Gong 2012 Marine-Structures

8/13/2019 Gong 2012 Marine-Structures

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Buckle propagation of offshore pipelines under external

pressure

Shunfeng Gong, Bin Sun, Sheng Bao*, Yong Bai

Institute of Structural Engineering, Zhejiang University, Hangzhou 310058, China

a r t i c l e i n f o

Article history:

Received 13 April 2012

Received in revised form 27 August 2012

Accepted 12 October 2012

Keywords:

Offshore pipeline

Buckle propagationCollapse

External pressure

a b s t r a c t

Accidental damage of offshore pipelines in the form of local

buckles induced by excessive bending deformation during deep-

water installation may severely lead to local collapse of the tube

and consequent buckle propagation along the pipeline. The

present paper describes experimental and numerical research

conducted to predict the magnitude of buckle propagation pres-

sure of offshore pipelines under external pressure. The experi-ments of buckle propagation for pipe specimens with different

initial geometric imperfections using 316 grade stainless steel

tubes are carried out under quasi-static steady-state conditions in

a sealed hyperbaric chamber. The stressstrain characteristics in

the axial tensile test are measured for the tube material, and then

used to numerically calculate the buckle propagation pressure of

the pipe. The comparisons between experimental and numerical

results are conducted to establish the precise numerical simulation

technique. Based upon experimental and extensive numerical

results, a more reasonable empirical formula for buckle propaga-

tion pressure of offshore pipeline with various values of diameter-

to-thickness ratio as well as different strain hardening modulus

and yield stress is proposed.

2012 Elsevier Ltd. All rights reserved.

1. Introduction

In recent years, the increasing demand for oil and natural gas resources has prompted the industry

to extend exploration and production to deepwater or ultra-deepwater regions. Pipelines play a very

* Corresponding author. Tel.: 86 571 88208728.

E-mail address: [email protected](S. Bao).

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Marine Structures

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / lo c a t e /

m a r s t r u c

0951-8339/$ see front matter 2012 Elsevier Ltd. All rights reserved.

http://dx.doi.org/10.1016/j.marstruc.2012.10.006

Marine Structures 29 (2012) 115130
mailto:[email protected]://www.sciencedirect.com/science/journal/09518339http://www.elsevier.com/locate/marstruchttp://www.elsevier.com/locate/marstruchttp://dx.doi.org/10.1016/j.marstruc.2012.10.006http://dx.doi.org/10.1016/j.marstruc.2012.10.006http://dx.doi.org/10.1016/j.marstruc.2012.10.006http://dx.doi.org/10.1016/j.marstruc.2012.10.006http://dx.doi.org/10.1016/j.marstruc.2012.10.006http://dx.doi.org/10.1016/j.marstruc.2012.10.006http://www.elsevier.com/locate/marstruchttp://www.elsevier.com/locate/marstruchttp://www.sciencedirect.com/science/journal/09518339mailto:[email protected]


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important role in these challenging activities, and are the most efcient and economical means ofgathering and transporting oil and natural gas from subsea wells to offshore or coastal storage facilities.

The current maximum laying water depth has exceeded 3000 m [1], and with ever increasing water

depths, the pipes have to be designed to withstand the high loads dictated by the extreme water depth

during installation as the pipeline is empty and not subject to any internal pressure. Accordingly, the

development of higher strength and toughness steel pipes with thicker wall is required to resist

ambient hydrostatic pressure. In general, the diameter-to-thickness ratios (D/t) of offshore pipeline

range from 10 to 60. However, for deepwater scenarios, the pipe with a value ofD/tranging from 10 to

35 is more suitable. In addition, the yield stress of steel for typical offshore pipelines is commonly

between 276 and 448 MPa.

Local buckle is prone to occur in offshore pipelines due to excessive bending deformation nearby

the touch-down point in the process of deepwater installation. If the external hydrostatic pressure ishigh enough, the buckle could catastrophically propagate along the pipelines, or even threaten their

structural integrity[2,3]. The lowest pressure required to sustain the buckle propagation, is referred to

as the propagation pressure Pp, which is a characteristic pressure of the pipe and only 15% to 30% of

collapse pressurePcoof the intact pipe. In general, collapse pressure replaced by propagation pressure

to guide the design of offshore pipeline is deemed to be very conservative and uneconomical. There-

fore, a better choice is to install buckle arrestors at regular intervals along the pipeline, with the

purpose of conning the extent of pipeline damage to a limited length between two adjacent arrestors.

