Received date： 2011-04-27Foundation item: Supported by China National Natural Science Foundation (No.40776055) and foundation of State

Key Laboratory of Ocean Engineering (1002)Biography: GAO Xi-feng(1975-), male, Ph.D., lecturer of Tianjin University; Corresponding author: LIU Run(1974-),

female, Ph.D., Professor, E-mail: liurun@tju.edu.cn.

Overview of Upheaval Buckling Theoretical Studies forSubmarine Buried Pipeline

GAO Xi-feng 1, LIU Run1, DU Zun-feng 1, TAN Zhen-dong 2

(1 State Key Laboratory of Hydraulic Engineering Simulation and Safety, Tianjin University, Tianjin 300072, China;2 Military Transportation Institute of the General Logistics Department, Tianjin 300161, China)

Abstract:Pipelines constitute a vital means of transporting and distributing liquids and gases suchas oil, gas and water over large geographical areas. A pipeline containing hot oil (and possibly insu-lated in order to prevent the solidification of wax in the pipe) will expand longitudinally on accountof the rise in temperature; and if such expansion is resisted, for example by frictional affects over akilometer or so of pipeline; compressive axial stress will be set up in the pipe-wall.The compressiveforces are frequently large enough to induce lateral buckling of untrenched pipelines,or verticalbuckling of trenched pipelines. The recent growth in interest in high temperature pipelines in Chinaand some failures in Bohai Gulf have led to an explosion of interest in these phenomena for Chineseocean engineers. This paper describes the history of the theoretical studies on the submarine buriedpipeline vertical buckling and the author’s experiences with this problem.Key words: submarine pipeline; trenched; buried; upheaval bucklingCLC number: TE973 Document code: A

1 Introduction

Since the early seventies of the last century pipelines have also become one of the mainmeans of transporting oil and gas offshore in many parts of the world.To ensure minimal in-terference with other marine activities, such as fishing nets, ship anchors, etc and to ensuresafety of the structure and the environmental,the pipelines are usually buried in a trench. Thepresent paper will pay more attention on the pipeline vertical,upheaval buckling. The first pa-per on pipeline buckling in the open literature appeared in 1974. Palmer and Baldry[1] demon-strated that the constraint of expansion of a pipeline on account of raised internal pressurecould induce buckling through a small-scale test. Hobbs[2-3] presented a summary of the basicmodels of buckling in a long pipeline in 1981 and 1984. An explosion of interest has been in-duced in pipelines thermal buckling in the early eighties of the last century as some upheavalbuckling incidents occurred in the North Sea.As Guijt[4] pointed out that at least five upheavalbuckling incidents occurred in the North Sea, three of which occurred in 1989, and all withsignificant cost penalties. The first incident, associated with Maersk Oilog Gas AS’Rolf

第 15 卷第 6 期 船舶力学 Vol.15 No.62011 年 6 月 Journal of Ship Mechanics Jun. 2011

Article ID： 1007-7294（2011）06-0678-10

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pipeline, occurred in 1986[5]. The cost of a failure in such a pipeline is so high, not only forrepairs but also for lost production. Therefore from then on, the question of thermally inducedbuckling is of considerable economic importance,and a substantial literature has built up inthe following 30 years. The papers presented at a special session of the 1990 Offshore Tech-nology Conference are perhaps the best explanation that the pipeline buckling research is ahot topic in this area.The 22nd Annual OTC also implied that the subsea pipeline bucklingresearches reach to peak at that time.Century 90’s up to date, the theoretical and experimen-tal studies on the subsea buried pipeline buckling are continued. The basic models presentedby Hobbs[3] have been modified and refined with considering the pipeline imperfection[6-12] andthe elastoplastic behaviour of buckling pipelines[13-14]. Furthermore, the numerical method hasbeen applied to analyze the pipeline buckling behavior[15-17]. This paper describes its develop-ment, and its relation to today’s applications for upheaval studies.

