Global Buckling Behavior of SuGlobal buckling behavior of submarine unburied pipelinesbmarine Unburied Pipelines

8/9/2019 Global Buckling Behavior of SuGlobal buckling behavior of submarine unburied pipelinesbmarine Unburied Pipelines

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DOI: 10.1007/s11771-013-1707-4

Global buckling behavior of submarine unburied pipelines under

thermal stress

GUO Lin-ping(郭林坪), LIU Run(刘润), YAN Shu-wang(闫澍旺)

State Key Laboratory of Hydraulic Engineering Simulation and Safety (Tianjin University), Tianjin 300072, China

© Central South University Press and Springer-Verlag Berlin Heidelberg 2013

Abstract: Buckling of submarine pipelines under thermal stress is one of the most important problems to be considered in pipeline

design. And pipeline with initial imperfections will easily undergo failure due to global buckling under thermal stress and internal

pressure. Therefore, it is vitally important to study the global buckling of the submarine pipeline with initial imperfections. On the

basis of the characteristics of the initial imperfections, the global lateral buckling of submarine pipelines was analyzed. Based on the

deduced analytical solutions for the global lateral buckling, effects of temperature difference and properties of foundation soil on pipeline buckling were analyzed. The results show that the snap buckling is predominantly governed by the amplitude value of initial

imperfection; the triggering temperature difference of Mode I for pipelines with initial imperfections is higher than that of Mode II; a

pipeline with a larger friction coefficient is safer than that with a smaller one; pipelines with larger initial imperfections are safer than

those with smaller ones.

Key words: submarine pipeline; lateral buckling; analytical solution; initial imperfection; subsoil friction resistance

1 Introduction

Since 1970’s, submarine pipelines gradually

become the main way in the offshore engineering totransport gas and oil all over the world. In-service

hydrocarbons must be transported at high temperature

and high pressure to ease the flow and prevent

solidification of the wax fraction. Thermal stress together

with Poisson effect will cause the steel pipe to expand

longitudinally. If such expansion is resisted, for example,

by friction effects of the foundation soil over a kilometer

or so of pipeline, compressive axial stress will be set up

in the pipe-wall. Once the value exceeds the constraint of

foundation soil on the pipeline, sudden deformation will

occur to release internal stress, which is similar to thesudden deformation of strut due to stability problems,

and lateral or vertical global buckling may occur. Studies

show that lateral modes will be dominant in pipelines

unless the line is trenched or buried [1]. Since the pipe

holds a great deal of hydrocarbon, once the pipeline

buckles or even yields, the hydrocarbon will leak out. This

will not only waste resources but also endanger the living

conditions of halobios and human beings. Therefore,

study on global buckling of submarine pipelines under

thermal stress has a great practical significance.

There is an early start on study of global lateral

buckling of unburied or semi-buried submarine pipelines.

LYONS [2] discovered that traditional Coulomb friction

model can be used to represent the force of sand to the

pipeline, while it can’t be applied to soft clay when the pipeline buckles in the lateral plane, which is obtained

from tests and numerical simulations. Based on

achievements of KERR on lateral global buckling of

continuously welded track, HOBBS [1] gave the

analytical solutions to lateral and vertical global buckling

of ideal submarine pipelines; TALOR and GAN [3]

provided analytical solutions to ideal submarine

pipelines based on the lateral soil resistance changes with

its displacement. SCHOTMAN [4] presented the

relationship of soil resistance versus pipeline

displacement by theoretical analysis and numericalsimulations; TAYLOR and GAN [5] studied effects of

initial imperfection on pipeline global buckling, and

pointed out limitations of the relationship between

temperature difference and buckling length proposed by

HOBBS [1]. PRESTON et al [6] presented a method to

control global lateral buckling by FEM, which applied

feed length on pipeline; PEEK and YUN [7] showed the

effects of flotation on lateral global buckling of

submarine pipelines. DUAN et al [8], ZHAO [9], and

LIU [10] did research on pipeline global buckling under

Foundation item: Project(51021004) supported by Innovative Research Groups of the National Natural Science Foundation of China; Project(NCET-11-

0370) supported by Program for New Century Excellent Talents in Universities of China; Project(40776055) supported by the National

Natural Science Foundation of China; Project(1002) supported by State Key Laboratory of Ocean Engineering Foundation, China

Received date: 2012−09−10; Accepted date: 2013−04−10

Corresponding author: LIU Run, Professor, PhD; Tel: +86−22−27404286; E-mail: [email protected]


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thermal stress. LIU et al [11], and GAO et al [12] studied

global buckling modes of submarine pipeline combining

practical engineering.