Buckle arrestors are the most commonly used devices that locally increase the wall bending rigidity of

the pipe in the circumferential direction, and thus provide an obstacle in the path of the propagating

buckle.

There are many different types of arrestors presented, respectively, by Johns et al. [4], and Kyr-iakides and Babcock[5]. For deepwater applications, the integral buckle arrestor with superior ef-

ciency is preferred to weld on to the pipe. It consists of a ring with certain length that has the same

internal diameter but is thicker than the pipe. The modied empirical design formula for arrestor

efciency is proposed by Lee et al. [6] based on extensive experimental and numerical results. Using

Nomenclature

D tube outer diameter

E Youngs modulus

E0 strain hardening modulusJ2 second invariant of deviatoric stress tensor

n RambergOsgood hardening parameter

Pco collapse pressure

Pp buckle propagation pressure

t tube wall thickness

D/t diameter-to-thickness ratio of the tube

V0 initial internal volume of the tube

dV volume change of the tube

w0 initial imperfection

x,y,z coordinatesD0 imperfection amplitude,D0 (Dmax Dmin)/(Dmax Dmin)

strain

q angular coordinate

n Poissons ratio

s stress

s0 API yield stress (0.5% strain offset)

sy RambergOsgood yield parameter

a strain hardening parameter,a E/E0

S. Gong et al. / Marine Structures 29 (2012) 115130116


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this formula to design buckle arrestors should predetermine buckle propagation pressure of the pipe.

In the past few decades, many researchers have addressed in such predictions and put forth some

empirical formula determined from theoretical and experimental results.

Palmer and Martin[3] rstly put forth an equation for propagation pressure as a function of both

diameter-to-thickness ratioD/t and material yield stress s0 based on the assumptions of the defor-

mation energy of the collapsed pipe cross-section equating to the work done by external pressure, andrigid-perfectly plastic material behavior of the pipe,

PPs0

p

t

D

2: (1)

The results underestimated the observed values of tests in that the mode of deformation considered

is purely inextensional so as to all deformation is assumed to be concentrated on four plastic hinges

equally spaced around the perimeter, and the surface stretching and strain hardening of the pipe are

neglected, thus it is only suitable for thin-walled pipes with larger values ofD/t. As to deepwater thick-

walled pipes, a propagating buckle will inevitably result in plastic and extensional circumferential

deformations as well as surface stretching.Johns et al.[4] rstly conducted the experimental studies on the arresting capability of different

types of buckle arrestors conceived to stop a propagating buckle along the pipeline. Based upon the

experimental results of the pipes with the values ofD/tranging from 21 to 71, an empirical formula was

proposed to evaluate the propagation pressure as follows:

PPs0

6

2t

D

2:5: (2)

The dynamic behavior of a propagating buckle for the purpose of designing effective arresting

devices was initially investigated through experiments by Kyriakides and Babcock [7], who subse-

quently implemented theoretical study in association with experimental results for aluminum tubes

and steel alloy tubes to develop an empirical formula for buckle propagation pressure of offshorepipeline, taking into account the effects of post-yield characteristics of the pipe material behavior

represented by the strain hardening modulus E0 and geometric parameter represented by the

diameter-to-thickness ratioD/tof the pipe[8, 9],

PPs0

10:7 0:54

E0

s0

t

D

2:25: (3)

The theoretical study was conducted to predict the propagation pressure of offshore pipeline based

on internal energy dissipation of a quadrant of ring model including the inuence of the material strain

hardening behavior by Steel and Spence [10], which developed the following expression to evaluate the

propagation pressure through some simpli

ed assumptions,

PPs0

p

4

1:0 2:07

E0

s0

0:122tD

0:352tD

2: (4)