2 Theoretical solutions for perfect straight pipelines

The compressive stress required to buckle a straight, pin-ended column (‘Euler buck-ling’) is inversely proportional to the square of the slenderness of the column; and it is obvi-ous that all pipes are very slender over sufficiently large lengths.Submarine pipelines oftencarry products which are much hotter than the surrounding seawater.The potential thermal ex-pansion is restrained by friction between the pipeline and the seabed,causing the developmentof large compressive axial forces in the line,which can lead to Euler buckling of the pipeline.Similar problems occurred in railway tracks morethan 70 years ago, and the percipient analysis byMartinet and the extensive literature summarizedby Kerr[18-19] are very closely related to the thermalbuckling problems in pipelines. Hobbs[2-3] present-ed a summary of the basic models of buckling ina long pipeline (Ref. to Fig.1).

In the model, denoting the cross-sectional area of the pipe by A,Young’s modulus by E,the coefficient of linear thermal expansion by α and the temperature change by △T. The forcecreated by full restraint of thermal expansion is,

P0=EAα△T (1)If the axial strain ε is completely restrained,the axial compressive force generated and

available to participate in buckling is,

P0=Aprt 0.5-! "υ (2)

where υ is Poisson’s ratio, p is the internal pressure, and t and r are the pipe wall thicknessand radius respectively.

A linear differential equation has been established to describe the deflected shape of the

Fig.1 Force analysis of a pipeline sectionwith vertical buckling

680 船舶力学 第 15 卷第 6 期

buckled part of the pipeline which is treated as a beam column under uniform lateral load.Itis assumed that the bending moment at the lift-off point is zero. In the notation of Fig.1,

y″+n2y+ m8 4x

2-L

22 2=0 (3)

where m=w/EI, n2=P/EL and w is submerged weight of pipeline (including weight coat) per u-

nit length.The equation is solved and the following results are obtained:

P=80.76EIL

2 (4)

P0=P+wLEI 1.597×10-5EA准wL

5-0.25 准E2 2I

22 21/2 (5)

where P is the axial load in the buckle and P0 is the axial load away from the buckle (Fig.1),准 is the coefficient of friction between pipe and subgrade.

The maximum amplitude of the buckle

vm=2.408×10-5wL4

EI (6)

And the maximum bending moment, at x=0, is

2

M=0.069 38wL2

(7)A further result of practical interest is the size of the slipping length adjacent to the

buckle,

Ls=P0-P准w -0.5L (8)

Fig.2 is the vertical buckling result for atypical pipeline with a practical range of frictioncoefficients,0.2≤准≤0.4,calculated by author withHobbs’s model in Bohai Bay.The pipe has anoutside diameter of 323.9mm,and a wall thicknessof 12.7mm giving a design internal pressure 4.65MPa and design temperature rising 85℃.

Fig.2 shows that there exists the largest ‘safe’temperature change to avoid vertical buckling in this particular pipeline is about 45℃ if thecoefficient of friction 准 is taken as 0.4.The curves in Fig.2 look like U-shape,that is to say,there are two possible buckle lengths for one temperature change. Hobbs [3] explained that theequilibrium path from A to B is unstable,which is a consequence of the assumption of fullymobilized friction even for vanishingly small displacement.The equilibrium path from B to Cdescribes the relationship between the temperature and the buckle length for a pipeline withsmall imperfection.The temperatures against length of buckle and amplitude curves are theclassical results in pipeline thermal buckling analysis until now. Nonetheless, as Hobbs [3] re-marked,the model no account has been taken of the initial out-of-straightness and furthernumerical work on the effects of initial imperfections would be valuable.

