12-0781.pdf

Appl. Math. Mech. -Engl. Ed., 33(6), 781–796 (2012)DOI 10.1007/s10483-012-1586-6c©Shanghai University and Springer-Verlag

Berlin Heidelberg 2012

Applied Mathematicsand Mechanics(English Edition)

Engineering measures for preventing upheaval buckling of buried

submarine pipelines∗

Run LIU (4 d)1, Wu-gang WANG (�Éf)1, Shu-wang YAN (A1!)1,

Xin-li WU (Ç#|)2

(1. State Key Laboratory of Hydraulic Engineering Simulation and Safety,

Tianjin University, Tianjin 300072, P. R. China;

2. School of Engineering Design, Pennsylvania State University,

University Park, PA 16802, Pennsylvania, USA)

Abstract In-service hydrocarbons must be transported at high temperature and highpressure to ease the flow and to prevent the solidification of the wax fraction. The hightemperature and high pressure will induce the additional stress in the pipeline, whichresults in the upheaval buckling of the pipeline. If such expansion is resisted, e.g., by thefrictional effects of the foundation soil over a kilometer or of a pipeline, the compressiveaxial stress will be set up in the pipe-wall. When the stress exceeds the constraint ofthe foundation soil on the pipeline, suddenly-deforming will occur to release the internalstress, similar to the sudden deformation of the strut due to stability problems. Theupheaval buckling may jeopardize the structural integrity of the pipeline. Therefore,effective engineering measures against this phenomenon play an important role in thesubmarine pipeline design. In terms of the pipeline installation and protection measurescommonly used in Bohai Gulf, three engineering measures are investigated in great details.An analytical method is introduced and developed to consider the protection effect of theanti-upheaval buckling of the pipeline. The analysis results show that the amplitudeof the initial imperfection has a great effect on the pipeline thermal upheaval buckling.Both trenching and burial and discrete dumping are effective techniques in preventing thepipeline from buckling. The initial imperfection and operation conditions of the pipelinesdetermine the covered depth and the number of layers of the protection measures.

Key words submarine buried pipeline, thermal stress, upheaval buckling, protectionmeasure

Chinese Library Classification TE542010 Mathematics Subject Classification 32G05

1 Introduction

In-service hydrocarbons must be transported at high temperature and high pressure toease the flow. The thermal stress together with the Poisson effect will cause the pipeline toexpand longitudinally. However, due to the soil restraint, e.g., by the frictional effects over a

∗ Received Jan. 18, 2011 / Revised Mar. 5, 2012Project supported by the Foundation for Innovative Research Groups of the National NaturalScience Foundation of China (No. 51021004), the National Natural Science Foundation of China(No. 40776055), and the Foundation of the State Key Laborary of Ocean Engineering (No. 1002)Corresponding author Run LIU, Professor, Ph.D., E-mail: [email protected]

782 Run LIU, Wu-gang WANG, Shu-wang YAN, and Xin-li WU

kilometer or of a pipeline, the pipeline cannot expand freely, and consequently, a compressiveaxial stress will be set up in the pipe wall. When the compressive force reaches or even exceedsthe constraint of the foundation soil, suddenly-deforming will occur to release the accumulatedinternal stress in the pipe wall. This phenomenon is similar to the theoretical stability analysisof a strut with two ends pinned and compressed by the anti-symmetry concentrated force. Theaccumulated compressive forces is frequently large enough to induce either the lateral bucklingor the vertical buckling of the buried pipelines[1]. Liu et al.[1], Taylor and Tran[2], Hobbs[3],and Maltby and Calladine[4] showed that vertical buckling was the main failure mode for theburied heated pipelines. Once the integrity of the pipeline is destroyed, the oil and gas will leakout, which will not only cause a great waste of resources, but also bring dangers to the livingenvironment of human beings. Therefore, upheaval buckling studies of the submarine pipelineplay an import role in the pipeline design.

Many researches have been carried out in this area since the early eighties of the last century.In 1985, Palmer and Davies[5] pointed out that the principal techniques for the trenching ofsubmarine pipelines comprised jetting, fluidization, milling, and ploughing. Each method hadits own particular features and constraints. By comparing with each other, it was concluded thatploughing was a less risky trenching technique. In 1985 and 1989, Schaap[6–7] studied the naturalbackfill of the trench, which was considered to be an economical and acceptable solution forcovering the pipeline after trenching. He presented a practical calculation method to determinethe natural backfill as a function of time for the specified trench profile and local naturalenvironment, and proposed a reliable empirical natural burial prediction model in conjunctionwith the deformation capacity of pipelines. In 1988, Pedersen and Jensen[8] expounded that, fora given pipeline, a given temperature, and a given burial depth, the mathematical model couldbe used to impose the restrictions on the allowable imperfections of the unloaded pipe centerline (plastic deformations) combined with the restrictions on the allowable imperfections of thetrench. They also obtained the partial analytical solutions. In 1991, Gokce[9] concluded thata deeper burial depth of a pipeline could greatly improve the capacity of the pipeline to bearthermal stresses. In 2004, Sahel and Hamdan[10] analyzed the stability of the anchored andtrenched pipeline with initial imperfections, and discussed the optimal position of the anchor.