DNV[11]in the Offshore Standard DNV-OS-F101 dened the calculating expression of the propa-

gation pressure for the pipe with the values ofD/tless than 45 as follows:

PPs0

35

t

D

2:5: (5)

Similarly, API[12]in the Offshore Pipeline Criterion RP 1111 recommended the following formula

for the propagation pressure,

PPs0

24

t

D

2:4: (6)

S. Gong et al. / Marine Structures 29 (2012) 115130 117


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Lately, Albernami et al. [13] proposed a modication to the lower bound solution of Eq. (1) by

accounting for the circumferential membrane and exural effects in the pipe wall, and obtained the

following expression:

PPs0 1:193

p

tD2

: (7)

Through nite element analysis they further discovered that a faceted pipe has a substantial

increase in initiation and propagation buckling capacities in contrast with the cylindrical pipe with the

same value ofD/t.

The indeed great differences exist in predicting the buckle propagation pressures of offshore pipeline

fromthe foregoing formula. The theoretical study [3,810] based upon the ring model generally neglects

the effects of stretching and bending along the length of the proles of propagation, and entirely

concentrates on the ring-bending. In addition, the results predicted from Eqs. (2), (5) and (6) are almost

very close for the values of D/t larger than 35. However, the drawbacks are very obvious as these

equations only take into account the diameter-to-thickness ratio and overlook the post-yield charac-

teristics of the pipe material behavior. Especially for the pipes with the values ofD/tless than 35, theplastic buckle will be predominant. Therefore, a more reasonable empirical formula for predicting the

buckle propagationpressure of offshore pipelines withvariousvalues ofD/tshould be developed further.

The numerical simulation is no doubt a very effective means to predict the buckle propagation

pressure for offshore pipelines. Jensen [14] rstly carried out theoretical and numerical study for a long

circular cylindrical shell to obtain the propagation pressure based on thin shell theory and small strain

approximation. The comparison of the calculated results for different elasticplastic material

descriptions shows that the J2 ow theory [15] predictions of propagation pressure are in better

agreement with the experimental results. Dyau and Kyriakides [16] systematically reviewed the

experimental aspects of propagation pressure of the tubes, then established a numerical analysis

model based upon the Sanders nonlinear shell kinematics with small strains and large displacements,

and the elasticplastic material behavior was modeled through the J2 ow theory[15]with isotropicstrain hardening. The results as compared with experimental values from small-scale aluminum tubes

and steel tubes with the values ofD/tbetween 18.2 and 37.2 show that the proposed numerical analysis

method may provide good estimates of the propagation pressure. Pasqualino and Estefen [17]devel-

oped a three-dimensional model based on the thin shell theory proposed by Sanders [18]with nite

strains and nite displacements to predict the propagation pressure of deepwater pipelines, and the

equilibrium equations were solved numerically using the nite difference method combined with the

explicit dynamic relaxation technique. The correlation between numerical and experimental results for

six small-scale steel pipe models with the values ofD/t, respectively, equal to 16, 21 and 24 presented

relatively good agreements. Xue and Hoo Fatt [19], respectively, conducted rigid-plastic theoretical

analysis andnite element analysis using the software ABAQUS for the steady-state buckle propagation

scenario in a corroded pipeline subjected to external hydrostatic pressure, and revealed the bucklepropagation pressure and the buckling mode depending on the depth and angular extent of the

corrosion. Furthermore, the buckle propagation phenomenon in pipe-in-pipe systems under external

pressure was studied through combined experimental and analytical efforts by Kyriakides[20], and

Kyriakides and Vogler [21], which put forth an empirical formula for propagation pressure of the

system and design recommendations.

Lately, in combination with experiments and theoretical analyses, Zara[22]investigated buckling

propagation and failure causes of the pressurized double-piping system under the thermo-

conditioning installation in the sewage treatment plant. Showkati and Shahandeh [23] carried out

the experimental studies on the buckling behavior of ring-stiffened thin-walled pipelines under

external hydrostatic pressure, and appraised the inuence of ring stiffeners on buckling strength,

buckling propagation, development of yield line and

nal collapse of the pipeline. In association withpractical pipe-laying circumstances, the buckling responses of thick-walled tubes under complicated

load combinations were systematically investigated by Gong et al. [24,25].