Fig.2 Buckle wavelength v.s. frictioncoefficient of foundation soil

3 Studies on imperfect pipelines

Engineering practice shows that the in-situ shape of a buried pipeline in the field is farfrom being straight.The imperfections concerned may include a crest in the sea bed profile, ora prop,as when isolated rock is located immediately below the line or another pipeline is to becrossed.Other less obvious possibilities include the free span gap or trench step, or an angu -larly mismatched field joint.With the initial imperfection, the pipeline may develop initial de -formation. Under the temperature rising, upheaval may take place in the pipeline because ofthe existence of initial imperfection.Many researches have been done on imperfect pipelines,such as, Taylor and Gan[20], Boer et al[21], Friedman[22], Richards and Andronicou[23], Ju and Kyr-iakides[24], Pedersen and Jensen[25], Ballet and Hobbs[6], Taylor and Tran[7-8], Maltby and Calla-dine[9], Croll[11], Hunt and Blackmore[12]. Among them, Taylor and Gan[20], Taylor and Tran[7-8],Maltby and Calladine [9] and Hunt and Blackmore [6] should be mentioned. Taylor and Tran(1996) summarized three basic types of initial imperfection for subsea buried pipeline, whichare illustrated in Fig.3.

The initial imperfection is denoted by amplitude vom and wavelength L0 or Li as shown.Whilst Li is determined from simple statics, L0 is subject to individual engineering judgment.

Fig.3 Typical imperfection configurations (Taylor and Tran, 1996)

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682 船舶力学 第 15 卷第 6 期

The physical problems together with the key conceptual and mathematical models for eachtype of initial imperfection have been set out based on a stress-free-when-initially-deformeddatum in the paper.And all models presume system symmetry and seabed trench-bottom rigid -ity, together with relatively small deformations and linear elastic properties.Similar analysis toperfect pipeline, internally generated temperature and pressure rises over ambient suffered bythe pipe, △T and p respectively, in terms of the so-called pre-buckling force P0 can be cal-culated by Eq.(1) and Eq.(2).

In the first case, so-called empathetic model, the pipeline remains in continuous contactwith some vertical undulation in an otherwise idealized horizontal and straight line.The em -pathetic model employs idealized buckling mode to propose

v0=v0m0.707-0.261 76 π

2x2

L0

2 +0.293cos 2.86 πxL0� ��

���

����

(9)

where v0 denotes imperfection displacement above the horizontal datum surface at x, 0<x<L0,

with v0m =v0 x=0 , in conjunction with the unique amplitude/wavelength relationship also origi -

nating from idealized studies:v0m

L0

4 = 2.407×10-3wEI = vm

L4 (10)

where w denotes the pipe’s submerged self-weight (or effective download) per unit length andI represents the pipe’s cross-sectional second moment of area.

Presuming fully mobilized friction resistance to be developed, the model definition iscompleted by means of the longitudinal equilibrium and compatibility expressions:

P0-Pe=准wL2 +准wLs (11)

andP0-Pe� �L2AE -uf +

准wLs

2AE =0 (12)

In accordance with the requirements of the potential energy theorem,the pipe bucklingforce Pe in Eq.(11) can be obtained from

Pe=P 1- ψ1

75.6Lo

L� �2� � (13)

where P denotes the idealized buckling force.

P=80.76EIL

2 =3.962EIwvm

� �1/2 (14)

In Eq.(12) Ls is the slip length and uf is the flexural end shortening

uf =12

L/2

0乙 y′2dx-

L/2

0乙 yi′2d� �x (15)

From Eqs.(1), (2) and (11)~(15), the relationship between temperature and buckle ampli -tude can be obtained.

The isolated prop alternatively features a sharp vertical irregularity such that voids existto either side. The prop represents the undercrossing of a non-parallel pipe or the presence ofan intervening rock; stop-start trenching procedures can also be responsible. Five key stageshave been proposed in buckling development (Ref. to Fig.4).

As the temperature of the pipeline rises due to routine operation, the initial span or im-perfection wavelength Li suffers a reduction as the pipeline tightens up under compressive ac-tion P.The wavelength L reduces to some specific value Lu whereupon the pipeline lifts off theprop. Post-upheaval buckling initially involves wavelength Lu<L<Li, with L>Li ensuing if cir-cumstances so dictate.The associated differential equation for the initial imperfection takes theform

Fig.4 Isolated prop topologies (Taylor and Tran, 1996)

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EIy″- wLi

2 -Fi� � Li

2 -� �x +w

Li

2 -� �x 2

2 =0 (16)

where Fi is the shear force acting at the crown.Computational manipulation with boundary conditions gives the vertical deflection vi as

vi=w

72EI 2LiLi

2 -� �x 3

-3 Li

2 -� �x 4� � (17)

In addition to the static determination of Li in terms of imperfection amplitude vi, equa-tions for longitudinal equilibrium and compatibility together with an appropriate buckling/flexural expression are required for each of the phases (b), (d) and (e) shown in Fig.4.