In China, Hu et al.[11] carried out a model test to study the hydrodynamic force incurred bythe wave current of the unprotected smooth pipeline and the pipeline protected by mattressesin 1991. They discussed the shape and interval of mattresses. In 1997, Ma[12] explained the keypoints of the mattress protection measure, including the calculation method of the mattressweight and the mattress installation method. In 2001, Jin[13] carried out a model test to studythe hydrodynamic force of the submarine pipeline protected by mattresses. The drag forceand the lift force of the pipeline with a mattress and the mattress itself were measured. In2002, Cao[14] described the characteristics of the mattress and the imperfection of the mattressstructure. The suitable conditions of mattresses with the shapes of diamond, trapezoid, ands-type were discussed. In 2005, Liu et al.[1] made a research on the upheaval buckling of asubmarine pipeline with initial imperfections in accordance with the case study, and pointedout that increasing the covered depth was an effective way to prevent the pipeline from upheavalbuckling. Literatures have shown that there are fewer studies on the upheaval buckling ofsubmarine pipelines with the initial imperfection under protection. As a result, analyzing thecharacteristics of the pipeline thermal buckling and comparing the effectiveness of differentprotection measures are very important in offshore pipeline engineering.

According to different protection measures, the thermal buckling analysis of a submarinepipeline is carried out with the consideration of the initial imperfection. An analytical methodis introduced and developed to consider the protection effect of the anti-upheaval buckling ofthe pipeline. This paper is organized as follows. Section 2 presents some pipeline protectionmeasures. Section 3 analyzes the upheaval buckling. Some cases are studied in Section 4.Finally, some conclusions are summarized in Section 5.

Engineering measures for preventing upheaval buckling of buried submarine pipelines 783

2 Pipeline protection measures

In terms of the offshore engineering practice in China, pipeline protection measures can besummarized in three categories.2.1 Natural backfill of trench

Since the natural backfill is an economical and convenient measure for covering the pipelineafter trenching, it is widely used in offshore engineering (see Fig. 1). The natural backfill isusually a slow procedure, and the backfill degree of the trench is difficult to be determined.If the backfilled soil is clay, the strength recovery will take a long time because of the lowpermeability of this type of soil. Therefore, for the conservative purpose, the soil submarineweight and the resistance above the pipeline under this protection state are not taken intoaccount in the pipeline thermal buckling analysis.

Fig. 1 Natural backfill after trench

Nevertheless, the resistance force of the trench is considered in the analysis, and it can bedetermined by

Fv1 = q sin θ + ϕLq cos θ, (1)

where Fv1 is the restraint force of the trench to the pipeline. q is the submerged self-weight ofthe pipeline per unit length, kN·m−1. ϕL is the lateral friction coefficient between the pipelineand the soil. θ is the angle of the trench slope.2.2 Trenching and burial

In general, burying pipelines by machine after trench ploughing is widely adopted to preventthe pipeline from ocean current scouring and to ensure the on-position stability. This protectionmeasure also can help the pipeline to avoid damages of ship anchoring and fishing activities.The design sketch of this protection measure is illustrated in Fig. 2.

Fig. 2 Typical burial topology

The soil resistance (Fv2) above the pipeline is taken into account with this kind of protectionmeasures and the soil resistance can be calculated as follows[15].

(i) Cohesive soil foundation

Fv2

γ′HD= 1 + 0.1

D

H+

2cu

γ′H

(H

D+

1

2

)

, (2)

where γ′ is the submerged unit weight, kN·m−3. D is the external diameter, mm. H is thedepth of embedment, m. cu is the average undrained shear strength on the vertical slip planes,kPa.


(ii) Sand soil foundation

Fv2

γ′HD= 1 + 0.1

D

H+ K tanϕ

(H

D

)(

1 +D

2H

)2

, (3)

where ϕ is the angle of the internal friction, and K is the lateral soil pressure coefficient.The total uplift resistance per unit length of the pipeline Fv2t is

Fv2t = Fv2 + q. (4)

In terms of the soil disturbance due to installation activities, such as trenching, paving, andburying, the soil strength decreases, especially for cohesive soil. In the analysis, the remoldingstrength of the subsoil should be adopted.2.3 Discrete dumping

The discrete dumping protection measure, which installs concrete blocks on the pipelineintermittently or continuously, can be adopted when the trenching and burial protection isdifficult or uneconomical to be used. The principle of this protection measure is to add theweight to the pipeline until it is heavy enough to avoid buckling. There are two main formsof discrete dumping. One is the single concrete block, which is put on the pipeline accordingto the design after the pipeline installation. The other is the prefabricated concrete mattress,which is laid on the pipe after the pipeline installation.

The sketches of the discrete dumping measure are illustrated in Fig. 3, in which Lm denotesthe space between the two adjacent concrete blocks, and Lmat is the design width of the concretemattress.

Fig. 3 Discrete dumping protection

Available results show that the vertical resistance provided by the single concrete block isequal to the negative buoyancy itself[12]. When the space between different concrete blocks isequal to zero, the resistance is about 20% to 30% of the weight of the pipeline. The concretemattress is widely used in offshore engineering in recent years, and the vertical resistance canbe calculated as follows[16]:

Fv3 = Wmat = 2Dghρ(30

35

)2

, (5)

where Wmat is the effective weight of the concrete mattress per unit length, kN·m−1. ρ is theconcrete density. h is the thickness of the concrete mattress.

