karampour 2013 engineering-structures

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On lateral and upheaval buckling of subsea pipelines

Hassan Karampour a, Faris Albermani a,, Julien Gross b

a School of Civil Engineering, The University of Queensland, Australiab ENSTA ParisTech, 32 Bd Victor, 75015 Paris, France

a r t i c l e i n f o

Article history:

Received 10 October 2012

Revised 25 February 2013

Accepted 28 February 2013

Available online 9 April 2013

Keywords:

Upheaval buckling

Lateral buckling

Subsea pipelines

a b s t r a c t

The lateral and upheaval buckling of subsea pipelines is investigated in this paper. For lateral buckling,

analytical and numerical studies are conducted and compared and a new interpretation of localisation

is given based on an isolated half-wavelength model. For upheaval buckling, a tabulated analytical solu-

tion based on a long heavy elastic beam resting on a rigid frictional foundation is given and the response

under three types of localised initial imperfection is compared. A comparison of the lateral and upheaval

responses of a subsea pipeline is made and indicates that excessive bending stress can be induced, par-

ticularly under upheaval buckling. The paper also highlights some differences between the current and

previously published results.

Crown Copyright 2013 Published by Elsevier Ltd. All rights reserved.

1. Introduction

Lateral and upheaval buckling are two possible buckling modes

that can occur in long pipelines (both onshore and offshore).

Although these two buckling modes are not essentially failuremodes, they can precipitate failure through excessive bending that

may lead to fracture, fatigue or propagation buckling [1]. The

1.3 million litres oil spill in Guanabara Bay (Brazil, January 2000)

was caused by lateral buckling of offshore pipeline that eventuated

in local buckling and rupture of the pipe wall [2]. A similar incident

involving upheaval buckling of onshore inclined pipeline during

construction was reported in Colombia that resulted in a number

of fatalities and injuries[3]. These examples highlight the vulner-

ability of pipelines to these buckling modes and the resulting

disastrous human, environmental and economic consequences.

One of the first experimental and analytical studies on upheaval

buckling of an axially compressed frictionless strip was conducted

by Allan[4]who observed the sensitivity of the problem to initial

imperfections. A decade later, a number of studies motivated bylateral buckling of railway tracks were presented by Kerr [5] and

Tvergaard and Needleman[6]who accounted for buckle localisa-

tion and the nonlinear forcedisplacement relationship of the track

foundation. Probably Hobbs [7]was the first one to study upheaval

and lateral buckling specifically in pipelines. A number of experi-

mental and analytical studies on upheaval and lateral buckling of

subsea pipelines appeared since then, most notably are Ju and

Kyriakides [8], Maltby and Calladine[9,10], Taylor and Tran[11],

Croll [12], Miles and Calladine [13], Cheuk et al. [14] and Peng

et al.[15].

A pipeline is a slender structure that travels long distances,

the hydrocarbon contents in the pipeline usually are at high

temperature (80 C or higher) and high internal pressure(10 MPa or higher). Both the rise in temperature and internal

pressure result in longitudinal expansion of the pipeline. The

seabed friction acts to restrain this expansion which results in

the build-up of compression stress in the pipe wall that will

eventuate in buckling. A pipeline resting on the seabed will

buckle laterally (in the horizontal plane) while a trenched pipe-

line will undergo upheaval buckling (in the vertical plane). The

axial compression force in the pipeline due to restrained longitu-

dinal expansion is given by

N EAaDTe 1

where the effective temperature change, DTeaccounts for the com-

bined effects of temperature and internal pressure

DTe DT qD1 2t4tEa

2

For example, the restrained expansion of 650 15 mm steel

pipe will induce 67 kN axial compression due to a temperature in-

crease of 1 C. Furthermore, 1 MPa increase in internal pressure in

this pipe will result in additional 133 kN of axial compression, a to-

tal of 200 kN. Using Euler buckling, the 200 kN compression force

corresponds to an effective length of only 130 m which is a small

fraction of a typical length of practical pipeline.

The paper presents a tabulated analytical and numerical study

of lateral and upheaval buckling of a pipe resting on the seabed.

The lateral and upheaval buckling are compared and the effects

of various parameters such as the amplitude and half-wavelength

0141-0296/$ - see front matter Crown Copyright 2013 Published by Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.engstruct.2013.02.037

Corresponding author. Tel.: +61 7 33654126.

E-mail address:[email protected](F. Albermani).

Engineering Structures 52 (2013) 317330

Contents lists available atSciVerse ScienceDirect

Engineering Structures

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of initial imperfection, friction, effective weight, initial stress and

presence of anchorage points are investigated and discussed. Cer-

tain results that are different from previously published results

are also highlighted.

