Thin-Walled Structures 72 (2013) 113–120

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Thin-Walled Structures

0263-82http://d

n CorrE-m

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Buckle interaction in deep subsea pipelines

Hassan Karampour a, Faris Albermani a,n, Martin Veidt b

a School of Civil Engineering, The University of Queensland, Australiab School of Mechanical and Mining Engineering, The University of Queensland, Australia

a r t i c l e i n f o

Article history:Received 29 October 2012Accepted 3 July 2013Available online 24 July 2013

Keywords:Buckle-interactionBuckle propagationUpheaval bucklingSubsea pipelines

31/$ - see front matter Crown Copyright & 20x.doi.org/10.1016/j.tws.2013.07.003

esponding author. Tel.: +61 7 336 541 26.ail address: f.albermani@uq.edu.au (F. Alberm

a b s t r a c t

The paper investigates the interaction between propagation buckling and upheaval or lateral buckling indeep subsea pipelines. The upheaval and lateral buckling are two possible global buckling modes in longpipelines while the propagation buckling is a local mode that can quickly propagate and damage a longsegment of a pipeline in deep water. A numerical study is conducted to simulate buckle interaction indeep subsea pipelines. The interaction produces a significant reduction in the buckle design capacity ofthe pipeline. This is further exasperated due to the inherent imperfection sensitivity of the problem.

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

1. Introduction

The relentless demand for energy resources has shiftedhydrocarbon exploration to deep frontier subsea regions. Exam-ples of recent deep subsea fields are the Perdido fold belt oil fieldsat depth of 2300 m in the Gulf of Mexico and Lula–Mexilhão gasfields at a depth of 2145 m at Santos basin off the coast of Brazil. Itis expected that 25% of offshore petroleum production will be indeep water by 2015. Hydrocarbon production in deep waterrequires long pipelines (several hundred kilometres) and thedesign of such pipelines poses many engineering challenges.

A long pipeline may experience global buckling through lateralor upheaval buckling modes. Although these two buckling modesare not essentially failure modes, they can precipitate failurethrough excessive bending that may lead to fracture, fatigue orpropagation buckling. In deep water, the catastrophic propagationbuckling can quickly transform the pipe cross-section into adumb-bell shape that travels along the pipeline, as long as theexternal pressure is high enough to sustain propagation. A numberof experimental, analytical and numerical studies have beenconducted by many researchers on; upheaval buckling [1–3],lateral buckling [4–6] and propagation buckling [7–9] of pipelines.

So far, the buckle interaction between lateral or upheavalbuckling and propagation buckling has received very limitedattention [10]. Buckle interaction is a possible scenario in deepwater. The current trends towards deep water operations justify anassessment of the effects of this interaction on the integrity of thepipeline, which is the subject of this paper.

Global buckling of subsea pipelines is presented in Section 2.An analytical approach is used to study upheaval buckling and a

13 Published by Elsevier Ltd. All r

ani).

numerical study is conducted for lateral buckling. Section 3 isdevoted to propagation buckling and its imperfection sensitivity.The interaction between upheaval or lateral buckling and propa-gation buckling is presented in Section 4. The effect of theinteraction is summarised in an interaction curve that shows thepercent reduction in buckling capacity of the pipeline. Two modelaluminium pipes with diameter-to-thickness ratio (D/t) of 28.57and 42.86 are used for comparison. The nominal properties ofthese model pipes are given in Table 1. A third pipe given inTable 1 (D/t¼34.9) is used for verification in Section 4.

2. Global buckling of subsea pipelines

A pipeline is a slender structure that travels long distances. Thehydrocarbon contents in the pipeline usually are at high tempera-ture (80 1C or higher) and high internal pressure (10 MPa orhigher). Both the rise in temperature and internal pressure resultin longitudinal expansion of the pipeline. The seabed friction actsto restrain this expansion which results in the build-up of axialcompression in the pipe that may eventuate in buckling. A pipelineresting on the seabed will buckle laterally (in the horizontal plane)while a trenched pipeline will undergo upheaval buckling (in thevertical plane). The axial compression force, N, in the pipeline dueto restrained longitudinal expansion is given by

N¼ EAαΔTe ð1Þ

where the effective temperature change, ΔTe accounts for thecombined effects of temperature ΔT and internal pressure ρ

ΔTe ¼ΔT þ ρDð1�2υÞ4tEα

ð2Þ

ights reserved.