With this protection measure, the total ultimate resistance (Fv3t) can be obtained by

Fv3t = Fv3 + q. (6)

3 Upheaval buckling analysis

3.1 Initial imperfection of submarine pipeline

The pipelines usually have an initial deformation during the manufacture and installationprocedure, e.g., the departure of the center during the manufacture, a crest in the sea bedprofile, a soil platform caused by the rooter, or a prop appeared when the isolated rock is


located immediately below the line or another pipeline is to be crossed as required in thedesign. Other less obvious possibilities that can cause initial deformations include the freespan gap, the trench step, and the angularly mismatched field joint. Such initial deformationsof the pipeline are defined as initial imperfections. The continuous prop mode is the mostcommon case among all kinds of pipeline initial imperfections[2]. This type of modes occurswhere the seabed is uneven, e.g., there are some heaves or bulges and the voids between theseabed and the pipeline are infilled with soil under the current action. With the infilled prop,the pipeline is approximately laid on the undulation foundation, as shown in Fig. 4. Since thefurther deformation always develops at an initial imperfection, the initial out-of-straightness willweaken the pipeline resistance against the upheaval buckling under thermal stresses. Therefore,the continuous prop initial imperfections are taken into account in analyzing the validity ofthe pipeline protection measures. Based on the research findings of Talor and Tran[2], therelationship between the amplitude and the buckle length can be depicted as follows:

L0 = 5.825 9(VomEI

q

)1

4

, (7)

where Vom is the maximum vertical amplitude of the continuous prop imperfection topology,mm. L0 is the buckle length of the initial imperfection topology, m. q is the submerged self-weight of the pipeline per unit length, kN·m−1. E is the Young modulus. I is the secondmoment of the area of the cross section.

Fig. 4 Continuous prop initial imperfection

The upheaval buckling of the submarine pipeline with the continuous prop initial imper-fection always develops at the initial imperfection under the actions of the thermal stress andinternal pressure. The upheaval buckling is composed of the following two main stages.

(i) The buckling length is no more than the initial imperfect wave length, and the amplitudeof the upheaval segment increases.

(ii) The buckling length is larger than the initial imperfect wave length, and the amplitudeof the upheaval segment increases.

The sketches and force analysis of these two stages are shown in Fig. 5.

In Fig. 5, Rv is the vertical resistance force of the buckled pipeline at the peel point. Rv =L2 Fv, in which Fv is the vertical restraint force of the peel point. Rh is the horizontal axial

resistance force of the buckled pipeline at the peel point, Rh = L2 ϕAFv. Rs is the slide friction

resistance between the pipeline and the subsoil, Rs = ϕAqLs.

3.2 Natural backfilled pipeline

In this case, the linear differential equation and the solution procedure condition for thedeflected shape of the buckled part of the pipeline are as follows.

In the first stage of upheaval buckling, Lu < L < Li, the following bending moment equilib-rium equation can be established (see Fig. 5(a)):

Mx = EI(V′xx − Vi′xx) = P (Vm − V ) −Fv1x

2

2+ M, (8)


Fig. 5 Force analysis for continuous prop model

and the quintessential buckling model employs a potential energy W approach with

W =

∫

L0

2

0

EI

2(V′xx − V0′xx)2dx +

∫ L

2

L0

2

EI

2(V′xx − V0′xx)2dx +

∫

L0

2

0

q(V − V0)dx

+

∫ L

2

L0

2

q(V − V0)dx −

∫

L0

2

0

P

2(V 2

′x − V 20′x)dx −

∫ L

2

L0

2

P

2(V 2

′x − V 20′x)dx, (9)

where Mx is the bending moment of the buckle segment pipeline at x. P is the axial force ofthe buckle segment. Vm is the maximum vertical amplitude of the buckled pipe, mm. V is thevertical displacement of the buckle segment at the section x. Fv1 is the restraining force of thetrench to the pipeline. M is the bending moment at the crown in the continuous prop model.Lu is the buckle length at the upheaval. Li is the buckle length of the imperfection topology.Other notations remains the same as forgoing.

In accordance with the corresponding boundary conditions

V |0 = Vm, V |L

2

= Vi|L

2

, V′x|0 = 0, V′x|L

2

= Vi′x|L

2

, V′xx|L

2

= Vi′xx|L

2

together with the assumption n2 = P/(EI). The expression of the buckle amplitude and themaximum bending moment of the pipeline can be written as

V = C1 +q

EIn4

(

C2 cos(nL

2− nx

)

+ C3 sin(nL

2− nx

)

− C2

−(nL)2

12

(

2Li

L− 3

q + Fv1

2q

)

+n2Li

3x − n2x2 q + Fv1

2q

)

, (10)

M =q

n2

(

−C2 cos(nL

2

)

− C3 sin(nL

2

)

+(nLi)

2

24− 2

)

. (11)

Substituting x = 0 into Eq. (10) yields the maximum in the first stage of upheaval bucklingas follows:

Vm = C1 + C4qL4

EI, (12)


where

C1 =qL4

1 152EI

(Li

L− 1

)3(Li

L+ 3

)

,

C2 = −1 −Fv1

q+

(nL)2

24

(Li

L− 1

)(Li

L− 3

)

,

C3 =nL

3

(Li

L− 3

q + Fv1

2q

)

+(nL)3

48

(Li

L− 1

)2

,

C4 =1

nL4

(

C2 cos(nL

2

)

+ C3 sin(nL

2

)

− C2 −(nL)2

12

(

2Li

L− 3

q + Fv1

2q

))

.