In the present study, the buckle propagation experiments of steel pipes with the value ofD/tequal

to 15 are conducted in a sealed hyperbaric chamber to identify the mechanism of a propagating buckle



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in a long thick-walled pipe, which is suited to deepwater pipelines. In combination with experimental

observations and results, a three-dimensional nite element model for the experimental steel pipe

within the frame of ABAQUS is developed to model the buckle propagation phenomenon. The nite

element model is validated by comparing numerical results with experimental observations. Subse-

quently, the parametric analysis for the propagation pressure is performed. Based on the extensive

numerical simulations and experimental results, a more reasonable empirical formula for predictingthe buckle propagation pressure of offshore pipeline is proposed taking into account geometric

parameter with the values ofD/tbetween 10 and 60, and material properties such as yield stresss0 and

strain hardening modulus E0.

2. Experiments

2.1. Experimental set-up and procedure

The purpose of this experiment is to initiate a propagating buckle in a long thick-walled pipe under

quasi-static steady-state conditions to measure the propagation pressure and observe the post-

buckling conguration of the pipe. For a propagating buckle to occur in a long pipe, two importantconditions are necessary. One is that the exterior surface of the pipe has to be subjected to high

enough external pressure, and the other is that a large initial geometric imperfection at one end of the

pipe should be imposed to initiate the propagating buckle. The experiments are conducted in

a specially designed and fabricated pressure cylinder shown in Fig. 1(a). The cylinder has an inner

diameter of 0.2 m and a length of 1.75 m, and its work pressure may attain 50 MPa. The external

pressure applied on the pipe specimen is achieved by using a single-piston manual pump shown in

Fig. 1(b) to pressurize water into the cylinder, in which the pressure is real-time monitored by an

electrical transducer shown inFig. 1(c) as well as the digital pressure gage shown inFig. 1(b) with

a precision of 0.01 MPa.

Fig. 1. Experimental facility.



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A section of pipe, welded two solid end-plugs on each end, usually more than 20 diameters long, is

completely sealed at one end and placed into a stiff pressure cylinder, and the other end with a hole in

the end-plug is connected with a container placed on the electronic scale through a rigid conduit, as

shown inFig.1(d) and (e). Therefore, the internal pressure of the pipe specimen is all along maintained

atmospheric pressure. The pressure cylinder, container, pipe specimen, and rigid conduit are

completely lled with water. It is ensured by vertically placing the pressure cylinder in a specially builtsteel box xed at the underground base with 2.0 m depth and venting water at the upper hole in the lid.

The gap between pressure cylinder and steel box is lled up with ne sand so as to guarantee the

stability of experimental set-up. The watertight test of the system is conducted by pressurizing water

into the cylinder, and keeping a relatively long time and then observing the pressure change inside the

cylinder through the digital pressure gage. To initiate a propagating buckle under external pressure, the

pipe specimen over a short length about 23 diameters away from the end-plug is intentionally

inicted to form a dent.

In the process of experiments, the pumping rate is always maintained at a relatively low, constant

velocity about 1.0 MPa per minute, therefore any material rate effects may be negligible. Along with

gradual increase of the pressure in the cylinder, local collapse is sure torstly occur at the imperfection,

which will lead to a sudden drop of the pressure in the cylinder to the level below the buckle prop-agation pressure of the pipe. Then, the buckle will gradually spread along the length while the pressure

in the cylinder remains at Pp. From a hole in the upper end-plug of the pipe specimen the displaced

water is directed to the container, the weight variation in the electronic scale is proportional to the

displaced water and therefore to the variation of the specimen inner volume. A schematic diagram of

the experimental apparatus assembly is shown inFig. 2.

2.2. Experimental results

The experiments are carried out using seamless stainless steel tubes, and the steel grade is SS316,

which has a very good elongation. In such case, a propagating buckle will not result in fracturing of thewall or ooding of the tube, and the buckle can be propagated in tubes with much lower values ofD/t.