H. Karampour et al. / Thin-Walled Structures 72 (2013) 113–120114

in the above equations, E is elasticity modulus, A is cross-sectionarea, D is the pipe's outer diameter, t is the wall thickness, υ isPoisson's ratio and α is the coefficient of thermal expansion.

2.1. Upheaval buckling

A long heavy beam resting on a rigid frictional foundationmodel is used for investigating upheaval buckling of pipelines.Previous studies have shown that upheaval buckling is sensitive tolocal initial imperfection while foundation stiffness has a smalleffect on the overall response [1–3]. A small deflection approach is

Table 1Nominal properties of the studied model pipes.

D/t D(mm)

t(mm)

E(GPa)

Et(MPa)

sy(MPa)

syE Py

(kN)Mp

(kN/mm)

42.86 38.10 0.90 69.0 1500 90 0.013 9.35 110.7928.57 25.40 0.90 69.0 1500 90 0.013 6.16 48.0734.90 31.70 0.91 186.0 2000 259 0.014 11.57 222.75

Fig. 1. Upheaval buckling of a pipe restin

Fig. 2. Axial force distributio

adopted here similar to [2]. Different types of local initial imper-fection can be considered such as; point imperfection (isoprop),fully contact imperfection and infilled prop. For sake of brevity, apoint imperfection (isoprop) is assumed here. Isoprop imperfec-tion gives a more critical response than fully contact imperfectionand is easier to model than infilled prop, which usually yieldscomparable result. Coulomb friction model is adopted with a coeffi-cient of friction μ.

A pipe with diameter D, wall thickness t and submerged self-weight m resting on a point imperfection of height Δ0 is shown inFig. 1. The initial, pre-buckled and post-buckled configurations ofthe pipe are shown in the figure with half lengths εo, εu and ε,respectively, and δ is uplift amplitude at the crown. The axialcompression force distribution of the buckled pipe (half model) isshown in Fig. 2. The compression axial force varies from P in thebuckled region to fully mobilised N away from the buckle with asliding length L1 controlled by axial friction.

Prior to temperature rise (P¼0), the equilibrium profile of thesuspended pipe is

w0x ¼� mx4

24EIþmε0x3

9EI�mε0

2x2

12EIþ Δ0 ð3Þ

g on a point imperfection (isoprop).

n in upheaval buckling.

H. Karampour et al. / Thin-Walled Structures 72 (2013) 113–120 115

with I is the second-moment of area, x≤εo and the length of thesuspended span

εo ¼ 72EIΔo

m

� �1=4

Axial compression is induced in the pipe as the effectivetemperature rises (Eq. (2)). Before lift-off, the bending momentat x is given by

Mo þ PðΔ0�wÞ þ Rx�12mx2 ¼ EI wx

00 �wox00½ � ð4Þ

where R and Mo are the vertical reaction and bending moment atthe crown (x¼0). Previous researchers [1,2] had neglected theeffect of initial curvature w0x

00 in Eq. (4) which was shown to beinconsistent with experimental results [3]. Solving Eq. (4) underpre-upheaval boundary conditions given in Table 2, the governingcharacteristic equation (pre-upheaval) is obtained as shown in thesame table and the prop reaction R is calculated:

R¼ 2mk

ð sin ðkεÞ�kε cos ðkεÞÞ1� cos kεð Þ �2

3mε0 ð5Þ

where k2¼P/EI with a characteristic length equal to 1/k.At lift-off (R¼0), the uplifted half-wavelength εu is obtained

from Eq. (5) which gives εu≅0:75εo . This indicates that the pipeshrinks before it lifts off the prop. Following lift-off, Eq. (4) (withR¼0) is solved subject to post-upheaval boundary conditions inTable 2 to yield the governing characteristic equation and theresulting buckle profile wx, both are given in Table 2.