The axial force P consists of the internal axial force P0, the lateral resistance of the peelpoint Rh, and the friction resistance between the pipeline and the subsoil Rs, which can bedetermined by[19]

P = P0 − Rh − Rs, (13)

Rh =L

2ϕAFv1, (14)

Rs = ϕAqLs, (15)

where P0 is the so-called pre-buckling force caused by the constrained expansions set up bythermal and internal pressure actions. A is the cross-sectional area. α is the coefficient of thelinear thermal expansion. ϕA is the axial friction coefficient. Ls is the axial slip length.

For computational convenience, the internal pressure actions can be changed into the pressure-equivalent temperature. The free axial strain ε, due to a positive pressure difference p betweenthe oil and the sea, is given in terms of the thin wall axial and hoop stresses in the pipe by

ε =1

E

(pr

2t− ν

pr

t

)

.

If ε is completely restrained, the axial compressive force generated and available to the partic-ipate in buckling is

EAε =Apr

t(0.5 − ν),

in which p the internal pressure, t is the wall thickness of the pipe, and ν is the Poisson ratio.It can be obtained by

P0 = AEα∆T = AEα(T + T ′), (16)

where AEαT is the pre-buckling force caused by the temperature difference, AEαT ′ is thepre-buckling force caused by the internal pressure, and

AEαT ′ =Apr

t(0.5 − ν).

With dWdVm

= 0 and Eqs. (11)–(12), the relationship between the upheaval amplitude and theaxial force can be established. By using the axial force and the bending moment in the upheavalpipeline segment, the maximum compressive stress can be obtained as follows:

σm =P

A+

MD

2I. (17)


Comparing σm with the material yield stress σyld, whether the pipe enters the state of yielddamage can be judged.

In the second stage of the upheaval buckling, L > Li, and the bending moment equilibriumcondition can be expressed as the following two equations (see Fig. 5(b)):

(i) 0 < x < Li/2

Mx = EI(V′xx − Vi′xx) = P (Vm − V ) −Fv1x

2

2+ M, (18)

(ii) Li/2 < x < L/2

Mx = EI(V′xx) = P (Vm − V ) −Fv1x

2

2+ M. (19)

From n2 = PEI and the boundary conditions V |0 = Vm, V′x|0 = 0, and V |L/2 = V′x|L/2 = 0,

the vertical displacement of different parts in the upheaval segment can be obtained as follows:(i) 0 < x < Li/2

V =q

EIn4

(

C5 cos(nx) −nLi

3sin(nx) + C6 +

n2Li

3x −

q + Fv1

2qn2x2

)

, (20)

(ii) Li/2 < x < L/2

V =Fv1

EIn4

(

C7 cos(nx) + C8 sin(nx) + 1 +(nL)2

8−

n2x2

2

)

. (21)

And the maximum bending moment at x = 0 is

M =q

n2

(

−C7 +nLi

6sin

(nLi

2

)

+ cos(nLi

2

)

+(nLi)

2

24− 2

)

, (22)

where

C5 =Fv1

q

(

−nL

2sin

(nL

2

)

− cos(nL

2

))

−nLi

6sin

(nLi

2

)

− cos(nLi

2

)

,

C6 = 1 +Fv1

q

(

1 +(nL)2

8

)

−(nLi)

2

24, C7 = −

nL

2sin

(nL

2

)

− cos(nL

2

)

,

C8 =nL

2cos

(nL

2

)

− sin(nL

2

)

.

Substituting x = 0 into Eq. (20) yields the maximum vertical amplitude in the secondupheaval buckling stage as follows:

Vm =q

EIn4

(

1 + C9 +Fv1

q

(

1 +1

8(nL)2

)

−1

24(nLi)

2)

, (23)

where

C9 =Fv1

q

(

−nL

2sin

(nL

2

)

− cos(nL

2

))

−nLi

6sin

(nLi

2

)

− cos(nLi

2

)

.

Based on the former point, from dWdVm

= 0 and Eqs. (13)–(16) and (22)–(23), the relationshipamong the axial force, the buckle amplitude of the upheaval segment, and the temperaturedifference can be established. The maximum compressive longitudinal stress σm in the pipe-wall can be defined with the axial force obtained by Eq. (13) and the bending moment obtainedby Eq. (11). Comparing σm with the material yield stress σyid, whether the pipe material entersthe state of yield damage can be judged.