On the basis of the characteristics of the initial

imperfections, the global lateral buckling of submarine

pipelines is analyzed. Based on the deduced analytical

solutions, mode I and mode II, for global lateral buckling,

effects of temperature difference and properties of

foundation soil on pipeline buckling are investigated in

great details.

2 Analytical solutions of global lateralbuckling

2.1 Global lateral buckling modes

The straight pipeline, with uniform cross section

and without initial imperfection, is the ideal pipeline.Probable global lateral buckling modes of ideal pipelines,

according to Ref. [2], are shown in Fig. 1. In practice,

lateral mode I is the most significant symmetrical lateral

mode, lateral mode II is the most significant

skew-symmetrical lateral mode, and lateral mode III and

VI are subordinate forms of lateral modes I and II,

respectively. Approximately, all four modes can be

considered to be a part of lateral mode ∞. Therefore, in

this work, mode I and mode II are mainly analyzed by

employing small deformation theory, and assuming that

the stress strain relationship obeys Hook’s law.

2.2 Analysis of pipeline on lateral mode I

2.2.1 Ideal pipeline

Global lateral buckling of submarine pipelines

under different temperature and pressure conditions is an

analogy with stability problem of strut. Figure 2

describes the topology and axial force distribution of the

global lateral mode I.

The lateral bending moment equilibrium equation

for the idealized pipelines is given by

08

)4(dd

22

L2

2

L xq Pv x

v EI (1)

Fig. 1 Lateral buckling modes: (a) Mode I; (b) Mode II;

(c) Mode III; (d) Mode VI; (e) Mode V

where EI denotes flexural rigidity, m4; v is the lateral

deformation of the buckling region, m; q is the

submerged weight of pipeline per unit length, kN; P is

the axial force in the buckling region, kN; L is the buckling length, m; L is the fully mobilized lateral

friction coefficient of foundation soil to pipeline [13].

EI

P n 2 (2)

Equation (1) has the solution of

2

22L2L

8

8

2sincos

n

Ln

P

q x

P

qnx Bnx Av

(3)

According to boundary conditions: ,00' x xv

,02/ L xv ,02' L x xv we may write

Fig. 2 Deformation and force distribution for first lateral buckling mode


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18

2cos

1 22

4

L0m

Ln

nL EI n

qvv x

EI qL

4

L310407.2 (4)

where vm is the buckling amplitude of the buckling

region, m.

Using Eqs. (2) and (4), the relationship of P versus

vm in the buckling region can be obtained by

21

L2

962.376.80

mv

q EI

L

EI P

(5)

The reduction in axial force in the pipeline equals

friction force in axial direction of the buckling region:

sAA

02

qLqL P P (6)

where Ls is the slip length of the buckling region, m, and

the relationship of Ls versus P can be obtained as

72

L602sA 103988.7

2

)(

2 L

EI

q

AE

L P P

AE

qL

(7)

And the relationship of L versus P can be rewritten

as

2/1

52

L

A

5A0 1106390.6

2

L

EI

AEqqL P P

(8)

The axial force P 0, caused by temperature and

internal pressure, is the reason for buckling. To

conveniently analyze this problem, internal pressure is

converted to temperature difference, which is: T =

);2/()]5.0)(2([ t E t D p therefore, the relationship

of P 0 versus temperature difference is

)(0 T T AE P (9)

where D is the outer diameter of the pipeline, m; E is the

elastic modulus, kPa; α is the coefficient of linear

thermal expansion, °C−1; )( T T is the temperaturedifference; p is the positive pressure difference, kPa;

is the Poisson ratio, which is taken as 0.3 generally; t is

the pipe wall thickness, m.