For phase (b) the buckling/flexural expression is

EI y″-yi� �″ -P v0m-� �y -M-Fx+ wx2

2 =0 (18)

where M denotes the crown moment and shear force F represents half the prop force.Theboundary conditions take the form y x=L/2=y′ x=L/2=y″ x=L/2=y′ x=0=0 and y x=0=v0m .

The expression for phase (d) is as follows with the boundary conditions y x=L/2 =y′ x=L/2 =y″ x=L/2=y′ x=0=0 and y x=0=vm.

EI y″-yi� �″ -P vm-� �y -M+ wx2

2 =0 (19)

And the differential equations for phase (e) are

EI y″-yi� �″ -P vm-� �y -M+ wx2

2 =0 (20)

for 0≤x≤Li /2

EIy″-P vm-� �y -M+ wx2

2 =0 (21)

for Li /2≤x≤L/2The boundary conditions appertaining to Eq.(20) and Eq.(21) are y x=0 =vm, y′ x=0 =0 and

y x=L/2=y′ x=L/2=y″ x=L/2 =0, respectively.The general solution to the buckling/flexural expression takes the following form

y=c1cosnx+c2sinnx+k1+k2x-wx

2

EIn2 (22)

where the constants of integration c1 and c2 can be determined in accordance with the bound-ary conditions of each phase.

The characteristic equation reveals the relationship between nL and Li /L, which can beobtained by the further computational manipulation.Model definition of each phase is com -pleted by means of longitudinal equilibrium and compatibility expressions of similar form toEqs.(11) and (12).

684 船舶力学 第 15 卷第 6 期

The third case in Fig.3 occurs where the above voids become infilled with leaching sandand represents a special sub -case of the first.If marine conditions permit and particularlywhere continuous burial is involved, the prop-attendant voids discussed above can become in -filled to the extent that the in -service,pre -buckling flexural stage is no longer effectivelyavailable. The post-upheaval buckling is divided into two stages. With L<Li in the early post-buckling phase,a vectorial equilibrium compatibility analysis is employed here by using themoment curvature relationship

EI y″-yi! "″ -P vm-! "y -M+ wx2

2 =0 (23)

This equation is as the same as Eq.(20), whereas the different boundary conditions, y x=0

=vm, y x=L/2 =yi x=L/2 , y′ x=0 =0, y′ x=L/2 =yi′ x=L/2 and EIy″ x=L/2 =EIyi″ x=L/2 induce the different

characteristic equation. For the later stage L>Li, requires similar analysis to that given aboveand typified by Eqs.(20) and (21). It is also necessary to employ longitudinal equilibrium andcompatibility expressions to relate P to the temperate rise.

In addition to establishing the above upheaval model for imperfection pipelines,Taylorand Tran (1993) developed the buckle model for protected pipeline, such as trenching andburial, continuous or discrete, together with the employment of fixed anchorages.

In all the calculations above,the buckle has been assumed to be completely symmetric.Ballet and Hobbs[6] investigated the possibility of asymmetric buckling in the prop case andfound the results. Hunt and Blackmore[12] studied the effects of asymmetric bed imperfections,typified by a step and adopted a shooting method to solve the standard fourth-order linear or -dinary differential equation. Comparison between two typical types, the prop and the step, im -plied that more profound destabilizing role can be attributed to the step than the prop. Maltbyand Calladine[9] proposed a simple formula for the axial load at which the localization of buck-ling occurs based on sinusoidal imperfection assumption.

p= EIQ0 /Y0! "1/2

(24)where p is the compressive force in pipe,Q 0 is themaximum value of soil resistance per unit length,and Y 0 represents the amplitude of initial imperfec-tion.