3.3 Pipeline with trenching and burial


In the first stage of the upheaval buckling, Lu < L < Li, and the following bending momentequilibrium equation can be established (see Fig. 5(a)):

Mx = EI(V′xx − Vi′xx) = P (Vm − V ) −Fv2tx

2

2+ M, (24)

where Fv2t is the total uplift resistance per unit length of the pipeline, which can be defined byEq. (4).

Similarly, in accordance with the corresponding boundary conditions V |0 = Vm, V |L

2

= Vi|L

2

,

V′x|0 = 0, V′x|L

2

= Vi′x|L

2

, and V′xx|L

2

= Vi′xx|L

2

together with the assumption n2 = PEI , the

expression of the buckle amplitude and the maximum bending moment of the pipeline can bewritten as follows:

V = C10 +Fv2t

EIn4

(

C11 cos(nL

2− nx

)

+ C12 sin(nL

2− nx

)

− C11

−(nL)2

12

(

2Li

L− 3

)

+n2Li

3x − n2x2

)

, (25)

M =Fv2t

n2

(

−C2 cos(nL

2

)

− C3 sin(nL

2

)

+(nLi)

2

24− 2

)

. (26)

Substituting x = 0 into Eq. (25), it yields the maximum amplitude in the first stage of theupheaval buckling as follows:

Vm = C10 + C13Fv2tL

4

EI, (27)

where

C10 =Fv2tL

4

1 152EI

(Li

L− 1

)3(Li

L+ 3

)

,

C11 = −2 +(nL)2

24

(Li

L− 1

)(Li

L− 3

)

, C12 =nL

3

(Li

L− 3

)

+(nL)3

48

(Li

L− 1

)2

,

C13 =1

nL4

(

C11 cos(nL

2

)

+ C12 sin(nL

2

)

− C11 −(nL)2

12

(

2Li

L− 3

))

.

Similarly to Eqs. (13)–(16), the axial compressive force in the buckling segment P in thiscase can be determined by

P = P0 − Rh − Rs, (28)

Rh = ϕAFv2tL

2, (29)

Rs = ϕAFv2tLs, (30)

where P0 can be calculated by Eq. (16). Combining with dWdVm

= 0 and Eqs. (27)–(28), therelationship among the axial force, the buckle amplitude of the upheaval segment, and thetemperature difference can be established. The maximum compressive longitudinal stress σm

can be defined by the axial force determined by Eq. (28) and the bending moment determinedby Eq. (26). Compare σm with the material yield stress σyid, whether the pipe material entersthe state of yield damage can be judged.



(i) 0 < x < Li/2


2

2+ M, (31)

(ii) Li/2 < x < L/2

Mx = EI(V′xx) = P (Vm − V ) −Fv2tx

2

2+ M. (32)

From n2 = PEI and the boundary conditions V |0 = Vm, V |L

2

= V′x|L

2

= V′xx|L

2

= 0, and

V′x|0 = 0, the vertical displacement of different parts in the upheaval segment can be obtainedas follows:

(i) 0 < x < Li/2

V =Fv2t

EIn4

(

C14 cos(nx) −nLi

3sin(nx) + C15 +

n2Li

3x − n2x2

)

, (33)

(ii) Li/2 < x < L/2

V =Fv2t

EIn4

(

C16 cos(nx) + C17 sin(nx) + 1 +(nL)2

8−

n2x2

2

)

. (34)


M =Fv2t

n2

(

−C7 +nLi

6sin

(nLi

2

)

+ cos(nLi

2

)

+(nLi)

2

24− 2

)

, (35)

where

C14 = −nL

2sin

(nL

2

)

− cos(nL

2

)

−nLi

6sin

(nLi

2

)

− cos(nLi

2

)

,

C15 = 2 +(nL)2

8−

(nLi)2

24, C16 = −

nL

2sin

(nL

2

)

− cos(nL

2

)

,

C17 =nL

2cos

(nL

2

)

− sin(nL

2

)

.

Substituting x = 0 into Eq. (33) yields the maximum vertical amplitude in this upheavalbuckling stage as follows:

Vm =Fv2t

EIn4

(

2 + C14 +1

8(nL)2 −

1

24(nLi)

2)

. (36)

Combining with Eqs. (28)–(30) and (36), the relationship among the axial force, the buckleamplitude of the upheaval segment, and the temperature difference can be established. Themaximum compressive longitudinal stress σm can be defined by the axial force determinedby Eqs. (28)–(30) and the bending moment determined by Eq. (35). Comparing σm with thematerial yield stress σyid, whether the pipe material enters the state of yield damage can bejudged.


3.4 Pipeline with discrete dumping


In the first stage of the upheaval buckling, Lu < L < Li, and the following bending momentequilibrium equation can be established:


2

2+ M, (37)

where Fv3t is the total ultimate resistance including the pipeline weight and the concrete blockaction determined by Eq. (6).

