J. Cent. South Univ. (2013) 20: 2054−2065 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 to transport 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 the sudden 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 numerical simulations; 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: liurun@tju.edu.cn

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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 lateral

buckling 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(

d

d 22L

2

2

Lxq

Pvx

vEI

(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

Pn 2

(2)

Equation (1) has the solution of

2

22L2L

8

8

2sincos

n

Ln

P

qx

P

qnxBnxAv

(3)

According to boundary conditions: ,00' xxv

,02/ Lxv ,02' Lxxv we may write

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

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18

2cos

1 22

4L

0mLn

nLEIn

qvv x

EI

qL4L310407.2

(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

qEI

L

EIP

(5)

The reduction in axial force in the pipeline equals

friction force in axial direction of the buckling region:

sAA

0 2qL

qLPP

(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 103 988.7

2

)(

2L

EI

q

AE

LPP

AE

qL

(7)

And the relationship of L versus P can be rewritten

as

2/1

52

L

A

5A0 1106 390.6

2

L

EI

AEqqLPP

(8)

The axial force P0, 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([ tEtDp therefore, the relationship of P0 versus temperature difference is

)(0 TTAEP (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; )( TT is the temperature difference; 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 )( TT versus L can be obtained as

2

76.80)( A2

qL

L

EITTAE

2/12L

A

55 1106 390.6

EI

AEqL

(10)

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

obtained:

4/1

L

m7 514.4

q

EIvL

(11)

Thus, the relationship of vm versus )( TT is

)( TTAE

4/1

L

mA

2/1

m

L 4 257.2962.3q

EIvq

v

qEI

2/14/5

L

m2

L

A

19 119.0

q

EIv

EI

AEq

(12)

The bending moment M reaches the maximum at

x=0:

2Lm 37 069.0 qLEIvM xx (13)

And the maximum total stress σm in the buckling

region can be obtained as

I

DM

A

P

2m

m (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 strain energy, for members with length of l, when pure bend occurs, the strain energy is

xEI

xMV

ld

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:

xEA

FV

ld

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]

xvEI

xvvEI

VL

L xx

/L

xxxx d)(2

d)(2

2/

2/

2'

2

0

2'0'

0

0

xqvxvvqL

L

Ldd)(

2/

2/ A2/

0 0A0

0

xvP

xvvP L

L xxx

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 the axial load in the buckling region is

201

2 6.75176.80

L

LR

L

EIP (18)

where

6 301.2)/4 493.4sin(1 603.4 01 LLR

}1/

)/1(4 493.4sin

1/

)/1(4 493.4sin

0

0

0

0

LL

LL

LL

LL

Based on the force analysis of the slipping part, the

relation between P and P0 can be

sAA

0 2qL

qLPP

(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

201

2 60.75176.80

L

LR

L

EITAE

2/12A7

07

2LA5

2)(1066 597.1

qL

LLI

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

43

22L

212

sincos AxAxEIn

qnxAnxAv

(21)

Taking boundary conditions:

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

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00 xv , 00' xxxv , 0Lxv , 0' Lxxv

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

EIn

qA

nLnLnL

nLnLnLnL

nL

EIn

qA

nLnLnLnL

nLnLnLnL

EIn

qA

EIn

qA

4L

4

3L

3

2

4L

2

4L

1

sincos

)cos1(sincos2

cossin2

)()cos1(sin

(22)

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

antisymmetric condition [17−18]: 0' Lxxxv 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 π2

π2sinπ

π2cos1

π16 L

x

L

x

L

x

L

x

EI

qLv

,

0≤x≤L (24)

x=0.346 4L, 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 105 531.5 (25)

Introduce Eq. (23) into P=n2EI, and the axial force

in the buckling region is

2478.39

L

EIP (26)

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

versus vm can be obtained:

2/1

m

L2 936.2

v

qEIP

(27)

The axial deformation at x=L is

72

L602sL 10715.8

)(

2L

EI

q

AE

LPP

AE

qL

(28)

The equilibrium of axial force is

sAA0 qLqLPP (29)

With Eqs. (28) and (29), the relationship of P0 versus P can be obtained:

5

2

L4

A01

10743.1 LEI

qAEqLPP

L

A

2/1

L

A2L

2A 12

(30)

The relationship of )( TT versus L can be

obtained:

qLL

EITTAE A2

478.39)(

2/12

L

A52

L4 11

110743.1

L

EIqAE

L

A1

(31)

The maximum bending moment Mm can be obtained with the boundary condition 0' xxxv :

2

L'm 8 108.0 qLEIvM 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:

xvEI

xvvEI

VL

L xx

L

xxxx d)(2

d)(2

2'

0

2'0'

0

0

xqvxvvqL

L

Ldd)(

A

0 0A0

0

xvP

xvvP L

L xxx

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:

20

32

2

2

ππ35

31

π4

L

LR

L

EIP (34)

where

LL

LL

LLLLLLR /π2

)/(1

)/π)(/()/π2sin( 02

0

02

002

)/()/π2cos(1π 00 LLLLL

Based on the force analysis of the slipping part, the relation between P and P0 can be obtained:

sAA0 qLqLPP (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:

20

32

2

4

ππ35

31

π4

L

LR

L

EITAE

2/1

2A

70

732

LA4 )()(10743.1

qLLLq

IE

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([ tEpDT the equivalent temperature 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 employed to 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

Consolidatedquick shear test

Compression test Cv/

(10−4cm2·s−1) Number of soil

Soil e Ip IL

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

Coefficient of permeability,k/(10−7cm·s−1)

Thicknessof 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 Mucky

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

2 Mucky

clay 1.225 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 the designed 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 global

buckling 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 friction coefficient, 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 σm

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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 stress

In 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 on temperature 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 the ideal 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, the amplitude 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 °C Triggering 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

Pipeline with 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) Mode

II

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 design temperature 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 be eliminated 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. References [1] HOBBS R E. In service buckling of heated pipelines [J]. Journal of

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(Edited by YANG Bing)