Using Eq. (8), the relationship of )( T T versus L can be obtained as

2

76.80)( A2

qL L EI T T AE

2/12

L

A

55

1106390.6

EI

AEqL

(10)

With Eq. (7), the relationship of vm versus L can be

obtained:

4/1

L

m7514.4

q

EI v L

(11)

Thus, the relationship of vm versus

)( T T

is

)( T T AE

4/1

L

mA

2/1

m

L 4257.2962.3q

EI vq

v

q EI

2/14/5

L

m

2

L

A

19119.0

q

EI v

EI

AEq

(12)

The bending moment M reaches the maximum at

x=0:

2Lm 37069.0 qL EIv M xx (13)

And the maximum total stress σ m in the buckling

region can be obtained as

I

D M

A

P

2

mm (14)

Comparing this bending plus axial stress with the

yield stress, it can be known whether the pipeline yields.

2.2.2 Pipeline with initial imperfection

Pipeline with imperfection will keep on deforming

from the imperfection position [14]. Topology and axial

load distribution of a buckling pipeline with single-arch

imperfection are shown in Fig. 3.

Fig. 3 Deformation and load analysis for first lateral buckling mode


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The global buckling of pipelines is induced by

external forces. According to the virtual work principle,

the work done by the external forces equals the strain

energy (V ) developing inside the loaded material.

According to the method to calculate the strainenergy, for members with length of l , when pure bend

occurs, the strain energy is

x EI

x M V

l d

2

)(

0

2

(15)

where I is the inertia; E denotes the elastic modulus; M ( x)

is the bending moment of the member.

The strain energy can also be expressed by the

tension force F :

x EA

F V

l d

2

0

2

(16)

where F denotes the axial force of the member; A is its

section area.

Strain in the pipeline is a function of the bending

moment, the axial friction force and the axial force in the

buckling region. Therefore, the strain energy in the

pipeline can be obtained by [4]

xv EI

xvv EI

V L

L xx

/ L

xx xx d)(2

d)(2

2/

2/

2'

2

0

2'0'

0

0

xqv xvvq L

L

Ldd)(

2/

2/A

2/

00A

0

0

xv P xvv P L L x x x

Ld)(

2d)()(

2

/2

2/

2'

2'0

2'

2/

0 0

0

(17)

where L0 is the length of the imperfection; v′ x is the

first-order derivative of the deformation; v0′ x is the

first-order derivative of the imperfection; v′ xx is the

second-order derivative of the deformation; v0′ xx is the

second-order derivative of the imperfection.

To determine the trigger temperature difference of

the global buckling, the minimum strain energy should

be determined first [15]. Solve the equation: dV /dvm=0,

and the relationship between buckling length and theaxial load in the buckling region is

2

012 6.75

176.80 L

L R

L

EI P (18)

where

6301.2)/4493.4sin(1603.4 01 L L R

}1/

)/1(4493.4sin

1/

)/1(4493.4sin

0

0

0

0

L L

L L

L L

L L

Based on the force analysis of the slipping part, the

relation between P and P 0 can be

sAA

02

qLqL

P P

(19)

The relationship between the axial load P and

temperature difference ∆T can be obtained from features

of slipping parts [16]. And the relationship between

temperature difference and the buckling length L is

2

01

2 60.75176.80

L

L R

L

EI T AE

2/12

A70

72

LA5

2)(1066597.1

qL L L

I

Aq (20)

Since L is associated with vm, based on the

relationship of temperature difference and buckling

length, the relationship of temperature difference and

buckling amplitude can be obtained.

2.3 Analysis on lateral mode II

2.3.1 Ideal pipeline

Since the lateral mode II is an antisymmetric mode,

a half of the buckled region is analyzed for simplification.

Figure 4 details the topology and axial force distribution

of lateral mode II [2].

The bending moment equation is the same as the

lateral mode I, therefore, the solution can be written as

432

2L

21

2

sincos A x A x

EI n

qnx Anx Av

(21)

Taking boundary conditions:

Fig. 4 Deformation and force distribution for second lateral buckling mode


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00 xv , 00' x xxv , 0 L xv , 0' L x xv

into Eq. (21), the constants A1 to A4 can be obtained:

EI n

q A

nLnLnL

nLnLnLnL

nL

EI n

q A

nLnLnLnL

nLnLnLnL

EI n

q A

EI n

q A

4

L4

3

L3

2

4

L2

4

L1

sincos

)cos1(sincos2

cossin

2

)()cos1(sin

(22)

Introduce Eq. (22) into Eq. (21), employ the

antisymmetric condition [17−18]: 0'

L x xx

v into

Eq. (21), and it can be obtained that,

2πnL (23)