In accordance with the corresponding boundary conditions V |0 = Vm, V |L

2

= Vi|L

2

, V′x|0 = 0,

V′x|L

2

= Vi′x|L

2

, and V′xx|L

2

= Vi′xx|L

2

together with the assumption n2 = PEI , the expression of

the buckle amplitude and the maximum bending moment of the pipeline can be written as

V = C10 +Fv3t

EIn4

(

C11 cos(nL

2− nx

)

+ C12 sin(nL

2− nx

)

− C11

−(nL)2

12

(

2Li

L− 3

)

+n2Li

3x − n2x2

)

, (38)

M =Fv3t

n2

(

−C2 cos(nL

2

)

− C3 sin(nL

2

)

+(nLi)

2

24− 2

)

. (39)

Substituting x = 0 into Eq. (38) yields the maximum amplitude in the first stage of theupheaval buckling as follows:

Vm = C10 + C13Fv3tL

4

EI. (40)

Similarly to Eqs. (13)–(16), the axial compressive force in the buckling segment P in thiscase can be determined by

P = P0 − Rh − Rs, (41)

Rh = ϕAFv3tL

2, (42)

Rs = ϕAFv3tLs. (43)

The method to get P0 in this equation is the same as Eq. (16). According to Eqs. (40)–(41),the relationship among the axial force, the buckle amplitude of the upheaval segment, and thetemperature difference can be established. The maximum compressive longitudinal stress σm

can be defined by the axial force determined by Eq. (41) and the bending moment determinedby Eq. (39). Compare σm with the material yield stress σyid, whether the pipe material entersthe state of yield damage can be judged.


(i) 0 < x < Li/2


2

2+ M, (44)

(ii) Li/2 < x < L/2

Mx = EI(V′xx) = P (Vm − V ) −Fv3tx

2

2+ M. (45)


From n2 = PEI and the boundary conditions V |0 = Vm and V |L

2

= V′x|L

2

= V′xx|L

2

=

V′x|0 = 0, the vertical displacement of different parts in the upheaval segment can be obtainedas follows:

(i) 0 < x < Li/2

V =Fv3t

EIn4

(

C14 cos(nx) −nLi

3sin(nx) + C15 +

n2Li

3x − n2x2

)

, (46)

(ii) Li/2 < x < L/2

V =Fv3t

EIn4

(

C16 cos(nx) + C17 sin(nx) + 1 +(nL)2

8−

n2x2

2

)

. (47)


M =Fv3t

n2

(

−C7 +nLi

6sin

(nLi

2

)

+ cos(nLi

2

)

+(nLi)

2

24− 2

)

. (48)

Substituting x = 0 into Eq. (46) yields the maximum vertical amplitude in the secondupheaval buckling stage as follows:

Vm =Fv3t

EIn4

(

2 + C14 +1

8(nL)2 −

1

24(nLi)

2)

. (49)

Based on the former point, from dWdVm

= 0 and Eqs. (41)and (49), the relationship among theaxial force, the buckle amplitude of the upheaval segment, and the temperature difference canbe established. The maximum compressive longitudinal stress σm can be defined by the axialforce obtained by Eq. (40) and the bending moment obtained by Eq. (48). Comparing σm withthe material yield stress σyid, whether the pipe material enters the state of the yield damagecan be judged.

4 Case study

4.1 Engineering description

The example is an API 5L X65 grade oil pipeline in Bohai Gulf. The design internal pressureis 4.65MPa, and the operational temperature change is 85◦C. Other design parameters of thispipeline are shown in Table 1, where ν is Possion’s ratio, T is the wall thickness, ρg is the steeldensity, ρo is the oil density, ρs is the seawater density, α is the coefficient of linear thermalexpansion. The soil properties are shown in Table 2, where w is the water content, e is the voidratio, Ip is the plasticity index, c is cohesion, al−2 and El−2 are the compaction indexes, andSu is the undrained shear strength. In term of the operational conditions, upheaval bucklingmay occur.

Table 1 Design parameters of pipeline

E/(N·mm−3) ν T/mm D/mm ρg/(kg·m−3) ρo/(kg·m−3) ρs/(kg·m−3) α/◦C

2.06E+5 0.3 0.012 7 0.323 9 7 850 800 1 120 1.1E−5

Table 2 Soil properties

Horizon H/m w/(%) γ′/(kN·m−3) e Ip c/kPa ϕ/(◦) al−2/MPa−1 El−2/MPa Su/kPa

Silt clay 0−2 38.8 7.8 1.05 14.3 18 18.6 0.63 3.26 4.0−7.5

Clay 2−3 45.8 9.4 1.27 21.5 10 15.9 0.89 2.58 7.5−16


The analysis results indicate that the minimum temperature difference causing the upheavalbuckling of the pipeline without protection measures is about 22.0◦C[17], which is much lowerthan the designed temperature difference. Therefore, the protection measure must be usedto prevent the pipeline from thermal buckling. The foregoing three protection measures areinvestigated separately. In the analysis, the internal pressure is converted into an equivalenttemperature difference determined by Eq. (16). In this case, the total equivalent tempera-ture difference is 90.7◦C. Considering the soil properties, the frictional coefficient between thepipeline and the subsoil is adopted as 0.2.4.2 Analysis with natural backfill measure