2. Lateral buckling

Maltby and Calladine[10]presented a simplified solution that

captures the main features of the detailed analysis presented by

Tvergaard and Needleman[16] for lateral buckling of a beam on

a rigid frictional foundation with nonlinear soil-resistance. Taking

a single half-wavelength, k, and using elastic small displacement

approach, a sinusoidal imperfection y0 is assumed

y0 Y0 sin pxk

3

The governing equation of the beam under a compression force

P(Fig. 1) is

EIy0000

y0000

0 Py00 Q 0 4

whereQrepresents a distributed lateral frictional drag force (soil-

resistance) which is a function of (yy0). An important conclusion

made by Maltby and Calladine [10]was that the condition for local-

isation of buckling in the presence of an imperfection depends on

the limiting value of Q but not much on the exact constitutive

law that governs Q. An exponential model for Q

Q Q0 1 eY

0Y

D

h i 5

is adopted withQ0= 3.2 N/m and D = 1.7 mm based on results ob-

tained from small-scale experiment[9]. Friction in the axial direc-

tion is not accounted for since lateral buckle bifurcation isentirely controlled by the lateral resistance [16].

For simplicity, a sinusoidal distribution of Q along the half-

wavelength was assumed which results in a sinusoidal lateral dis-

placementy

Nomenclature

A cross-sectional area of pipeD mean diameter of pipeE modulus of elasticityI second moment of areaL0 imperfection half-wavelength

M0 crown moment at x = 0N compression force in pipe far from the buckled region

(thermal load)P compressive force in the buckled region of the pipeQ lateral restoring force per unit lengthY amplitude of lateral displacementY0 amplitude of lateral initial imperfectionm effective weight of the pipet pipe wall thicknesswx upheaval buckle profilew0 initial imperfection in the vertical planey lateral position of the pipey0 lateral initial imperfectionDTe effective change in temperature

DTu uplift temperatureDTC critical temperatureDTM minimum temperatureD0 amplitude of vertical imperfectionDs geometric shortening effect

k half-wavelength of lateral bucklea coefficient of thermal expansion of pipe materialt Poissons ratio of pipe materiale uplifted half-wavelengthd uplifted buckle amplitudeea fixed anchorage point distanced0, l0 normalised amplitude and lengthq internal pressure in the pipefi fully contact imperfection shape functione0 suspended length of the pipeeu uplifted half-wavelength in point and infill prop imper-

fectionl coefficient of friction (Coulomb)

Fig. 1. Isolated half-wavelength model of a laterally buckled pipe under axial load Pand lateral soil resistance Q.

318 H. Karampour et al. / Engineering Structures 52 (2013) 317330


3/14

y Ysin pxk

6

A solution for the axial compression force Pin the buckled re-

gion of the pipe is obtained by substituting Eqs. (6) into (4)

P k

2

p2

! Q

Y

p2

k2

EI 1

Y0Y

7

The two load carrying mechanisms, membrane and flexure,

contributing to P are given by the first and second terms on the

rhs of Eq. (7), respectively. The contribution of these two mecha-

nisms is depicted inFig. 2against normalised crown displacement

for EI= 3.13 N m2 with two initial imperfection amplitudes,

Y0= 0.5 and 1 mm, two half-wavelengths, k = 600 and 1200 mm

and two Q0 values (Q0 and 2Q0). Initial imperfection which pro-

duces low initial curvature results in a response that is initially

dominated by membrane action, while higher initial curvature re-

sults in a more dominant flexure. A growth in crown displacement

Ycorresponds to a growth in flexure and decay in membrane ac-

tion. Higher initialY0 leads to higher rate of growth/decay in flex-

ure/membrane actions, respectively. While higher frictional drag

Q0 translates to lower rate of growth/decay in flexure/membrane

actions. The resulting overall response can be a bifurcation type

(with unstable post buckling response) or a stable response with

monotonic increase in Pand Y. A bifurcation type response is indic-

ative of the dominance of membrane mechanism, and is expected

when the paths of the two mechanisms intersect as in Fig. 2when

k= 1200 mm andY0= 1.0 mm for example.

