference, and the maximum compressive stress

occurs in the horizontal pipeline area with maximum deformation.Maximum stress gradually decreases with the increase in distancefrom the deformed symmetrical pipeline axis. The minimum stressvalue is reached when the distance reaches 20 m to 30 m, which is inaccordance with the reverse displacement position depicted in Fig. 6.The stress at the compressed side of the pipeline begins to decreaseand even tensile stress occurs when reverse pipeline displacementtakes place. The difference in the deformation results computedusing both the 2D and 3D models directly lead to a difference inpipeline stress. The stress value obtained in the 2D model issignificantly higher than that obtained in the 3D model. The increasein stress with the increase in temperature variation in the 2D modelalso varies significantly from that in the 3D model. This finding ismainly attributed to the difference in temperature variation, whichcorresponds to the stress mutation. The temperature differencecorresponding to the stress mutation is approximately 30 1C in the2D model and 60 1C in the 3D model. The absolute maximum stressvalue in the pipeline can be shown as |sd|¼195.608 MPa 4 |sb|¼280.73 MPa 4 |sa|¼201.95 MPa 4 |sc|¼166.70 MPa when the tem-perature difference of 60 1C is used as an example in the comparisonof the results obtained using the implicit Riks and the explicitdynamic methods. The absolute maximum stress value obtainedusing the explicit method is higher than the result obtained using theimplicit method.

3.3.3. Relationship between maximum pipeline buckling amplitudeand temperature difference

Fig. 8 shows the relationship between maximum lateral defor-mation amplitude and temperature difference as determinedusing the four computation methods. Maximum lateral displace-ment at the symmetrical axis is positioned along the horizontal

Fig. 7. Axial stress variation in different FEA models at different operational temperatures. (a) 2D-Implicit FEA model, (b) 2D-Explicit FEA model, (c) 3D-Implicit FEA modeland (d) 3D-Explicit FEA model.

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axis, and the temperature load is applied along the vertical axis ofthe pipeline.

The lateral deformation amplitudes obtained using the four com-putation methods all increase with the increase in temperaturedifference, as shown in Fig. 8. However, differences in the deformationdevelopment processes exist. The pipeline buckling deformationamplitudes obtained using the four different methods are significantlydifferent when modeling analysis is conducted under similar tem-perature variation. The 3D model results are apparently different fromthose of the 2D model, and this finding is reflected in the temperaturevariation differences corresponding to the occurrence of global buck-ling. The initial buckling temperature difference in the 2D model isapproximately 20 1C, whereas the temperature difference in the 3Dmodel is greater than 30 1C. The initial buckling temperature differencereaches 60 1C based on the 3D explicit method. This finding indicatesthat a low initial buckling temperature is obtained when the 2Dmodelis used because this model does not consider the situation in whichthe pipeline is partially embedded in the foundation soil. By contrast,the 3D model considers such a situation. Also, higher soil resistance isused in the 3D simulation; thus, the initial buckling temperaturedifference is greater than that shown by the 2D simulation.

Computation results obtained in the 2Dmodels are consistent whenthe temperature difference is less than 30 1C; however, a relatively largepipeline deformation occurs when the temperature difference isbetween 20 1C and 30 1C. Maximum pipeline buckling amplitudeincreases rapidly with the use of the 2D explicit method comparedwith the use of the 2D implicit method when the temperaturedifference is greater than 30 1C. The calculation results in the two 3Dmodels are consistent when temperature difference is less than 30 1C,but a large pipeline deformation occurs with the use of the 3D implicitmethod when the temperature difference is between 30 1C and 50 1C.When the temperature difference is greater than 60 1C, the maximumpipeline buckling amplitude increases linearly with the temperatureincrease, according to the 3D implicit method. The pipeline deformationincreases more rapidly when the 3D explicit method is used.

3.3.4. Relationship between maximum bucking amplitude and axialpipeline stress

Fig. 9 shows the relationship between maximum axial pipelinestress and the temperature differences obtained using the fourcomputation methods. Maximum horizontal displacement at thesymmetrical axis occurs along the horizontal axis, and maximumcompressive stress on the symmetrical axis occurs along thevertical axis.