In accordance with the pipeline practice of Bohai, the depth of the trench is about 1.5m,and the trench slope is 20◦. The restraint force of the trench to the pipeline can be calculated byEq. (1), which is 1.302kN·m−1. Therefore, the relationship between the temperature differenceand the upheaval amplitude with the continuous prop initial imperfection amplitude varyingfrom 50mm to 300mm can be obtained, which is shown in Fig. 6. The results show thatthe amplitude of the initial imperfection has a great impact on the upheaval buckling of thesubmarine pipeline. The larger the imperfection is, the smaller the temperature differenceneeded to lead to upheaval buckling is, which implies that the upheaval buckling is more likelyto occur in the the pipeline with large initial imperfections. Figure 6 shows that with relativelysmall imperfections, such as 50mm and 100mm, a peak value appears on the curve of thedifferential temperature against the buckling amplitude. The peak value on the curve indicatesthat the pipeline will suffer unstable deformations and a dynamic snap will occur subsequently.With a slight increment of the thermal stress at this point, the buckle amplitude of the pipelinewill increase greatly and suddenly to release the accumulated internal compressive axial stress.With the increase in the initial imperfection, the peak on the curve vanishes gradually. Itimplies that the deformation of the pipeline is continuous rather than sudden. The initialimperfection keeps increasing until the pipeline falls into failure. The broken line in the figureis the yield stress of the pipeline material. Figure 6 shows that the material of the pipeline willyield when the buckle amplitude is in excess of 750mm. The analyzing results imply that theprotection measure of the natural backfill can improve slightly the capacity of the pipeline tobear the thermal stress. For example, when Vom = 300mm, the pipeline buckling temperatureincreases from 22.0◦C to 22.7◦C. However, the increased temperature is still much lower thanthe designed value. It can be concluded that the natural backfill measure cannot achieve thepurpose of preventing upheaval buckling.

Fig. 6 Temperature difference with buckle amplitude for natural backfill of trench

4.3 Analysis with trenching and burial measure

With this measure, the trench will be filled with soil by machine, which increases the force toresist the pipeline uplift, and the soil resistance of different depths of cover can be determined


by Eq. (4). In terms of the soil disturbance, due to the installation activities such as trenching,paving, and burying, the soil strength decreases, especially for cohesive soil. Therefore, theremolding strength of the subsoil is adopted in the analysis, cu = 5kPa. Since the typicaldepth-to-diameter ratios are between 3 and 6 in Bohai Gulf, these depth-to-diameter ratios areemployed in the calculations. The calculated results are summarized in Table 3. In the analysis,different initial imperfections with continuous prop modes and different depth-to-diameter ratiosare investigated in great detail. The typical relationship between the upheaval amplitude andthe temperature difference is obtained, and one of them is shown in Fig. 7.

Table 3 Analysis results of depth-to-diameter

Embedment ratio Resisting uplift Temperature Vom/mm

(H/D) force /(kN·m−1) /◦C 50 100 200 300

3 15.6Tu 102.4 92.6 82.7 69.1

Tmax 123.9 101.9 — —

4 19.4Tu 116.0 106.9 93.8 82.4

Tmax 139.8 115.2 — —

5 23.2Tu 128.0 120.8 108.6 94.8

Tmax 154.0 127.4 — —

6 27.1Tu 141.1 130.1 113.2 103.4

Tmax 167.2 138.5 — —

Fig. 7 Temperature difference with buckle amplitude for trench-burial

Figure 7 depicts the relationship between the pipeline upheaval amplitude and the temper-ature difference when H/D = 5. It shows that the pipe material will yield when the buckleamplitude is in excess of 850mm.

Comparing Fig. 7 with Fig. 6, it can be seen that the pipeline with the trenching and burialprotection measure has the same buckling mode as the natural backfill protection measure,except the increase in the buckling temperature. For example, when Vom = 300mm, thebuckling temperature increases from 22.7◦C to 94.8◦C. The results suggest that the trenchingand burial protection is an effective measure to improve the capacity of the pipeline to bearthermal stresses. However, the validity depends mostly on the depth-to-diameter ratio and theinitial imperfection amplitude. Table 3 shows that the higher the designed temperature is , thelarger the depth-to-diameter ratio is required. That is to say, a deeper depth of cover is neededto prevent the pipeline from buckling when the initial imperfection amplitude is large. In thispractice, H/D = 5 is adopted. The engineering has been constructed in Bohai Gulf since 2005.The protected pipeline is under good conditions up to the present.


4.4 Analysis with discrete dumping measure

The concrete mattress is widely used in the pipeline installation in Bohai. The force, which isprovided by the concrete mattress to resist the pipeline uplift, can be calculated by Eq. (6). Thethickness of one layer concrete mattress is usually 300mm, and the corresponding resistance is3.43 kN·m−1. The relationship between the upheaval buckling amplitude and the temperaturedifference is illustrated in Fig. 8. Comparing Fig. 8 with Figs. 6 and 7, it can be concluded thatthe bucking mode of the pipeline under these three protection measures are almost the same.Figure 8 shows that the yield stress of the pipeline material corresponds to the buckle amplitudeof 750mm. If one layer of the concrete mattresses is laid on the pipeline, for Vom = 300mm,the initial buckling temperature will increase from 22.0◦C to 52.4◦C. Since it still cannot meetthe requirement of the designed conditions, more layers of concrete mattresses are needed. Byconsidering that the configuration and the force analysis of the concrete mattresses on thepipeline are similar to the catenary[18], the uplift resistance can be solved by

Fv3nt =n

∑

i=1

Fvi3 + q, (50)

where Fvi3 = k ai

2 (ex

ai − e−

x

ai ) is the resistance of the concrete mattress of the ith layer, inwhich ai is the shape factor of the concrete mattress and k is the weight of the mattress perunit area.