Use Eqs. (21), (22) and (23), and the buckling

deformation v can be obtained as

22

4

4L

π2π2

sinππ2

cos1π16 L

x

L

x

L

x

L

x

EI

qLv

,

0≤ x≤ L (24)

x=0.346 4 L, where the buckling amplitude occurs,

can be obtained from boundary condition 0' xv ( x (0,

L)). Take the value of x into Eq. (24):

EI

qLv

4L3

m 105531.5 (25)

Introduce Eq. (23) into P =n2 EI , and the axial force

in the buckling region is

2478.39

L

EI P (26)

Use Eqs. (25) and (26), and the relationship of P

versus vm can be obtained:

2/1

m

L2936.2

v

qEI P

(27)

The axial deformation at x= L is

72L602sL 10715.8)(

2 L

EI

q

AE

L P P

AE

qL

(28)

The equilibrium of axial force is

sAA0 qLqL P P (29)

With Eqs. (28) and (29), the relationship of P 0

versus P can be obtained:

5

2

L4

A0

110743.1 L

EI qAE qL P P

L

A2/1

L

A

2L

2A 12

(30)

The relationship of )( T T versus L can be

obtained:

qL L

EI T T AE A2478.39)(

2/12

L

A52

L4 11

110743.1

L

EI qAE

L

A1

(31)

The maximum bending moment M m can be obtained

with the boundary condition 0' xxxv :

2L'm 8108.0 qL EIv M xx (32)

2.3.2 Pipeline with initial imperfection

Topology and axial load distribution of a buckling

pipeline, with a double-arch imperfection, are shown in

Fig. 5 [19].

Fig. 5 Deformation and force analysis for second lateral buckling mode


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The strain energy in the pipeline can be obtained:

xv EI

xvv EI

V L

L xx

L

xx xx d)(2

d)(2

2'

0

2'0'

0

0

xqv xvvq L

L

Ldd)(

A

0

0A0

0

xv P xvv P L L x x x

Ld)(

2d)()(

2

2'

2'0

2'

0 0

0

(33)

To determine the trigger temperature difference of

pipeline global buckling, the minimum strain energy

should be determined first [20−21]. Solve the equation,

dV /dvm=0, and the relationship between buckling length

and the axial load in the buckling region can be deduced:

2

0

32

2

2

ππ35

31

π4

L

L R

L

EI P (34)

where

L L

L L

L L L L L L R /π2

)/(1

)/π)(/()/π2sin( 02

0

02

002

)/()/π2cos(1π 00 L L L L L

Based on the force analysis of the slipping part, the

relation between P and P 0 can be obtained:

sAA0 qLqL P P (35)

The relationship between the axial load P and

temperature difference ∆T can be obtained from the force

analysis of slipping parts [22−25]. And the relationship

between temperature difference and the buckling length

L can be obtained:

2

0

3

2

2

4

ππ35

31

π4

L

L R

L

EI T AE

2/1

2A

70

732

LA4 )()(10743.1

qL L Lq

I E

A

(36)

3 Case study

3.1 Case introduction

The outer diameter of the concerned pipeline is

355.6 mm, and the thickness of the pipe wall is 12.7 mm.

The designed internal pressure and temperature

difference are 7.6 MPa and 48 °C, respectively. With

equation ],2/)5.0([ t E pDT the equivalenttemperature difference can be determined as 57 °C. The

length of the analyzed part is 500 m, and the initial

imperfection amplitude is 300 mm. Design parameters of

pipeline and properties of foundation soil are presented

in Table 1 and Table 2.

3.2 Analysis results

The analytic method mentioned above is employedto predict the global lateral buckling of pipelines in this

case. The friction factor between the pipeline and subsoil

is 0.3 according to the geology condition. Figure 6 shows

the shape and values of the pipeline deformation. The

x-coordinate denotes the horizontal distance from

midpoint of pipeline. The analysis results show that with

the bending plus axial stress reaching the material yield

strength, the corresponding temperature differences are

57.8 °C and 65.4 °C for Mode I and Mode II,

respectively.

It can be known from Fig. 6 that, with the increase

of temperature difference, when temperature difference

is less than 20 °C, the buckling amplitude increases

slowly. Once the temperature difference is larger than 20

°C, the buckling amplitude will increase obviously. The

triggering temperature differences of Mode I and Mode

II are 20.89 °C and 20.75 °C, respectively. And it’s

obvious in Fig. 6 that, the increase of amplitude is not

uniform with the same increase interval of temperature

difference.