In a long pipe, the axial compression force N away from the

buckled region is obtained by accounting for the geometric short-

ening (bowing effect) in the buckled half-wavelength

N Pp2Y2

k2 EA 1

Y0Y

2" # 8

An elastic geometric nonlinear finite element model of the half-

wavelength shown inFig. 1is developed in ANSYS[17]using qua-dratic three-node pipe elements PIPE289. The contact between the

pipe and the foundation is modelled using CONTA177 elements

and one rigid TARGE170 element. For sake of comparison, the same

geometric and material properties used by Maltby and Calladine

[10] are adopted; k= 0.6m, EA= 650 kN, EI= 3.13 N m2 and Q

according to Eq. (5). The FE results for a range of initial imperfec-

tion amplitudes (Y0= 0.051.0 mm) are shown in Fig. 3 in terms

of axial force (Pand N) against crown displacement, Y. The FE re-

sults are in excellent agreement with the analytical results from

Eqs. (7) and (8) forPandNrespectively. These results do not agree

with Maltby and Calladine results (Fig. 9 in their paper[10]) which

we believe are not correct. The current results show no sharp peaks

at smaller imperfections and the magnitude ofP

andN

are lower

than their results. The axial force Pshows a rather smooth post

peak drop at lower Y0which is flattened out as Y0is increased. This

overall response is expected if we refer to Fig. 2 (k= 0.6 m) whereit

is clear that the response is dominated by flexure without intersec-

tion of the two mechanisms, hence no bifurcation is expected.

Fig. 3 also indicates that the response away from the buckled re-

gion show monotonic increase inNwithout any snap at lower Y0,

contrary to Maltby and Calladine observation[10].

To identify the most critical initial wavelength, the axial force, P,

from Eq. (7) using a fixed amplitude of imperfection (Y0= 0.5 mm)

and increasing half-wavelength length k is plotted against crown

displacement inFig. 4. In this figure, Pis normalised by the corre-

sponding Euler buckling load, Pe. It appears that Tvergaard and

Needleman [16] condition of localisation at vanishing peak in P

corresponds toP/Pe= 1 in the isolated half-wavelength model.

In the above study the buckle wavelength is constrained to be

the same as the wavelength of initial imperfection. To investigate

the localisation effect in a long pipe, a model of a long pipe with

properties shown in Table 1 (Pipe A) is generated in ANSYS. A local-

ised sinusoidal imperfection withk= 21 m and Y0= D/2 is assumed

at mid-length and a total length ofL = 20k is used. For computa-

tional efficiency the lateral frictional drag is approximated using

COMBIN39 axially sliding nonlinear spring elements instead of

contact/target elements. The lateral frictional drag is modelled

using exponential forcedisplacement relation similar to Eq. (5)

with D = D/2 and Q0= 2.955 kN/m corresponding to l = 0.5 andm= 5.91 kN/m.

Fig. 5 shows the evolution of the deformed lateral profile of

the pipe at three different levels of axial loading. In order toshow the lateral deformation, only the central region of the pipe

is shown in Fig. 5. Initially a periodic pattern emerges over the

entire length, this is followed by localised growth in a central

lobe leading to bifurcation. An invigorated growth in a second

generation of lobes flanking the central lobe is followed. The

Fig. 2. Contribution of membrane (A) and flexure (B) load carrying mechanisms in lateral buckling of isolated half-wavelength model.

H. Karampour et al./ Engineering Structures 52 (2013) 317330 319


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growth in the amplitude (Y Y0) of the central lobe during the

three stages shown in Fig. 5 is 0.22D, 0.87D and 9.34D respec-

tively, that is nearly 42 times. In comparison the growth of the

half-wavelength of the central lobe (no longitudinal friction) is

k= 15.2, 16.4 and 28.6 m respectively. The loaddisplacement re-

sponse of the axial force P in the central and flanking lobes is

shown inFig. 6. It is clear that the localisation of the central lobe

starts early in the pre-bifurcation path while the growth in the

flanking lobes only emerges after bifurcation. Although the iso-

lated half-wavelength model (Eq. (7)) do not account for growth

in wavelength, a lower bound estimate of the critical load P can

still be obtained by finding the critical k (for Y0= D/2) that

makes P/Pe = 1 (as in Fig. 4). This will lead to k= 14.2m and

hence P= 564 kN which is 80% of P from the long FE model(Fig. 5, P= 707 kN).

3. Upheaval buckling

Similar to lateral buckling, the upheaval buckling is sensitive

to local initial imperfection and the foundation stiffness has a

small effect on the overall response. Ju and Kyriakides [8] used

small displacement approach to investigate upheaval buckling.

A similar approach is adopted here where the pipe is modelled

as a long heavy elastic beam on a rigid frictional foundation.

Three types of local imperfections; fully-contact imperfection,

point imperfection (isoprop) and infilled prop as shown in Fig. 7

are investigated in this work. The geometry of the problem is de-

fined inFig. 8a and the axial compression force distribution along

the pipe is shown in Fig. 8b. The axial force distribution shows a

sliding lengthL1, controlled by axial friction, adjacent to the buck-led length e.

Fig. 3. Comparison ofPand Nresponses from FE analysis and Eqs. (7) and (8) for k = 0.6 m and increasingY0.