Parametric studies of Empathetic models,em-ploying the foregoing data in Bohai Bay,have beeninvestigated with graphical comparison illustrated byFig.5 for various initial imperfection amplitudes v0m ,ranging from 50mm to 300mm.With regard to the re-spective temperature rise/buckling amplitude lociand the results given in Fig.5,it can be seen that on-ly the relatively small imperfections typified by v0m=

Fig.5 Parametric studies of Empatheticmodel

第 6 期 GAO Xi-feng et al: Overview of Upheaval Buckling … 685

50 and 100mm display a maximum temperature together with the associated snap bucklingphenomenon. The remaining four cases, v 0m =150 to 300mm, generate stable post-buck-ling path.The safe temperature obtained by imperfect model is obviously lower than the onecalculated by perfect model.

4 The future

The economic importance of submarine pipelines has greatly increased in recent yearswith the development of offshore oil and gas fields in many parts of the world. The vulnerabil -ity of pipelines to thermal stress is well documented in many case studies and review papers.Analysis and numerical modelling of pipelines upheaval buckling have progressed over thelast thirty years, broadly from the classical analysis, to one covering initial imperfections, toone additionally covering material non-linearity, to one additionally including large pipe dis -placement and associated cover non-linearity.However, there are still some involved problemsneed to be dealt with.

The trend for offshore oil and gas extraction to take place in deeper and more remote wa -ters is leading to the construction of longer pipelines that operate at higher temperatures andpressures. In shallow water, it is common practice to bury a pipeline for protection from trawlgear and to prevent buckling. However, in deep water, there is no requirement for burial, andit is more cost-effective to simply lay pipelines on the seabed.Without the lateral restraint thehorizontal snaking is dominant for pipeline to relieve the thermal stresses.Therefore, the stud -ies of pipelines lateral thermal buckling and the soil lateral resistance will become a hot topicfor unburied submarine pipeline.

References

[1] Palmer A C, Baldry J A S. Lateral buckling of axially-compressed pipelines[J]. Journal of Petroleum Technology, 1974, 26(11): 1283-1284.

[2] Hobbs R E. Pipeline buckling caused by axial loads[J]. Journal of Constructional Steel Research, 1981, 1(2): 2-10.[3] Hobbs R E. In-service buckling of heated pipelines[J]. Journal of Transportation Engineering, ASCE, 1984, 110(2): 175-

189.[4] Guijt J. Upheaval buckling of offshore pipeline: overview and introduction[C]// In proceedings of the 22nd Annual OTC.

Houston, Texas, 1990: 573-578.[5] Nielsen N J R, Lyngberg B, Pedersen P T. Upheaval buckling failures of insulated burial pipelines-a case story[C]// In

proceedings of the 22nd Annual OTC. Houston, Texas, 1990: 581-592.[6] Ballet J P, Hobbs R E. Asymmetric effects of prop imperfections on the upheaval buckling of pipelines[J]. Thin- Walled

Structures, 1992, 13: 355-373.[7] Taylor N, Tran V. Prop-imperfection subsea pipeline buckling[J]. Marine Structures, 1993, 6: 325-358.[8] Taylor N, Tran V. Experimental and theoretical studies in subsea pipeline buckling[J]. Marine Structures, 1996, 9(2):

211-257.[9] Maltby T C, Calladine C R. An investigation into upheaval buckling of buried pipelines--i. Experimental apparatus and

some observations[J]. International Journal of Mechanical Sciences, 1995, 37(9): 943-963.[10] Maltby T C, Calladine C R. An investigation into upheaval buckling of buried pipelines--ii. Theory and analysis of ex-

perimental observations[J]. International Journal of Mechanical Sciences, 1995, 37(9): 965-983.

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[11] Croll J G A. A simplified model of upheaval thermal buckling of subsea pipelines[J]. Thin-Walled Structures, 1997, 29(1-4): 59-78.

[12] Hunt G W, Blackmore A. Homoclinic and heteroclinic solutions of upheaval buckling[J]. Phil. Trans. R. Soc. Lond.,1997, 355(4): 2185-2195.