The lateral deformation amplitude values obtained using thefour methods increase with the increase in axial-pipeline com-pressive stress, as shown in Fig. 9. The deformation developmenttrends are similar. However, the pipeline deformation amplitudesare different under similar axial stress. This difference appears toincrease with the increase in the axial load on the pipeline.Internal pipeline axial load obtained in the 3D model is greaterthan that obtained in the corresponding 2D model under similardeformation amplitudes as a result of the soil resistance effect.

Internal pipeline axial force computed using the Riks implicitmethod is significantly smaller than that obtained using theexplicit dynamic method under similar horizontal deformationamplitudes because dynamic analysis considers the influence ofspeed on the pipeline deformation process. Part of the workgenerated by internal force is converted into kinetic energy; thus,significant axial stress in the dynamic algorithm is required toattain the same deformation amplitude as that of the static Riksalgorithm.

Fig. 10 shows the relationship between deformation rate in thehorizontal symmetrical pipeline axis and the temperature loadsapplied to the pipeline in both 2D and 3D explicit dynamic models.

Peak velocity v2d¼0.23 m/s when the temperature difference is25 1C in the 2D explicit model, whereas the peak velocity v3d¼0.14 m/s when the temperature difference is 50 1C in the 3D explicit

Fig. 8. Midpoint pipeline buckling amplitude versus operational temperaturedifference.

Fig. 9. Pipeline buckling amplitude versus maximum axial stress.

Fig. 10. Horizontal midpoint pipeline speed at different operational temperatures.

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model as shown in Fig. 10. Buckling development reaches a turningpoint in the 2D explicit model when temperature variation is 25 1C,as shown in Figs. 8 and 10. Buckling deformation enters an unstabledevelopment stage with the increase in temperature difference. The3D explicit model results are similar to those of the 2D explicitmodel. However, the temperature difference corresponding to thisdeformation turning point is 50 1C, which indicates that peakdeformation rate is attained when the pipeline enters an unstabledeformation stage. The deformation rate obtained via dynamicanalysis can be used as a reference to determine when the entranceof a pipeline enters into unstable buckling stage.

4. Verification

The computation results obtained using the numerical simula-tion methods in this study are compared with classic analyticalsolutions (Talor and Gan, 1986) to lateral pipeline global bucklingto verify the accuracy of the proposed methods.

Fig. 11 illustrates the relationship between pipeline deforma-tion and the change in pipeline axial stress under temperaturestress, as well as the relationships among horizontal pipelinebuckling deformation amplitude, temperature difference, and axialcompressive stress. The figure shows the comparison of the resultsobtained using the 2D implicit method and those calculated usingthe analytical solutions because the 2D implicit model meets thebasic assumptions of analytical solutions.

The pipeline deformation obtained using the 2D implicitmethod has a clear inflection point according to the deformationdistribution along the pipeline shown in Fig. 11(a). The maximumbuckling amplitude of the forward/positive bending is 1.2 m and

the maximum buckling amplitude of the reverse curvature is0.56 m. Reverse curvature/bending does not exist in the firstbuckling mode in analytical solutions. The maximum bucklingamplitude of the forward/positive bending is 2.74 m. The differ-ence in deformed pipeline shapes is attributed to the assumptionheld by analytical solutions that deformed pipeline shapes aresimilar to initially imperfect pipelines.

Fig. 11(b) shows that the internal stress distribution along thepipeline is related to deformed pipeline shapes. Internal compres-sive stress as determined using the FEA method is 25.5 MPa in theinitial deformation segment, i.e., in the vicinity of the horizontalaxis x¼25 m. The result calculated using the analytical solution is30 MPa. Internal compressive stress as determined using the FEAmethod is 196 MPa in the area with maximum pipeline deforma-tion, i.e., at horizontal axis x¼125 m. The result obtained using theanalytical solution is 286 MPa, which is 1.45 times the result of theFEA method. This difference is directly related to the variations inthe deformation analysis results.