By prescribing the uplifted length ε, the characteristic equa-tion (Table 2) is solved for the compressive axial force, P, in theuplifted pipe. Due to loss of friction along the buckled length andgeometric shortening effect, the force P is less than the axialcompressive force, N, away from the buckle (Fig. 2). The two axialforces P and N can be related using compatibility at x¼ε. Assumingan infinitely long pipe

N¼ P þ 2mμEAΔs�ðmμεÞ2h i0:5

ð6Þ

where Δs is the geometric shortening in the buckle and is given by

Δs¼Z ε

0

12

wx0ð Þ2� wo

0ð Þ2h i

dx ð7Þ

Table 2Upheaval buckling: boundary conditions, characteristic e

Boundary conditionsPre-upheaval wð0Þ ¼Δ0 ; w

Post-upheaval wð0Þ ¼w0 ; wεu≤ε≤ε0

Post-upheaval wð0Þ ¼w0 ; wε0oε

Characteristic equationsPre-upheaval ðkε0Þ4½ cos ðkεÞ��Post-upheaval kε0 1� cos kεð Þ� �εu≤ε≤ε0

Post-upheaval kε cos kεð Þ� sinε0oε

Upheaval buckled profile w(x)Pre-upheaval w xð Þ ¼ Λ sin kxð

C1 ¼� mk2EI

; C

Post-upheaval w xð Þ ¼ A sin kxðC1 ¼� m

k2EI;

The above equations together with the equations given inTable 2 were implemented in Matlab [11] and solved numericallyby prescribing the uplifted length ε.

Using the two pipes shown in Table 1 with D/t 42.86 and 28.57,and assuming soil friction μ¼0.5, the submerged self-weightm¼14.1 N/m (with coating, for both pipes) and imperfectionamplitude Δ0¼D, the upheaval buckling response of these twopipes is calculated using this approach. Fig. 3 shows the normal-ised axial compression force P/Py in the buckle and the normalisedcrown's bending moment M/Mp against normalised crown's upliftdisplacement (δ�Δ0)/D, where

Py ¼ syπDot ð8Þ

Mp ¼ syD20t ð9Þ

D0 ¼D�t ð10Þand sy is the yield stress.

The early drop in the moment curve shown in Fig. 3 correspondto shrink in the pipe before lift-off. While the axial force reachesits peak at lift-off the bending moment shows monotonic increasewith uplift. The normalised upheaval response (axial and bending)for the two pipes (D/t¼28.57 and 42.86) nearly coincide at higheruplift amplitudes.

2.2. Lateral buckling

Finite element modelling of lateral buckling is more amenablethan upheaval buckling since there is no loss of contact between thepipe and the seabed during buckling. For this reason, nonlinear FEshell modelling of the two pipes used in the previous section(Table 1) is conducted using ANSYS [12]. Thin shell (Shell-181)elements with 5 through-thickness integration points and von-Mises elastoplastic material definition (Table 1) with isotropic hard-ening are used. The model accounts for possible ovalization of thepipe's cross-section under lateral buckling. Nonlinear spring elements(COMBIN39) are used to account for the lateral drag and verticalstiffness of the seabed. Fig. 4 shows the bilinear constitutive modeladopted for the seabed lateral drag where the peak force Fy isassumed to be mobilised at a lateral displacement equal to the pipe'sdiameter D. Assuming rigid seabed, the vertical springs wereassigned a substantially higher stiffness than lateral springs.

quations and buckled profiles.