[13] Friedmann Y, Debouvry B. Analytical design method helps prevent buried pipe upheaval[J]. Pipeline Industry, 1992, 11:63-68.

[14] Villarraga J A, Rodríguez J F, Martínez C. Buried pipe modeling with initial imperfections[J]. Journal of Pressure VesselTechnology, 2004, 126(5): 250-257.

[15] Zhou Z J, Murray D W. Behaviour of buried pipelines subjected to imposed deformations[C]// 12th Int. Conference onOffshore and Arctic Engineering. ASCE, 1993, II: 115-122.

[16] Pasqualino I P, Alves J L D, Battista R C. Failure simulation of a buried pipeline under thermal loading[C]// Proceedingsof the 20th International Conference on Offshore Mechanics and Arctic Engineering (OMAE), 2001, OMAE2001-4124.Rio De Janeiro, Brazil, 2001.

[17] Einsfeld R A, Murray D W, Yoosef-Ghodsi N. Buckling analysis of high-temperature pressurized pipelines with soil-structure interaction[J]. Journal of the Brazilian Society of Mechanical Sciences and Engineering, 2003, 25(2): 164-169.

[18] Kerr A D. On the stability of the railroad track in the vertical plane[J]. Rail Int., 1974, 5: 131-142.[19] Kerr A D. Lateral buckling of railroad tracks due to constrained thermal expansion[M]. Railroad Track Mechanics and

Technology, Pergamon Press, Oxford, 1978.[20] Taylor N, Gan A B. Submarine pipeline buckling imperfection studies[J]. Thin-Walled Structures, 1986, 4: 295-323.[21] Boer S, Hulsbergen C H, Richards D M, et al. Buckling considerations in the design of the gravel cover for a high tem-

perature oil line[C]// Proc. 18th OTC, 1986, May, 5294. Houston, Texas, 1986: 1-8.[22] Friedmann Y. Some aspects of the design of hot buried pipelines[C]// Offshore Oil and Gas Line Technology Conference,

1986. Paris, France, 1986: 1-34.[23] Richards D M, Andronicou A. Seabed irregularity effects on the buckling of heated submarine pipelines[C]// WEMT Ad-

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1988, 110(4): 355-364.[25] Pedersen P T, Jensen J J. Upheaval creep of buried heated pipelines with initial imperfections [J]. Marine Structures,

1988, 1: 11-22.

海底埋设管线屈曲剧变理论研究综述

高喜峰 1， 刘 润 1， 杜尊峰 1， 谭振东 2

（1 天津大学 水利工程仿真与安全国家重点实验室， 天津 300072； 2 总后军事交通运输研究所， 天津 300161）

摘要: 海上生产的烃类必须在高温高压下运输以使流体舒缓而避免蜡组分的凝固。 由于内外压差和温差引起强制膨胀

导致轴向压缩力，此时便产生了海底管线的屈曲。 此压缩力既能引起管线在海床平面内的侧向屈曲，也能造成垂向屈

曲。 通常将管线埋于沟槽内从而确保与其它海洋活动的干扰最小化。 在此情况下，侧向土约束超过了由管线浮重造成的

垂向隆起约束。 因此，要特别注意挖沟埋设管线的垂向屈曲。 最近对中国高温管线和渤海湾几起事故的关注引发中国海

洋工程师对此现象的强烈兴趣。 文章综述了海底埋设管线垂向屈曲的理论研究历史，以及作者对此问题的一些经验。 希

望海洋工程师能从本领域的文献综述和总结中受益。 文中同时研究了管线的防护层设计，最后明确了未来发展的方向。关键词: 海底管线； 沟槽； 埋设； 屈曲剧变

中图分类号: TE973 文献标识码: A作者简介： 高喜峰（1975-），男，博士，天津大学建筑工程学院讲师；

刘 润（1974-），女，博士，天津大学建筑工程学院教授；杜尊峰（1984-），男，博士，天津大学建筑工程学院讲师；谭振东（1979-），男，博士，总后勤部军事交通运输研究所高级工程师。

第 6 期 GAO Xi-feng et al: Overview of Upheaval Buckling … 687