5/14

3.1. Fully contact imperfection

The local geometric imperfection of the seabed is described as:

w0 fix 0 6 jxj fi0 D0

0 jxj > L0

9

Since exact profile of the seabed is not always available, four

different imperfection shape functions (f1f4) were assumed to ac-

count for possible undulations of the seabed. These shape functions

are given in Eq. (10) and are symmetric aboutx= 0. The functionf2represents the fundamental buckling mode of a fix-ended column

while f4 is the empathetic model with a unique amplitude/wave-

length relationship[11].

Fig. 4. Normalised axial forceP (Eq. (7)) against normalised lateral buckle amplitude for various values ofk.

Table 1

Pipe properties.

Pipe A

Pipe mean diameter D 214mm

Pipe wall thickness t 14.3 mm

Modulus of elasticity E 207GPa

Poissons ratio t 0.3Linear thermal coefficient of expansion a 11 106/C

Pipe B

Pipe external diameter D 9.53 mm

Pipe wall thickness t 1.60 mm

Modulus of elasticity E 195GPa

Effective self-weight m 0.00341 N/mm

Friction coefficient l 0.2Linear thermal coefficient of expansion a 11 106/C

Fig. 5. Amplified lateral profiles at mid-span of a long pipeline model.



6/14

f1 D08

3

x

L0 2

3 x

L0 1" #

1 x

L0 3

10a

f2 D0

2 1 cos

pxL0

10b

f3 D0 4x

L0 1

x

L0 1

410c

f4 D0 0:707 0:26176p2x2

L20

0:293 c os 2:86pxL0

" # 10d

If the temperature is sufficiently raised, part of the pipe (Fig. 8a)

lifts off the seabed inxzplane with an axial compression force P, a

crown displacement d= w(0) and uplifted length e. The bending

moment at a distance x (x 6 e) from the centre of imperfection isgiven by:

Mx Pdwx M01

2mx2 11

Two initial conditions are possible; unstressed and stressed

pipe. The former (conservative) corresponds to local imperfection

in the pipe itself with a conforming foundation. The later repre-

sents local imperfection in the foundation with a conforming pipe

(a straight pipe has to bend under self-weight to conform to the

undulation of the foundation). Using a small deflection theory,the bending moment at x is obtained from

Mx EIw00x w

000x 12

for initially unstressed pipe, and

Mx EIw00x 13

for initially stressed pipe. Substituting Eqs. (12) or (13) into (11)

gives

w00xk2w w00

0xk2d

M0EI

1

2

mx2

EI 14

for unstressed pipe, and

w00xk2w k

2d

M0EI

1

2

mx2

EI 15

for stressed pipe, andk2 = P/EIwith 1/k is characteristic length.

The governing characteristic equation for fully contact imper-

fection is obtained by solving Eqs. (14) and (15) subject to the rel-

evant boundary conditions given in Table 2. For fully contact

imperfection, the governing characteristic equation is given inTa-

ble 3in terms of three characteristic functions (G,FandH) and the

resulting buckle profile wx is listed in Table 4. The characteristic

functions (G,Fand H) are listed in Table 5 for each of the imperfec-

tionfifunctions (Eq. (10)). For initially stressed pipe,Hk;e 0 for

allfi. It is worth noting that for f1 initially unstressed pipe, the de-

rivedH

k;

e function (Table 5) is different from the one obtainedby Ju and Kyriakides[8]which was independent ofe.

Fig. 6. Axial force, P, response in the central and flanking lobes under lateral

buckling of a long pipeline model.

Fig. 7. Idealised local initial imperfections for upheaval buckling.



7/14

3.2. Point imperfection (isoprop)

A pipe having an isoprop imperfection with a height D0 is

shown inFig. 7b, the pipe is suspended under its self-weight, m.

Before any temperature rise (P= 0), the equilibrium profile of the

suspended pipe is

w0x mx4

24EIme0x3

9EI

me20x2

12EI D0 16

Withx 6 e0 and the length of the suspended spane0 72EID0

m

1=4.

As the temperature increases, axial compression is induced inthe pipe. Prior to lift-off, the bending moment is given by

M0PD0w Rx1

2mx2 EIw00x w

000x 17

whereR is the vertical reaction at the point imperfection x = 0. Ju

and Kyriakides[8]and Croll[12]had neglected the effect of initial

curvature w000x in (17) which was shown to be inconsistent with

experimental results[11].

Solving (17) subject to point imperfection pre-upheaval bound-

ary conditions (Table 2), the pre-upheaval governing characteristic

equation can be obtained (Table 3) with the prop reaction R as:

R

2m

k

sinke ke coske

1 coske

2

3me0 18

Fig. 8. Geometry and axial force distribution for fully contact imperfection.