Table 1 Properties of pipeline

Elastic

modulus,

E /kPa

Poisson

ratio,

External

radius,

r /m

Wall

thickness,

t /m

Thermal

expansion coefficient,

α/(m·°C−1)

Bulk

density/

(kN·m−3)

Yield stress of

steel/MPa

Seawater

density/

(kg·m−3)

2.07×1011 0.3 0.177 8 0.012 7 1.17×10−5 7 850 448 1 025

Table 2 Physic-mechanical properties of soils

Consolidated

quick shear testCompression test

C v/

(10−4cm2·s−1) Number

of soilSoil e I p I L

c/kPa /(°) a1−2/MPa−1 E s/MPa 100 kPa 200 kPa

Coefficient

of permeability,

k /(10−7cm·s−1)

Thickness

of soil/m

1 Clay 1.747 25.91 1.55 11.10 9.90 1.52 1.86 2.90 2.71 2.760 3.0−9.5

1 Muckyclay 1.149 17.10 1.25 15.82 13.8 0.83 2.99 10.33 14.47 29.540 1.0−4.0

2Mucky

clay1.225 20.43 1.15 16.92 12.2 0.94 2.45 12.9 11.62 0.390 4.3−6.2


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Figure 7 presents the characteristics of the pipeline

global buckling with the friction coefficient of 0.3 and

the initial imperfection amplitude of 300 mm in the case.

The temperature difference corresponding to the

point, as marked in Fig. 7(a), is defined as the safe

temperature difference or triggering temperature

difference. Once the temperature difference is higher

than this value, the global buckling will happen

continuously until the stress in the pipe wall reaches the

yield stress. And the pipeline would not suffer the global

buckling failure if the design temperature difference is

lower than the triggering temperature difference.

It can be known from Fig. 7(a) that the triggering

temperature difference of Mode I is higher than that of

Mode II, which means that, Mode II is easier to occur

than Mode I. The two ending points in Fig. 7(b) imply

that the maximum bending plus axial stresses under thedesigned temperature difference and pressure are

444 MPa and 334 MPa for Mode I and Mode II,

respectively. This also verifies that Mode II is easier to

occur than Mode I and Mode II is much safer than Mode

I while the global buckling occurs.

In order to investigate more detailed regularity of

pipeline global buckling, the influencing factors, such as

the soil friction resistance and the initial imperfection,

are analyzed in following section.

4 Influencing factors on pipeline globalbuckling

4.1 Influence of subsoil friction resistance

4.1.1 Impact on triggering temperature difference

In order to analyze the effect of the subsoil friction

resistance on pipeline global buckling, temperature

difference of pipelines with imperfection amplitude of

300 mm, which is common in practice and different

pipe-subsoil friction coefficient of 0.1, 0.2, 0.3 and 0.5

are applied in pipeline global buckling analysis. Figure 8

shows the main results. It can be seen that, as to pipeline

with the same initial imperfection amplitude and

different friction coefficients, the larger the frictioncoefficient, the larger the triggering temperature

difference. This indicates that with large subsoil

resistance, the pipeline will not be easy to occur global

buckling.

4.1.2 Impact on buckling shape

In order to investigate the effect of the subsoil

friction resistance on the shape of the pipeline global

Fig. 6 Horizontal distance from midpoint of pipeline: (a) Mode I; (b) Mode II

Fig. 7 Buckling characteristics of pipeline: (a) vm vs ∆T ; (b) vm vs σ m


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buckling, the same imperfection amplitude and different

temperature differences are used in the analysis. A half of

analyzed model of the pipeline with the global buckling

is drawn in Fig. 9 with the temperature difference of

30 °C.

It can be seen that, as to pipeline with the same

initial imperfection amplitude under the same

temperature difference conditions, the larger the

pipe-subsoil friction resistance is, the smaller the

buckling amplitude and buckling length of the pipeline

are. This suggests that if the pipe-subsoil friction

resistance is large enough, the length of the pipeline

global buckling segment will be very small. On the

contrary, with the small friction resistance, the pipeline

global buckling segment will be quite long under the

same temperature difference conditions.