x0 ð0Þ ¼ 0 ; wðε0Þ ¼ 0 ; wx

0 ðε0Þ ¼ 0; wx00 ðε0Þ ¼ 0

x0 ð0Þ ¼ 0; wðεÞ ¼ 0; wx

0 ðεÞ ¼ 0; wx00 ðεÞ ¼ 0

x0 ð0Þ ¼ 0 wðεÞ ¼ 0; wx

0 ðεÞ ¼ 0; wx00 ðεÞ ¼ 0

1� þ 72 kεð Þ2½1þ cos ðkεÞ��288kε sin ðkεÞ ¼ 0

�3 sin kεð Þ�kε cos kεð Þ� �¼ 0

kεð Þ þ kε03 cos kε0ð Þ� sin kε0ð Þ þ 2kε0

3 ¼ 0

Þ þ Γ cos kxð Þ þ C1x2 þ C2xþ C3

2 ¼ 1k2EI

Rþ 23mε0

� �; C3 ¼ 2m

EIk4þ Δ0 þ M 0ð Þ

k2EI� mε20

6k2EI

Þ þ B cos kxð Þ þ C1x2 þ C2xþ C3

C2 ¼ 2mε03k2EI

; C3 ¼w0 þ Mð0Þk2EI

þ mk2EI

2k2� ε20

6

�

Fig. 4. Assumed lateral drag force of the seabed under lateral buckling.

0.00

0.05

0.10

0.15

0.20

0 1 2 3 4 5 6 7

P/Py

D/t=42.86 UpheavalD/t=28.57 UpheavalD/t=28.57 lateralD/t=42.86 Lateral

0.0

0.2

0.4

0.6

0.8

1.0

0 1 2 3 4 5 6 7

M/Mp

D/t=42.86 UpheavalD/t=28.57 UpheavalD/t=28.57 lateralD/t=42.86 Lateral

Fig. 3. Normalised upheaval and lateral buckling response: (a) axial force vs. crown displacement and (b) crown moment vs. crown displacement.

H. Karampour et al. / Thin-Walled Structures 72 (2013) 113–120116

An initial geometric imperfection with amplitude Δo¼4D anda sinusoidal half wave length λ¼200D is assumed (Δo/λ iscomparable to the upheaval buckling case in Section 2.1). In orderto have adequate thermal feed-in length for the evolution oflateral buckling, the length of the FE model of the pipe is takenas 4λ.

Due to symmetry, a half-model (2λ) is used in the analysis. Theaxial compression force is applied through incrementing thelongitudinal displacements at the far end of the pipe. The resultingaxial force and bending moment are obtained by integrating theinduced reactions at the near end (the crown). Fig. 3 shows thenormalised axial compression force P/Py in the buckle and thenormalised crown's bending moment M/Mp against normalisedcrown's lateral displacement (δ�Δ0)/D for the two pipes. FromFig. 3, it is clear that the axial compression force that initiateslateral buckling is substantially lower than that necessary forupheaval buckling. Unlike upheaval buckling response, the lateralresponse remains distinguishable for both pipes at higher lateraldisplacements. A gentle change in axial load following buckling isobtained under lateral buckling in comparison to the sharp dropunder upheaval buckling. At higher lateral displacements, thebending response for D/t¼42.86 approaches that for upheaval

buckling. However, for lower D/t (28.57) which is more suitable fordeep subsea application, substantially higher bending moment isinduced under upheaval rather than lateral buckling.

3. Buckle propagation in deep subsea pipelines

Buckle propagation is a snap-through phenomenon that can betriggered by a local buckle, ovalization, dent or corrosion in the pipewall. The resulting buckle quickly transforms the pipe cross-sectioninto a dumb-bell shape that travels along the pipeline as long as theexternal pressure is high enough to sustain propagation. Fig. 5 shows atypical buckle propagation response obtained from testing a 3 m longaluminium pipe with D/t¼25 in a hyperbaric chamber [7]. Theresponse shown in Fig. 5 is depicted in terms of the appliedhydrostatic pressure against the pipe's volume change (ΔV/V) and ischaracterised by; the pressure at which the snap-through takes place(the initiation pressure PI) and the pressure that maintains propaga-tion (the propagation pressure Pp) which is a small fraction of PI.

The elastic collapse pressure, Pc, represents an upper-bound onPI while Palmer and Martin [9] pressure, PPM, gives a lower-bound

0.00

0.25

0.50

0.75

1.00

1.25

0.0 0.2 0.4 0.6 0.8 1.0

P/Pc

D/D

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

0.0 0.1 0.1 0.2 0.2

P (MPa)

D/D

Fig. 6. Propagation buckling response for intact (Ω¼0.1%) and dented (Ω¼1.0% and Ω¼1.5%) pipes: (a) normalised pressure; and (b) absolute pressure.