Table 2

Boundary conditions for upheaval buckling.

Fully contact imperfection (initially unstressed)

0 6 e 6 1 w0 d; w0x0 0; we w0e; w0

xe w00xe;

w00

x e w000xe

eP 1 w0 d; w0x0 0 we w0e 0; w0

xe w00xe 0;

w00x e w000xe 0

Fully contact imperfection (initially stressed)

0 6 e 6 1 w0 d; w0x0 0; we w0e; w0

xe w00xe;

w00x e w000xe

eP 1 w0 d; w0x0 0 we w0e 0; w0

xe w00xe 0;

w00x e w000xe 0

Point imperfection

Pre-upheaval w0 D0; w0

x0 0; we0 0; w0

xe0 0; w00

x e0 0Post-upheaval w0 w0; w

0x0 0; we 0; w

0xe 0; w

00x e 0

eu 6 e 6 e0Post-upheaval w0 w0; w

0x0 0; we 0; w

0xe 0; w

00x e 0

e0< e

Infilled prop

e< e0 w0 wm; w0

x

0 0; we w0e; w0x

e w00x

e;w00x e w

000xe

e0< e w0 wm; w0

x0 0; we 0; w0

xe 0; w00

x e 0

Table 3

Characteristic equations for upheaval buckling.

Fully contact imperfection

0 6 e 6 1 sinke:Gk;e coske:Fk;e 0

eP 1 sinke kecoske Hk;ek kL0;e eL0 ;

mmL 40EID0

Point imperfection

Pre-upheaval ke04coske 1 72ke21 coske

288ke sinke 0Post-upheaval ke01 coske 3sinke ke coske 0eu 6 e 6 e0Post-upheaval kecoske sinke ke03 coske0 sinke0

2ke03

0

e0< e

Infilled prop

e< e0 sinke 2 ke2

6e0e 1

e0e 3

h i coske 2ke3

e0e 3

ke3

6e0e 1

23 ke0 0

e0< e kecoske sinke ke03 coske0 sinke0 2ke0

3 0



8/14

The uplifted half-wavelength eu at lift-off (R= 0) is obtainedfrom Eq. (18) which gives euffi 0.75e0. This indicates that the pipe

will shrink before it lifts off the prop. Following lift-off, Eq. (17)

(withR = 0) is solved subject to the point imperfection (post-up-

heaval) boundary conditions in Table 2 to yield the governing char-

acteristic equation (Table 3) and the resulting buckle profile wxshown inTable 4.

3.3. Infilled prop

The void between the seabed and the pipe with point imperfec-

tion can be filled with sand and other infill material present on the

seabed (Fig. 7c). This leads to a different scenario where the pre-

buckling suspended configuration is no longer exists and the bend-

ing moment atx (x 6 eu) is obtained from

M0Pdwx 1

2mx2 EIw00x w

000x 19

where w0 is from Eq. (16). Using the boundary conditions for infilled

prop shown inTable 2, the governing characteristic equation and

the resulting buckle profile wx for the infilled prop case are listed

inTables 3 and 4 respectively.

3.4. Compatibility

The characteristic equations developed so far (Table 3), yield the

compressive axial force, P, in the uplifted pipe. Due to loss of fric-

tion along the buckled length and geometric shortening effect, the

forceP is less than the axial compressive force, N, away from the

buckle. A schematic of the compression axial force distribution

along the pipe is shown in Fig. 8b. The two axial forces P andN

can be related using compatibility. Two practical cases are investi-

gated; infinitely long pipe and a pipe with anchorage points.

3.4.1. Infinitely long pipe

Using the coefficient of Coulomb friction, l, the length L1 over

which the axial compression is mobilised to Nis given by:

L1 NPmle

ml 20

The compatibility of axial displacement atx = e gives:

Table 4

Upheaval buckled profile w(x).

Fully contact imperfection (initially unstressed f1)

wx K sinkx C coskxC1x3 C2x

2 C3xC4

C1 160D03k

2L50

; C2 60D0

k2

L40 m

2k2

EI; C3

320D0k

4L50

; C4 120D0

k4

L40 m

k4

EI 20D0

3k2

L20 dM0

k2

EI

Fully contact imperfection (initially stressed f1)

wx K sinkx C coskxC11x2 C22xC33

C11 m2k2EI; C22 0; C33 M0

k2EI d mk4EI

Point imperfection

Pre-upheaval wx K sinkx C coskx C1x2 C2xC3

C1 m

k2

EI; C2

1

k2

EI R 23 me0

; C3 2m

EIk4 D0

M0

k2

EI

me20

6k2

EI

Post-upheaval wx A sinkx B coskx C1x2 C2xC3

C1 m

k2

EI; C2

2me03k

2EI; C3 w0

M0

k2

EI m

k2

EI

2

k2

e20

6

Infilled prop

wx K sinkx C coskx C1x2 C2xC3

C1 m

k2

EI; C2

2me03k

2EI; C3 wm

m

k2

EI

2

k2

e20

6 M0

m

h i

Table 5

Characteristic functions for different fi.