Therefore, the steel pipes with lower diameter-to-thickness ratio (D/t) and larger initial geometric

imperfection can be adopted in experiments. The geometric characteristics of the pipes tested as well

Pressure cylinder

Pipe specimen

Initial imperfection

End-plug

Lid

Pressure gageTransducer

Electronic scale

Container

Water pump

Rigid conduit

Fig. 2. Schematic diagram of experimental apparatus.



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as their propagation pressures measured are listed in Table 1.The local collapsed and nal deformed

congurations of the steel pipe specimen No. 2 are shown inFig. 3, and the pipe after buckle propa-

gation is completely attened. In addition, the axial test coupon inFig. 4cut from the pipe is used to

measure the engineering stressstrain response of the material. The measured stressstrain response

is further transformed into true stresslogarithmic strain curve, as shown in Fig. 5. The elastic modulus

Eof the material is 188.9 GPa. The API [26] denition of yield stresss0, i.e., the stress at a strain of 0.005,

is used.

To conveniently conduct numerical simulation for different strain hardening parameter in pre-

dicting the propagation pressure, it is required a good representation of the material stress

strainbehavior for the complete strain range of interest. Apparently, a bilinear approximation of the stress

strain behavior is too rough. A more accurate, three-parameter t of the stressstrain behavior is

provided by the well-known RambergOsgood model[15]given by

s

E

1

3

7

ssy

n1: (8)

Table 1

Geometric parameters and experimental results

No. L(mm) D(mm) t(mm) D/t D0 Pco(MPa) Pp (MPa) s0(MPa) Pp/s0 103

1 1200 60.0 4.0 15 2.0% 27.6 12.3 319.2 38.5

2 1200 60.0 4.0 15 4.7% 21.4 11.6 319.2 36.3

Fig. 3. Deformed congurations of steel pipe specimen No. 2.



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where syis the effective yield stress, andnis the material hardening parameter. This material model in

nature cannot t the typical experimental results for strain levels larger than 12%.Fig. 5 similarly

shows the tted stressstrain curve through Eq. (8). As can be seen, a good t is achieved for strain

levels less than 0.015. However, for higher strains, it will cause a larger deviation. Due to this inade-

quacy of Eq.(8), the following modication is adopted in subsequent parametric analyses. The Ram-

bergOsgood model is still adopted for strain levels less than 0.015. Whereas for strain levels larger

than 0.015, the stressstrain curve is approximated with a straight line having a slope given by

Fig. 4. Material property test.

0.00 0.03 0.06 0.09 0.12 0.150

100

200

300

400

500

600

700

/MPa

Modified R-O Fit

R-O Fit

Experimental

Fig. 5. Measured and tted uniaxial stressstrain curves for SS316.



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E0 ds

d

0:015

E

1 3

7n

ssy

n1 : (9)

where E0

, in fact, is the strain hardening modulus. The modied RO t is also drawn inFig. 5, and isseen to be very close to the experimental curve. The t parameters for the material SS316 are given as

sy 291 MPa,n 10.1,E0 2680 MPa.

3. Comparison of experimental and numerical results

3.1. Finite element model

A nite element model is particularly developed in parallel with the experimental effort

within the framework of ABAQUS to simulate numerically the quasi-static steady-state buckle

propagation scenarios in the steel pipe specimen No. 2 presented in Table 1 under external

pressure. The actual geometric characteristics and material behavior of the pipe are used toestablish the numerical model. A three-dimensional, eight-node brick element with incompat-

ible mode, C3D8I, is chosen to model the pipe. Since this type of element is enhanced by

incompatible modes to bending behavior, it is well suited for the present large deformation

problems [27,28].

The general geometric characteristics of the model are shown inFig. 6.Guided by the experimental

observations, the deformation of the pipe cross-section is assumed to have two planes of symmetry,

namely, planesx-yandx-z. A local imperfection is added at the location ofx 2D3D. This is the only

way of initiating local collapse without affecting subsequent buckle propagation. The tubes are

assumed to be circular and the wall thickness to be uniform along the length. The local imperfection in

the form of ovality is described in the following form:

Fig. 6. Mesh and imperfection ofnite element model.