4.1.3 Impact on maximum compressive stressIn order to investigate the effect of the subsoil

friction resistance on the maximum compressive stress in

the pipe wall, the same imperfection amplitude and

different temperature differences are used in the analysis.

The analyzed results with temperature difference of

50 °C are shown in Fig. 10.

It can be seen from Fig. 10 that, as to pipeline with

the same initial imperfection and different pipe-subsoil

friction coefficients under the same temperature

difference condition, the larger the friction coefficient is,

the lower the maximum compressive stress is. This

indicates that pipeline with larger pipe-subsoil friction

resistance is safer than the pipeline with the smaller one.

All the above analysis results are tabulated in Table

3. Table 3 shows that with the increase of pipe-subsoil

friction resistance, the triggering temperature difference

increases, and the buckling length, amplitude and

bending plus axial stress all decrease, which indicates

that the large subsoil friction resistance will make the

pipeline safer.

4.2 Influence of initial imperfection

4.2.1 Impact on temperature difference

To analyze the impact of initial imperfection ontemperature difference, temperature difference of

pipelines with different initial imperfections amplitudes,

100 mm, 200 mm, 300 mm, 500 mm, are introduced in

the analysis. Figure 11 shows the analyzing results. In

the analysis, the friction coefficient between the pipeline

and subsoil is 0.3, which is commonly used in practice.

It can be seen from Fig. 11 that the snap buckling

Fig. 8 Comparison of temperature difference: (a) Mode I; (b) Mode II

Fig. 9 Horizontal distance from midpoint of pipeline: (a) Mode I; (b) Mode II


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Fig. 10 Comparison of bending plus axial stress

phenomenon is predominantly governed by vom. Only theideal pipeline or pipelines with relatively small

imperfections, such as 100 mm of Mode I, display the

maximum temperature difference together with the

associated snap buckling phenomenon [1]. When the

curve coincides with the left branch, the buckling shape

of the pipeline is equivalent to the first kind of stability

problem. This kind of global buckling can occur only if

the foundation soil can provide enough restraint to the

pipeline.

For the unburied pipeline, the restraint of

foundation soil comes from friction force only, therefore,

the global buckling shape of pipeline corresponding to

the left branch cannot occur or it’s an unstable buckling

stage. When the curve coincides with the right branch of

the curve, the global buckling regulation of the pipeline

is equivalent to the second kind of stability problem of

Euler buckling for a slender column. This means that

with the increase of temperature difference, theamplitude increases or is even damaged. This kind of

buckling also indicates that the increase of global

buckling segment length is the only way to increase the

friction resistance between the subsoil and pipeline for

unburied condition. However, once the imperfection

exceeds some value, such as 200 mm in Mode I and 100

mm in Mode II, the snap buckling phenomenon will

disappear, and the stable post-buckling only takes place.

Figure 11 indicates that the triggering temperature

difference of ideal pipeline is larger than that of pipeline

with imperfection. There is also an interesting

phenomenon that once the temperature difference

exceeds some value, such as 30 °C in this analysis,

buckling amplitudes of pipeline with different initial

imperfections are nearly the same under the same

temperature difference.

4.2.2 Impact on buckling shape

To analyze the regularity of impact of initial

imperfection on buckling shape, half of buckling shape

of pipelines with pipe-subsoil friction factor of 0.3 and

different initial imperfections of 100 mm, 200 mm,

300 mm, 500 mm under temperature difference of 25 °C

are analyzed. Figure 12 shows the analyzing results.

As to the impact of initial imperfection on buckling

Table 3 Buckling characteristics of pipeline under same temperature difference

Triggering temperature difference/°C Length/m Amplitude/m Bending plus axial stress/MPaFriction

coefficient Mode I Mode II Mode I Mode II Mode I Mode II Mode I Mode II

0.1 12.56 11.59 192 137 11.69 6.90 263.07 198.46

0.2 17.63 16.73 140 99 11.09 3.79 265.00 195.33

0.3 20.89 20.75 114 81 4.32 2.54 254.75 185.67

0.5 27.84 27.22 85 60 2.25 1.26 220.60 152.14

Fig. 11 Comparison of temperature difference: (a) Mode I; (b) Mode II


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Fig. 12 Horizontal distance from midpoint of pipeline:

(a) Mode I; (b) Mode II

shape, with the increase of initial imperfection amplitude

under the same temperature difference, there are

increases in buckling amplitudes, however, it is not large

at all. And the higher the temperature difference is, the

smaller the difference of global buckling amplitude will

be.