f1 (initially unstressed)

Gk;e f320sinke 203k38e3 9e2 1 403k 8ke mkg

Fk;e 320cos ke 20k2 8e2 6e 16k2

203k4ee 122e 1 mk2e

n o

Hk;e 40mk

3ek2 4k2e2 8 coske 3k 8ke sinke 8

f1 (initially stressed)

Gk;e 203k28e3 9e2 1 m

Fk;e 203k3ee 122e 1 mke


Gk;e p2k2 cospe 2 m

Fk;e pk3 sinpe 2 mke


Gk;e m 20k24e 1e 12

F

k;

e

m

k

e 20

k3

e

e 13


Gk;e 0:293k22:86p2 cos2:86pe m 2 0:2617p2k2

Fk;e 0:293k32:86p sin2:86pe ke m 2 0:2617p2k2

Fig. 9. Geometry and axial force distribution of a pipe with fixed anchorage points.



9/14

Z e0

NP

EA dx

Z eL1e

NPx

EA dx Ds 0 21

The first term in Eq. (21) corresponds to the tensile extension

within the buckled length e, the second term represents the tensilerelief in the slip length L1, and the last term, Ds, is the geometric

shortening or bowing effect in the buckle:

DsZ e

0

1

2 w0x2 w002h i

dx 22

Using Eqs. (20)(22), the axial compression force away from the

buckle,N, can be obtained

N P 2mlEADs mle20:5 23

3.4.2. Pipe with fixed anchorage point

The expected axial force distribution along a pipe with a point

imperfection and restrained from axial displacement at a distance

eafromthe centre of imperfection (x= 0)is shown in Fig. 9. This sit-uation relates to practical case where rocks are dumped on the

pipe or represents a small-scale experimental setup with fixed

ends. The compatibility equation in this case leads to

N PEADs

ealm2ea

e2a e2 24

3.5. Solution method

Solution of the upheaval buckling problem is implemented in

MATLAB [18]. The solution starts by prescribing the uplifted length

e and solving the relevant characteristic equation (Table 3) for k

and hence P. Using the remaining boundary conditions (Table 2),

M0, d, K and C (Table 4) are evaluated and the buckle profile wxis obtained as inTable 4. Havingwx, the geometric shortening Ds

is calculated from (22) followed by calculating the axial force N

(Eqs. (23) and (24)) and hence the temperature rise in the pipe

(DTe) from Eq. (1). The lift-off temperature DTUcan be easily calcu-

lated from

DTUmEIw0000

00

aEAw000

0 25

For fully contact imperfection (Eq. (10)), functionsf1andf4are close

to each other whilef3gives the highest curvature atx= 0 and hence

is expected to give lower DTU. However, unlike the other types of

imperfection considered in this study, f3 leads to partial lift-off

away from x = 0 which requires a different solution strategy. For

this reason and for sake of comparison with point imperfection

and in-filled prop,f1is chosen to represent full contact imperfection

in this study.

4. Results and discussion

The effects of various parameters on the upheaval buckling re-

sponse of a pipe resting on the seabed are evaluated and summa-

rised in this section. The parameters considered are; local

geometric imperfection, longitudinal friction and the effective

weight of the pipe. The effective weight of a subsea pipe comprises

the submerged weight of the pipe, coating material and any soil/

rock cover in case of a buried pipe. The geometric and material

properties of the pipe used in Sections4.14.4are listed inTable 1

(Pipe A). These properties are similar to those used in [8].

Fig. 10. Axial force and thermal response of initially stressed pipe (fully contact) with increasing half-wavelength.



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A typical upheaval response of a pipe on a frictional foundation

subjected to temperature increase is characterised by three impor-

tant temperature rises; a barrier temperature to initiate uplift DTU,

a critical or limit temperature DTCand minimum temperature DTM(Fig. 10). As the temperature rise reaches DTUthe pipe start lifting

off the foundation with a stable pre-buckling response, further in-

crease in temperature to DTCresult in snap buckling with unstable

post-buckling response which corresponds to a theoretical drop in

temperature to the minimum temperature DTMthat is followed by

a stable post-buckling response as the temperature is increased be-

yond DTM. In practical situation, the service temperature is kept

constant and the actual response is characterised by a snap-

through from DTCto an adjacent point on the stable post-buckling

path.