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w0q

8>>>>>>>>>>>>>:

D0

D

2

exp

b

2:25Dx

D

2cos2q; x2:0D; 2:25D

D0

D

2

cos 2q; x2:25D; 2:75D

D0

D2

exp

b

x 2

:

75DD

2cos 2q; x2:75D; 3:0D

: (10)

where w0 is the radial displacement, q is the polar angular coordinate measured fromy axis, andx is the

axial coordinate. The imperfection parameters used are the initial ovality D0, and the constantb 0.5D,

which characterizes the extent of the imperfection.

The pipe model is discretized into 4 parts through the thickness, 20 parts around the quarter

circumference, and 240 parts along the length, which is found to be adequate through trial analysis.

The symmetrical boundary conditions are applied at planes x-yand x-z, i.e., in plane x-y, z-direction

displacement of the nodes is only constrained, and in planex-z,y-direction displacement of the nodes

is only constrained. However, at both ends of the tube, all displacements of the nodes are xed.

Since only one quarter of the cross-section is analyzed, an imaginary rigid surface is set along plane

x-zusing rigid elements (R3D4) to simulate the contact of inner surfaces of the tube wall. Contactbetween the walls of the collapsing tube is modeled through the surface-based contact pair. In this

scheme, the rigid surface is dened as the specied master surface, and the inner surface of the tube

wall is dened as the slave surface. The contact direction is always normal to the master surface, and

Fig. 7. A sequence of deformed congurations for steel pipe specimen No. 2.



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the slave nodes are constrained not to penetrate into the master surface. In addition, the collapsing

walls of the tube do not allow separation after contact, and the hard contact and nite sliding options

are used.

The J2 ow theory [15] of plasticity with isotropic strain hardening and nite deformation is

adopted to describe the elastoplastic behavior of the pipe material. The multi-linear approximations of

true stresslogarithmic strain curve and the modied RambergOsgood model are used to characterizethe material properties, respectively.

As similar calculations have been conducted by Kyriakides and Vogler[21], a volume controlled

loading procedure is adopted using the hydrostatic uid elements of ABAQUS (a combination of F3D3

and F3D4). This type of elements can indicate the change in volume inside a control region de ned

around the structure. Thus, the pressure becomes an additional unknown while the volume change is

enforced as a constraint via the Lagrange multiplier methods.

3.2. Simulation of a propagating buckle

The pipe model is loaded by external pressure. Buckling and collapse are initiated by a small, local,

initial geometric imperfection in the neighborhood of x 0. The response of such systems oftenexhibits limit load and turning point instabilities, thus Riks path-following method (arc length

method) is used to follow the loading history. Nlgeom option is also selected for this problem char-

acterized by several nonlinearities which severely varies during the loading history. An incremental

solution scheme with variable loading increments depending on the stage of the loading history is

developed empirically.

Fig. 7 shows that a sequence of deformed congurations of the steel pipe specimen No. 2 is

calculated by numerically simulating a propagating buckle. The main characteristics of the calculated

congurations are similar to those seen in the experiments shown inFig. 3. The initial conguration of

the structure is identied by the numbered I. The conguration II represents a pipe of local collapse at

the region of imperfections. When the opposite walls of the pipe come into contact in conguration III,

the collapse is arrested locally and the buckle starts to propagate along the downstream pipe. Theconguration IV illustrates the prole of buckle propagation of such a pipe, and the area of the pipe wall

in contact is seen to have increased. Eventually, as the buckle propagation travels to both ends of the

pipe and has to be terminated, the pipe is completely attened in conguration V.

Fig. 8 shows experimental and calculated pressurechange in volume responses for steel pipe

specimen No. 2 (V0is the initial internal volume of the tube, and dVis the absolute value of the change

of volume evaluated for each deformed conguration). As can be seen in the gure, local collapse of the

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

5

10

15

20

25

30

V/V0

P/MPa

Experiments

FEM(Multi-linear Model)

FEM(Modified R-O Model)

Fig. 8. Pressurechange in volume responses for steel pipe specimen No. 2.