Fig. 11(c) illustrates the changes in maximum lateral pipelinedeformation amplitude with the variation of the temperature differ-ence. The initial temperature difference in pipeline buckling defor-mation calculated using the 2D implicit method is significantly lessthan that determined using the analytical solution, as shown in thefigure. The pipeline enters an unstable deformation stage when thetemperature difference is greater than 20 1C, and deformation devel-ops rapidly with the increase in temperature difference, as the FEAmethod results prove. However, the results of the analytical solutionshow that initial temperature difference is no less than 30 1C whenpipeline deformation enters an unstable stage.

Fig. 11(d) illustrates the variation trend of maximum lateral-deformation amplitude versus internal axial pipeline stress.

Fig. 11. Comparison of FEA model and analytical solution. (a) lateral deformation, (b) axial stress, (c) vm vs. ΔT (d) vm vs. s.

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Internal axial pipeline stress as determined using the analyticalsolution is more significant than that determined using the 2Dimplicit method when the buckling amplitudes are similar, asshown in the figure. The axial stress development trend asdetermined using the 2D implicit method is less significant thanthat calculated using the analytical solution when buckling ampli-tude increases.

Differences between the computation results determined usingthe 2D implicit method and those obtained using the analyticalsolution exist because the calculations of the analytic solution arelimited by the assumption it holds, as summarized in the preced-ing paragraphs. However, the results of both methods can beeffectively compared, thereby indicating that the numerical meth-ods are reliable from the perspectives of globally deformedpipeline-shape development and internal pipeline stress.

5. Conclusions

This study describes in detail the 2D implicit, 2D explicit, 3Dimplicit, and 3D explicit methods to simulate global bucklingdeformation in pipelines under temperature stress. These fourmethods have been utilized in systematic analysis and comparisonof certain practical engineering projects. The following conclusionscan be drawn based on the comparison of the results computedusing the proposed methods and those obtained using classicanalytical solutions to the lateral buckling of initially imperfectpipelines.

Lateral global buckling in pipelines under temperature stresscan be simulated by either the 2D or 3D model and either theimplicit Riks or explicit dynamic method depending on thestructural characteristics and deformation process of the pipeline.The 2D analysis model is simple and is evidently advantageous inthe simulation of structures such as slender rods. The sinking of apipeline into soil and the foundation soil loading and deformationprocesses can be simulated in the 3D analysis model. The implicitRiks method can monitor the unstable deformation process of thestructure. The explicit dynamic method can capture the snappingphenomenon during pipeline global buckling.

The results of an analysis of a practical engineering projectdemonstrate significant differences among the pipeline globalbuckling results obtained using the four analysis methods, asreflected in the various mutational temperature differences inpipeline deformation and stress development obtained via the 2Dand 3D models. The pipeline buckling amplitude, internal pipelinestress, and initial buckling axial force computed using the implicitRiks method are also different from those determined using theexplicit dynamic method. The differences in the analysis resultsare mainly caused by the following: the 3D model considers thesinking of the pipeline into the foundation soil because of self-weight, that is, the pipeline is partially embedded in the founda-tion soil. Therefore, additional soil resistance to the pipeline isgenerated in the pipeline deformation process. The dynamicanalysis considers the influence of speed on the deformationprocess, that is, the partial work generated by internal forces isconverted into kinetic energy, which cannot be simulated by astatic method.

An analysis method based on modal analysis that introducesinitial pipeline imperfection is presented in this paper. Thismethod can be used to analyze thermal buckling in an initiallyimperfect pipeline. The temperature difference that correspondsto a high pipeline buckling deformation rate can be obtainedusing the explicit dynamic method. Thus, the deformation ratedetermined in the dynamic analysis can be used as a basis todetermine when a pipeline enters into an unstable buckling stage.

Differences are observed between the results obtained usingthe classic method and those using the FEA method because theanalytical solution is derived from simplified assumptions. How-ever, the results obtained using the 2D implicit method and theanalytical solution are comparable with respect to pipeline globalbuckling deformation development, buckling amplitude changes,and internal pipeline stress development because these twomethods have consistent assumptions.

Acknowledgment

The authors are grateful for the support provided by the Innova-tive Research Groups of the National Natural Science Foundation ofChina (51021004), the China National Natural Science Foundation(51322904, 40776055), and the Program for New Century ExcellentTalents in University (HCET-11–0370).

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