4.1. Effect of local imperfection

The two main parameters describing the local geometric imper-

fection are imperfection amplitude D0 and half-wavelength L0.

These parameters are normalised using the pipe mean diameter D

d0

D0

D ; l0

L0

D 26

4.1.1. Effect of imperfection half-wavelength L0While holding m, d0 andl constant, the response of initially

stressed pipe with fully contact imperfection profile f1 (Eq. (10))

using three different values of l0 (50, 75 and 100) is shown in

Fig. 10. The response in terms of normalised compression axial

forcePin the uplifted section and temperature rise DTare shown

against normalised uplifted length e/Dand buckle amplitude d/D. Itis clear from this figure that both DTU andDTC(for clarity, these

two temperatures are only marked on the curve corresponding to

l0= 100 inFig. 10) are increased as l 0 is increased. Another point

to note fromFig. 10is the presence of a local minimum between

DTU and DTC, although the difference between uplift and critical

temperatures is rather small. It is interesting to note that the min-

imum temperature DTMis insensitive to changes in l0. The uplift

temperature DTU is inversely proportional to the curvature of ini-

tial imperfection at x = 0 (Eq. (25)), this explain the rise in DTUas

l0 is increased (while d0 is held constant). Following DTMthe pipe

response is stable; further temperature increase results in mono-

tonic increase in uplifted length and crown displacement accompa-

nied by decrease in the axial force P due to loss of friction and

geometric shortening (bowing) effect. The three characteristic

temperatures DTU, DTCand DTMresponse is within the validity of

small displacement assumption, as indicated in Fig. 10 (slope


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Fig. 11that DTUand DTCare sensitive while DTM is insensitive to

changes ind0.

A somewhat similar trend is observed in Fig. 12with a pointimperfection. As the temperature increases the response is stable

and the suspended lengthe0 shrinks by 25% to eu as the pipe lift-off at DTU (for clarity e0 andeu are marked inFig. 12 on d0= 0.2curve only). In this case DTC degenerates to DTU and for d0< 1

the post-buckling response following this point is unstable and

corresponds to a theoretical drop in temperature to the mini-

mum temperature DTM. Any temperature increase beyond DTMcorresponds to stable response with monotonic increase in up-

heaval span and crown displacement. Again under point imper-

fection DTM is insensitive to changes in d0. For d0P 1 both pre

and post buckling response are stable. When comparing the

curves for d0= 0.5 in Figs. 11 and 12 (since for this case

L0= 10.7m and e0= 10.275 m are comparable), it appears that

the point imperfection is more critical than fully contact imper-fection since it leads to lower DTC although DTM is about the

same.

The response of the pipe with infilled prop initial imperfection

is shown inFig. 13. For the same d0 in Figs. 12 and 13, the initial

(suspended) lengthe0 is the same.Fig. 13shows that as the tem-perature is increased, the pipe lifts-off at DTU with a length euand that eu 0.214e0 which agrees with Taylor and Tran [11].When comparingFigs. 12 and 13, DTMis about the same but the

infilled prop leads to a reduction in DTCcompared to point imper-

fection. A peculiar feature of infilled prop imperfection is that

while there is a noticeable growth in eas the temperature is raised

from DTUto DTC, there is very small growth in crown displacementuntil DTCis reached.

Fig. 14shows the response in terms of the three characteristic

temperatures (DTU, DTC and DTM) against d0 for fully contact f1and point imperfection. For each d0 the length L0 for f1 is made

equal to e0 of the point imperfection. It is clear fromthis figure thatfor the samed0the point imperfection requires lower DTUand DTCthan fully contact imperfection while DTM is about the same for

both.

4.2. Effect of coefficient of friction

A Coulomb friction model with a friction coefficient l is usedthroughout this study. This friction model has been shown to be

accurate enough for practical application [11]. Fig. 15 shows theresults from a sensitivity study using different values of seabed

frictionl and two types of imperfection; fully contact and iso-prop. It is clear from this figure that seabed friction has a

stabilising effect that is only evident in the post-buckling

response.

4.3. Effect of effective weight of the pipe

Increasing the effective weight, m, of the pipe has been recogni-

sed as a mitigation provision against upheaval buckling. This can

be achieved by coating the pipe with a heavy material such as con-

crete and by partial or continuous trenching.Fig. 16shows the re-

sponse of fully contact and isoprop types imperfection. All the

three characteristic temperatures DTU, DTCand DTMare increasedas the effective weight is increased.

4.4. Initially stressed vs. initially unstressed pipes

The response presented so far was for initially stressed pipe.