12/16

pipe results in a precipitous decrease in pressure, and the subsequent pressure plateau represents

steady-state propagation of the buckle. Finally, two ends of the pipe are engaged by the buckle, the

pressure in the closed vessel once again starts rising. It is noted that these responses have a very similar

varying trend. The predicted and experimental results for collapse pressure and buckle propagation

pressure are listed inTable 2. The errors are larger for the collapse pressure Pco than ones for the

propagation pressurePp, because the collapse pressure is more sensitive to initial imperfections andmaterial properties of the tubes. In fact, it is unimportant in the present study and becomes negligible

for the buckle propagation. As to propagation pressure, the errors are minor and the maximum

difference is less than 8% for the specimen No. 2. However, for the specimen No. 1 listed inTable 1, the

maximum difference reduces to less than 3%. The sources of errors are mainly attributed to variations

of the tube thickness along the length, differences in material properties in the circumferential and

longitudinal directions, differences in material properties in tension and compression, etc. In addition,

the response calculated from the multi-linear approximation material model is compared to the one

obtained from the modied RO material model in Fig. 8. The predicted responses for two material

models are almost always identical, and the observed differences for collapse pressure and propagation

pressure listed inTable 2are both less than 2%. Therefore, the modied RO model is suitably used to

conduct parametric study on buckle propagation pressure of the tube. In conclusion, it has beendemonstrated that the aforementioned numerical model may offer a very satisfactory prediction on the

buckle propagation of a long tube with various geometric characteristics and material properties.

4. Results and discussion

The validity of numerical model proposed in the present study has been demonstrated in the

foregoing section. In what follows, a parametric study of the propagation pressure is carried out

numerically in order to illustrate the inuence of different material parameters on the results and

establish a more reasonable empirical formula.

Previous research ndings have revealed that the propagation pressure is primarily related to yield

stress and post-yield characteristics of the pipe material behavior as well as diameter-to-thickness

ratio. The analyses are, respectively, performed for the pipe models based the parameters of

L 6096 mm,D 304.8 mm (12 inch),E 206 GPa,n 0.3,D/t10, 12, 15, 18, 20, 22, 25, 30, 35, 40, 50

and 60,s0 272, 350, 400 and 500 MPa, and the strain hardening parameter a E/E0 25, 50, 102, 103

and 104.a 25 represents a very strong strain hardening material behavior, anda 104 represents the

elastic perfectly plastic case. If the nominal yield stress s0and the strain hardening parametera are

given, using Eqs.(7) and (8)the RambergOsgood model parameterssyandn may be solved, thus the

stressstrain curve for the modied RambergOsgood model is completely determined, and then used

to model the pipe material behavior.

Fig. 9shows the predicted propagation pressures including a total of 240 data points with varying

diameter-to-thickness ratio and material property through extensive numerical simulations based on

the software ABAQUS. It is clearly observed that for a given geometry, the tubes with smaller strainhardening parametera(stronger post-yield behavior) always possess larger propagation pressure, and

the effect of having a harder material on propagation pressure is essentially a shift of the straight line

upwards. Similarly, it is seen that the tubes with larger material yield stress s0 characterize larger

propagation pressure. In addition, it is worth noting that for the tubes with smaller values ofD/t, the

effect of yield stress is more pronounced compared with the case of larger values ofD/t. The values ofD/

tbetween 10 and 60 are carefully examined, because this is range of interest for pipeline application.

The predicted results for various values ofD/t indicate that the propagation pressure is a power law

Table 2

Comparison between experimental and numerical results for specimen No. 2.

Parameter Experimental value FEM multi-linear model Difference FEM modied RO model Difference

Pco (MPa) 21.4 25.8 17.05% 26.1 18.01%

Pp (MPa) 11.6 12.4 6.45% 12.6 7.94%



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dependence on this parameter. Summarizing the above-mentioned results, the propagation pressure is

heavily affected by the strain hardening modulus E0 and the yield stress s0. Therefore, it can be

concluded that the propagation pressure PPhas a parametric behavior as follows:

PP

s0 abE

0

s0 t

D

b

tD

g

: (11)

where the coefcientsa and b, and the exponentsb and g are to be tted.