Fig. 5. Buckle propagation in hyperbaric chamber test of a 3 m long aluminium pipe (D/t¼25).

H. Karampour et al. / Thin-Walled Structures 72 (2013) 113–120 117

on Pp. These two pressures, Pc and PPM, are given by

Pc ¼2E

1�ν2� � t

D

�3

ð11Þ

PPM ¼ πsytD

�2

ð12Þ

Nonlinear finite element analysis of propagation buckling wasconducted and shown to agree reasonably well with experimentalresults [7].

Buckle propagation of the two pipes used in Section 2 isconducted using nonlinear finite element analysis with ANSYSthin shell-181 element. Frictionless contact and target elements(ANSYS element 174 and 170) are used to define the contactbetween the inner surfaces of the pipe wall. A von-Mises elasto-plastic material definition with isotropic hardening was adopted.In order to control the nonlinear analysis, a small dent Ω¼0.1%over a small circular surface area (20 mm diameter) is introducedat the pipe's mid-length

Ω¼ΔD=D ð13ÞFig. 6 shows the predicted finite elements propagation

response of the two pipes with D/t¼28.57 and 42.86 (Table 1).

The response is shown in terms of normalised applied externalpressure (P/Pc, Fig. 6(a)) and the applied pressure (P, Fig. 6b)against normalised distortion of the pipe (ΔD/D). In order toclearly distinguish the response of each pipe at buckle initiation,Fig. 6(b) shows the response up to ΔD/D¼0.2. It is clear from thisfigure that propagation pressure Pp is a small fraction of initiationpressure PI (around 20% for both pipes according to Fig. 6(a)). Thisnecessitates substantial increase in material and installation costof deep subsea pipelines since the design is governed by Pp.

The initiation pressure PI represents a snap-through instability;it is expected to be very sensitive to imperfection (such as dent forexample). By increasing the initial imperfection from intact(Ω¼0.1%) to dented pipe with Ω¼1% and Ω¼1.5%, Fig. 6(b) shows a drastic reduction in PI of 17% and 33% for D/t¼42.86and 28.57 respectively. On the other hand, Fig. 6 shows that Pp isinsensitive to imperfection.

The catastrophic nature of buckle propagation and its acuteimperfection sensitivity highlights the importance of investigatingpossible buckle interactions between global (upheaval or lateral)and local (propagation) buckles in deep subsea pipelines. It is forthis reason that Albermani et al. [7,13] proposed a texturedpipeline that exhibits superior buckle propagation capacity andinsensitivity to imperfection.

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 0.1 0.2 0.3 0.4 0.5 0.6

M/Mp

/ c

Current Result

B

A

Experiment [14]

Fig. 7. (a) Exaggerated view of the assumed wrinkled initial imperfection in the vicinity of mid-length and (b) interaction of bending and external hydrostatic pressure forpipe D/t¼34.9 in Table 1.

Fig. 8. A crown segment of the pipeline under the combined actions of upheaval/lateral buckling (axial force P and bending moment M) and external pressure ρ.

H. Karampour et al. / Thin-Walled Structures 72 (2013) 113–120118

4. Buckle interaction

As shown in Sections 2 and 3, upheaval and lateral buckling caninduce excessive bending in pipelines. The resulting bending mayprecipitate catastrophic buckle propagation that damages a sub-stantial length of deep subsea pipeline. This is further exasperatedby the severe imperfection sensitivity of propagation response. Inthis section, a FE study is conducted to investigate buckle interac-tion for the two pipes with D/t¼28.57 and 42.86 shown in Table 1.

First, the FE modelling is verified against available experimen-tal study on the interaction between bending and externalpressure [14]. A steel pipe 1 m long (L) with D/t¼34.9 (L/D¼31.5, Table 1) was used in the reported experimental study. Usingsymmetry, a shell FE model of the pipe using half-length(500 mm) and half cross-section is generated in ANSYS [12] usinga total of 2250 SHELL-181 elements (18 elements in circumfer-ential direction and 125 elements along the length). A bilinearmaterial model (Table 1, D/t¼34.9) using von-Mises plasticitywith isotropic hardening is adopted.