Fig. 17 shows the response of initially unstressed pipe with f1fully contact imperfection, this figure should be compared with

Fig. 11 for the initially stressed pipe. To facilitate this compari-

son, the response for d0= 0.2 (stressed and unstressed) is repro-

duced in Fig. 17. It is clear that the initially unstressed

assumption yields conservative results for DTU and DTC while

DTM is rather unaffected. It is interesting to note that the entire

DT response of initially unstressed pipe with d0 = 1 is stable and

beyond DTMthe response is stiffer for larger d0. For the same d0the initially unstressed response is stiffer than the initially

Fig. 13. Thermal response of a pipe with infilled prop and increasing imperfection amplitude.

Fig. 14. Variation of the characteristic temperatures (DTU, DTC and DTM) withincreasing imperfection amplitude for fully contact and point imperfections.



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stressed one. This is different from Ju and Kyriakides [8]results

(Fig. 7a in their paper) and is attributed to the difference in the

derived Hk;e function (Table 5) which was reported indepen-dent of e in their work.

4.5. Pipe with fixed anchorage point

Taylor and Tran[11]conducted a small scale experiment on a

pipe with point imperfection (isoprop) and fixed anchorage points.

Fig. 15. Effect of increasing friction on temperature response under fully contact and point imperfections.

Fig. 16. Effect of increasing the pipes effective weight on temperature response under fully contact and point imperfections.

Fig. 17. Upheaval response comparison of initially stressed and unstressed pipe.



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The pipe parameters are listed in Table 1(Pipe B). The current re-

sults (temperature rise against half-wavelength and amplitude) are

compared with their experimental results inFig. 18. The current

analytical results are in good agreement with the experimental re-sults and for the range of D0 considered, the post-upheaval re-

sponse is stable.

5. Comparison of lateral and upheaval buckling

A comparison between lateral and upheaval response is con-

ducted using the pipes parameters in Table 1(Pipe A). In the up-

heaval case, a point imperfection (Section 3.2 and Tables 3 and 4)

is assumed with D0= 0.5D, l = 0.5 and effective weightm= 5.91 kN/m which gives a suspended spane0= 10.495 m. Simi-lar geometric parameters are used for the lateral buckling case

(Fig. 1and Eq. (5)) with Y0= 0.5D,k = 2e0 and Q0= lm= 2.955 kN/m and D = D/2.

The upheaval and lateral response is shown inFig. 19in termsof normalised axial force P/AE and the total strain at the crown

point (x= 0) against the normalised buckle amplitude. Lateral

buckling takes place at a lower axial compression force Pthan up-

heaval buckling. In the post buckling range the strain is dominated

by bending strain and upheaval buckling induces higher strain.Assuming yield strain of 0.2%, Fig. 19 shows that the resulting

strain at buckle amplitude 8D is nearly 1.5 times and 3.25 times

the yield strain under lateral and upheaval buckling, respectively.

This highlights the danger posed by lateral and particularly by up-

heaval buckling in triggering a catastrophic failure through exces-

sive bending that may lead to, for example, propagation buckling

[1].

6. Conclusion

For lateral buckling, analytical results based on isolated half-

wavelength model and FE results from a long pipe with nonlinearpipesoil interaction model were presented and compared. It was

shown that some of the previously published results do not agree

with the current analytical and numerical results. It was found that

the condition of localisation at vanishing peak in the compressive

axial forcePis synonymous to P/Pe= 1 for the isolated half-wave-

length model. That is, for a given imperfection amplitude, localisa-

tion will take place at a critical k at whichPequals toPe.

For upheaval buckling, a tabulated analytical solution using

three different types of local initial imperfection was presented.

The effect of various parameters on upheaval buckling is investi-

gated and the response under various initial imperfection types

is compared. The current results for initially unstressed pipe are

different from previously published results.

The lateral and upheaval response of a typical pipeline is inves-

tigated and compared. Higher compressive axial force P(and hence

higher temperature) is achieved under upheaval buckling in com-

parison to lateral buckling. Excessive bending stress is induced

during the post-buckling response particularly in upheaval buck-

ling. The excessive bending stress may trigger catastrophic failure,

such as propagation buckling of the pipeline [1].The coupling be-

tween lateral/upheaval buckling and propagation buckling is cur-

rently being investigated.

Acknowledgments

This research is being undertaken within the CSIRO Wealth

from Oceans Flagship Cluster on Subsea Pipelines with fundingfrom the CSIRO Flagship Collaboration Fund.

Fig. 18. Comparison of analytical and experimental results of a pipe with fixed anchorage points.

Fig. 19. Comparison of normalised axial force and crown point strain under lateral

and upheaval buckling.



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