4.2.3 Impact on maximum compressive stress

To analyze the regularity of impact of initial

imperfection on the maximum compressive stress in the

pipe wall, the maximum compressive stresses with the pipe-subsoil friction factor of 0.3, the different initial

imperfections of 100 mm, 200 mm, 300 mm and 500 mm

and the temperature difference of 30 °C are shown in

Fig. 13.

Fig. 13 Comparison of maximum compressive stress

It can be seen from Fig. 13 that, the larger the initial

imperfection amplitude is, the lower the maximum

compressive stress is. This means that the pipeline with

larger initial imperfection amplitude is safer than the

pipeline with smaller one.

All the above analysis results are tabulated in Table

4.

4.3 Comprehension influence analysis

The combined effects of vom and the pipe-subsoil

friction coefficient on thermal buckling for an unburied

pipeline are shown in Fig. 14. It presents comparison

results among the pipelines with different imperfection

amplitudes and different pipe-subsoil friction coefficients

for Mode I and Mode II.

Figure 14 shows that the curves of pipeline with the

same pipe-subsoil friction coefficient will reach a same

certain point, that is to say, pipelines with the same

friction coefficient and different initial imperfections will

have the same buckling amplitude while the temperature

difference exceeds a certain value. It can also be seen

that the larger the friction coefficient is, the higher the

merging point is, which means that the larger the friction

Table 4 Buckling characteristics of pipeline corresponding to different initial imperfections

Buckling characteristics under temperature difference of 30 °CTriggering temperature

difference/°C Length/m Amplitude/m Bending plus axial stress/MPaCondition of pipeline

Mode I Mode II Mode I Mode II Mode I Mode II Mode I Mode II

Ideal pipeline 22.55 21.71 113 81 4.20 2.49 389.84 325.08

100 mm 21.25 20.66 113 81 4.22 2.52 287.70 224.50

200 mm 20.01 19.12 114 81 4.26 2.56 268.52 201.99

300 mm 20.89 20.75 114 82 4.32 2.61 254.75 187.77

Pipelinewith initial

imperfection

500 mm 19.10 18.26 115 82 4.43 2.72 233.29 166.41


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Fig. 14 Comparison of buckling curves: (a) Mode I; (b) ModeII

coefficient is, the higher the triggering temperature

difference is.

5 Conclusions

1) For two global buckling modes, with the increase

of the temperature difference, both the buckling

amplitude and length increase. However, this kind of

increase is not uniform: buckling amplitude and length

increase slowly when the temperature difference is less

than a certain value. Once the temperature difference

exceeds this value, there will be a series of intense

increase in the buckling amplitude and length. This

value of temperature difference is defined as the

triggering temperature difference. The triggering

temperature difference of Mode I is higher than that of

Mode II, which indicates the lateral buckling is easier to

occur for Mode II than Mode I. The bending plus axial

stress will increase with the increase of temperature

difference, and bending plus axial stress of Mode I is

much higher than that of Mode II under the designtemperature difference, which indicates that Mode I is

more dangerous compared with Mode II once it occurs.

2) As to the impact of the pipe-subsoil friction

coefficient on pipeline global buckling, the larger the

friction coefficient is, the smaller the buckling amplitude

and buckling length of the pipeline for both global

buckling modes are. The triggering temperature

difference increases and the bending plus axial stress in

the pipe wall decreases with the friction between the

pipeline and subsoil increasing. This means that a

pipeline with larger pipe-subsoil friction coefficient is

safer than that with small one.

3) As to analysis on impact of initial imperfection

on pipeline global buckling, the snap buckling is

predominantly governed by the amplitude value of initial

imperfection. The snap buckling phenomenon only exists

on pipeline with small initial imperfection. For the same

temperature difference, the global buckling amplitude

increases with the initial imperfection amplitude of the

pipeline. However, this kind of increase will beeliminated with the temperature difference rising.

Meanwhile, the larger the initial imperfection amplitude

is, the smaller the bending plus axial stress is, which

indicates that actively creating larger initial

imperfections can effectively prevent submarine

pipelines from global buckling failure.

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(Edited by YANG Bing)

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