To control the numerical solution, a localised wrinkled initialimperfection ω [15] is imposed on the compression side of thepipe at mid-length as shown in Fig. 7(a).

ω¼�D2

ao þ ai cosπxNλ

� h icos ðπx

λÞ 0≤x≤Nλ ð14Þ

where the imperfection half-wave length λ¼0.165D, number ofhalf-waves N¼11, with a base amplitude ao¼0.0025 and 20%amplitude bias, ai/ao, towards mid-span.

The load is applied at two stages according to the experiment[14]. At the first stage, a couple is incrementally applied at the farend of the pipe until the desired bending/curvature is achieved.This is followed by incremental application of external hydrostatic

pressure while maintaining the desired curvature. Fig. 7(b) showsthe normalised moment-curvature response from the experimentand the current FE simulation. The curvature k is normalised bycritical curvature kc

κc ¼t

D20

ð15Þ

As seen in Fig. 7(b), the bending moment is incrementedbeyond the plastic moment capacity and held constant at anormalised curvature around 0.55 followed by the application ofthe external hydrostatic pressure. The onset of failure during theexperiment was reported at an applied hydrostatic pressure of0.3po accompanied by 20% drop in moment (Point B in Fig. 7(b)),where po is given by

po ¼ 2sytD0

�ð16Þ

The FE results are in good agreement with the experimentalresults with a predicted collapse at 0.28po (Point A) accompaniedby 14% drop in moment.

4.1. Interaction of upheaval and propagation buckling

According to Section 2.1, the highest bending moment underupheaval buckling is in the vicinity of the uplifted crown point. Forthis reason, the FE model used to investigate the interactionbetween upheaval and propagation buckling is based on a crownsegment of a dented pipe with a length L¼1000 mm (500 mm oneither side of the crown point, Fig. 8). The two pipes with D/t¼28.57 and 42.86 (Table 1) are used in the interaction study. Thisgives L/D of 26–39 which is comparable to that of the experi-mental study presented in Fig. 7(b).

Fig. 9. (a) FE model of the interaction between upheaval/lateral and propagation buckling (loading and constraints) and (b) FE result of buckle propagation due to theinteraction between upheaval/lateral buckling and external pressure.

Table 3Reduction in initiation pressure due to interaction of upheaval and propagationbuckling.

D/t M/Mp Initiation pressureþI (MPa)

Reduction in initiationpressure (PI�þI)/PI (%)

42.86 (Ω¼1%) 0 1.74 00.50 1.61 7.470.60 1.58 9.200.70 1.54 11.490.80 1.49 14.370.90 1.44 17.24

28.57 (Ω¼1.5%) 0 4.53 00.50 3.89 14.130.60 3.82 15.670.70 3.73 17.660.80 3.65 19.430.90 3.58 20.971.00 3.25 28.26

0

5

10

15

20

25

30

0 0.2 0.4 0.6 0.8 1

%

M/Mp

Fig. 10. Interaction of upheaval/lateral buckling and propagation buckling.

H. Karampour et al. / Thin-Walled Structures 72 (2013) 113–120 119

A half-length and half-section model is used as shown in Fig. 9(a). A shell FE model using SHELL-181 elements and a bilinearmaterial model (Table 1, D/t¼28.57 and 42.86) using von-Misesplasticity with isotropic hardening is adopted. Similar localisedwrinkled initial imperfection as described in Section 4 (Eq. (14)and Fig. 7(a)) is assumed.

Making use of symmetry (Fig. 9(a)), the lateral (X) displacementand the rotation about the longitudinal axis (Z) are restrainedalong L1 and L2. Similarly, the longitudinal displacement (Z) androtation about X-axis are restrained along L4 at mid-length and thevertical Y-displacement is restrained along L3 at the far end.