Fig. 8 Temperature difference with buckle amplitude for mattress

The calculated results show that when three layers of the concrete mattresses are laid on thepipeline, the vertical resistance can reach 16.31kN·m−1, and the corresponding initial bucklingtemperature is 104.1◦C, which is higher than the design conditions.

Based on the foregoing analysis, the protection measures of trenching and burial and discretedumping are both effective measures to prevent the upheaval buckling of the submarine pipeline.In the design, the depth of cover and the layer number of mattresses should be determined bythe initial imperfection configuration and the operation conditions of the pipeline.

5 Conclusions

By considering the commonly used engineering measures, the upheaval buckling analysis ofthe submarine pipeline with typical initial imperfections is carried out. The following conclu-sions can be obtained.

(i) Three engineering measures can be employed to prevent the submarine pipelines fromthermal upheaval buckling in practice, i.e., the natural backfill after trenching, the trenchingand burial, and the discrete dumping.

(ii) The amplitude of the initial imperfection has a great effect on the pipeline thermalupheaval buckling. The larger the imperfection is, the smaller the temperature needed to induce


the upheaval buckling of the pipeline is. The pipeline with a small initial imperfection will suffera dynamic snap procedure with thermal stresses. With the increasing initial imperfection, acontinuous deformation will develop instead of an unstable deformation procedure.

(iii) The protection measures of the natural backfill after trenching can raise the initialbuckling temperature slightly, while the other protection measures can raise the initial bucklingtemperature greatly. The increment of the buckling temperature depends on the depth of coverand the thickness of the mattress, separately. Considering the pipeline initial imperfection andoperation conditions, the covered depth and the number of layers can be determined in practice.

References

[1] Liu, R., Yan, S. W., and Sun, G. M. Improvement of the method for marine pipeline upheavalanalysis under thermal stress (in Chinese). Journal of Tianjin University, 38(2), 124–128 (2005)

[2] Taylor, N. and Tran, V. Experimental and theoretical studies in subsea pipeline buckling. Marine

Structures, 9(2), 211–257 (1996)

[3] Hobbs, R. E. In-service buckling of heated pipelines. Journal of Transportation Engineering,110(2), 175–189 (1984)

[4] Maltby, T. C. and Calladine, C. R. An investigation into upheaval buckling of buried pipelines-II.theory and analysis of experimental observations. International Journal of Mechanical Sciences,37(9), 965–983 (1995)

[5] Palmer, A. C. and Davies, P. Advances in submarine pipeline trenching by plough. Advances in

Offshore Oil and Gas Pipeline Technology (ed. De la Mare, R. F.), Gulf Publishing Co., Texas,61–75 (1985)

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ence on Offshore Mechanics and Arctic Engineering, 5(8), 33–38 (1989)

[8] Pedersen, P. T. and Jensen, J. J. Upheaval creep of buried heated pipelines with initial imperfec-tions. Marine Structures, 1, 11–22 (1988)

[9] Gokce, K. T. and Gunbak, A. R. Self burial and stimulated self burial of pipelines by waves.Proceedings of the First International Offshore and Polar Engineering Conference, Edinburgh,308–314 (1991)

[10] Abduljauwad, S. N., Al-Ghamedy, H. N., Siddiqui, J. N., Asi, I. M., and Al-Shayea, N. A. Stabilityof vertically bent pipelines buried in sand. Journal of Pressure Vessel Technology, 126(3), 382–390(2004)

[11] Hu, Y. X., Sun, S. X., and Xing, Z. Z. Study of hydrodynamic loadings on pipe near seabed inwave and current (in Chinese). Journal of Dalian University of Technology, 31(4), 445–453 (1991)

[12] Ma, L. The on-position stability of submarine pipeline (in Chinese). China Offshore Platform,12(2), 64–72 (1997)

[13] Jin, L. Hydrodynamic forces laboratory study of horizontal cylinder with masses near the oceanbottom (in Chinese). Journal of Henan Normal University (Natural Science), 29(3), 30–36 (2001)

[14] Cao, Q. Application of high strength concrete interlock block to yard and road of port (in Chinese).Port Engineering Technology, 2, 35–37 (2002)

[15] Bransby, M. F., Newson, T. A., and Brunning, P. The upheaval capacity of pipelines in jettedclay backfill. International Journal of Offshore and Polar Engineering, 12(4), 280–287 (2002)

[16] DEP 31.40.10.16-Gen. Upheaval Buckling of Pipelines, Shell DEP Mechanical Engineering Equip-ments (1998)

[17] Gao, X. F., Yu, J. X., and Liu, R. Anchor wire design for drop anchor near the pipeline (inChinese). Ocean Technology, 28(1), 97–100 (2009)

[18] Sun, X. F. and Fang, X. S. Mechanics of Materials, 2nd ed., Higher Education Press, Beijing,294–321 (1979)

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