The loading shown in Fig. 9(a) is applied in three steps. In thefirst step, axial compression load is incremented to the maximumupheaval buckling load obtained in Fig. 3(a). In the second loadingstep, the axial load is maintained (which is conservative) while acouple is incremented at the far end of the pipe. The couple isincremented to the desired M/Mp ratio (Fig. 3(b)). In the third stepof loading, while maintaining the axial and bending load achievedin the previous two steps, external hydrostatic pressure is incre-mented until a propagating buckle in initiated at þI (Fig. 9(b)). Theresulting initiation pressure þI is compared to the initiationpressure of a dented pipe (same amount of dent Ω) subjected toexternal hydrostatic pressure alone, PI, as discussed in Section 3. Itis worth noting that the effect of the axial compression load (step1) on the final results is negligible. This is expected since theupheaval response is dominated by bending.

The effect of the interaction between upheaval and propagationbuckling on buckle initiation þI for the two pipes (D/t¼28.57 and42.86) is summarised in Table 3 and Fig. 10. The initiation pressurePI of the dented pipes (no interaction, M/Mp¼0, Fig. 6(b)) is 1.74and 4.53 MPa for D/t 42.86 and 28.57, respectively. When the pipeundergoes upheaval buckling (due to restrained thermal expan-sion), the rapid growth in bending moment as the pipe liftsoff the seabed (Fig. 3(b)) results in a drastic reduction in initia-tion pressure þI. According to Table 3, when 90% of the pipe'sbending capacity is exhausted, the resulting reduction in initiationpressure is 17-21% for D/t¼42.86 and 28.57 respectively. As seenfrom Fig. 10, a steeper reduction in initiation pressure is obtainedas M/Mp approaches 1. Due to interaction, higher reduction inbuckle initiation capacity is expected at lower D/t (deep subseaapplications).

4.2. Interaction of lateral and propagation buckling

A similar model to that used in the previous section (Figs. 8 and9) is used to investigate the interaction between lateral andpropagation buckling of the two dented pipes with D/t¼28.57and 42.86 (Table 1). The restrained conditions along L1–L4 (Fig. 9(a)) were revised to reflect the symmetrical conditions underlateral buckling. Similar loading sequence (three steps) is followedas described before and according to the lateral buckling responseshown in Fig. 3(a) and (b). Due to the interaction between thelateral and propagation buckling, the resulting initiation pressureþI is lower than the initiation pressure, PI, of a similarly dentedpipe subjected to external hydrostatic pressure alone (Section 3).

H. Karampour et al. / Thin-Walled Structures 72 (2013) 113–120120

Interaction between lateral and propagation buckling for the twopipes (D/t 28.57 and 42.86) can be represented by the sameinteraction curves shown in Fig. 10 and in conjunction withFig. 3 (lateral case). Note that according to Fig. 3, the same crowndisplacement (lateral or upheaval) induces less moment underlateral rather than upheaval buckling. Accordingly, the resultingreduction in initiation buckling (Fig. 10) due to lateral/propagationinteraction is less than that due to upheaval/propagation interac-tion. As discussed in Section 3, buckle initiation is a snap-throughinstability that is very sensitive to geometric imperfection, thisimperfection sensitivity together with the possibility of buckle inter-action, impose sever design limitations on deep subsea pipelines.

5. Conclusions

The paper has presented an analytical solution to upheavalbuckling and a FE analysis of lateral buckling and buckle propaga-tion. It is shown that the snap-through propagation buckling isvery sensitive to initial imperfection. The upheaval and lateralbuckling response for two model pipes with different D/t ratios arepresented and compared. A pipe segment in the vicinity of thecrown point of global buckling (upheaval or lateral) is used tostudy buckle interaction between upheaval or lateral buckling andpropagation buckling. The calculated axial compression load andbending moment from upheaval/lateral buckling are fed to thissegment model followed by the incremental application of exter-nal hydrostatic pressure on the pipe to obtain propagationresponse. Due to the interaction between upheaval/lateral andpropagation buckling, a substantial reduction in initiation pressureis expected, particularly at lower D/t ratios. An interaction curve ispresented for each of the pipe models considered. Higher reduc-tion in initiation pressure is expected under upheaval/propagation

interaction in comparison to lateral/propagation interaction. Theacute imperfection sensitivity coupled with buckle interactionneed to be considered in the design of deep subsea pipelines.

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