It is more useful for design purposes to have a simple, single equation for the propagation pressure.

Based upon the numerical results, the unknown parameters in Eq. (11)are determined by nonlinear

least-squares estimation procedures that used the LevenbergMarquardt algorithm. Accordingly, the

expression of propagation pressure can be written as:

PPs0

23 13

E0

s0

t

D

t

D

2:4(12)

All curves shown inFig. 9are calculated by Eq.(12), it is seen that these data points are very close to

the corresponding curve, the difference of results between numerical simulation and empiricalformula is within 10% other than a few data points, but all differences are all less than 12%.

10 20 30 40 50 600

1

10

100

Pp

/MPa

D/t

1.0104

1.0103

1.0102

50

25

10 20 30 40 50 600

1

10

100

Pp

/MPa

D/t

1.0104

1.0103

1.0102

50

25

MPa2720 = MPa3500=

10 20 30 40 50 600

1

10

100

Pp

/MPa

D/t

1.0104

1.0103

1.0102

50

25

10 20 30 40 50 600

1

10

100

Pp

/MPa

D/t

1.0104

1.0103

1.0102

50

25

MPa4000 = MPa5000=

a b

c d

Fig. 9. Predicted propagation pressures as a function of ( D/t) for various values of strain hardening factor and yield stress.



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Fig. 10 shows the comparison between the experimental results collected from previous limited

information and the predicted values via Eq.(12), the abscissa of data points is the calculated value of

Eq.(12),and the ordinate represents the corresponding experimental result. Most of the data points

locate at the vicinity of the diagonal line, and their differences are very small and within 10% other than

several exceptional data points. The good correlation between predicted and experimental results

validates the empirical expression and the proposed numerical simulation technique.Illustrated inFig. 11is the propagation pressure of the tubes with varying values ofD/tpredicted

from different empirical formula. It is clearly shown that the predicted results through Eq. (4) proposed

by Steel and Spence [10]generate a relative larger deviation. For larger values ofD/t, this empirical

formula overestimates the propagation pressure, whereas for lowerD/tvalues, it contrarily underes-

timates the propagation pressure. For the tubes with the values ofD/t larger than 35, the predicted

results from Eq.(5)[11], Eq.(6)[12]and Eq.(12)are comparatively consistent. However, the empirical

0 5 10 15 200

5

10

15

20

4.2

00p

'

1323

+= Dt

D

tEP

PP

(ExperimentalResults)

Kyriakides

Gong

Estefen

Fig. 10. Comparison between experimental results and predicted values from empirical formula.

10 20 30 40 50 60

0

1

10

100

Pp

/MPa

D/t

Gong

KyriakidesSteel

API

DNV

Fig. 11. Comparison of the results predicted from different empirical formula. (s0 310 MPa, E0 2.06 GPa).



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formula, respectively, proposed by Kyriakides and Babcock[8], and Steel and Spence[10] apparently

overestimate the propagation pressure for this range ofD/tvalues. As to the values ofD/tbetween 15

and 35, the empirical formula, respectively, proposed by Kyriakides and Babcock[8], and DNV[11], as

well as Eq.(12)provide very close predictions of propagation pressure. But the results, respectively,

obtained from Eqs.(4) and (6)actually underestimate the propagation pressure. In addition, when the

values ofD/tare between 10 and 15, the predicted results for propagation pressure by Eq. (12) are largerthan ones of other empirical formula. In conclusion, through aforementioned comparative analysis of

predicted results for propagation pressure from different empirical formula, Eq.(12)proposed in the

present study is able to provide very credible results.

5. Conclusions

The experiments have been successfully conducted to identify the mechanism of a propagating

buckle in a long thick-walled pipe under quasi-static steady-state conditions in a sealed hyperbaric

chamber. In combination with experimental observations and results, a three-dimensional nite

element model for the experimental steel pipe within the frame of ABAQUS is developed to model the

buckle propagation phenomenon. The good correlation between numerical and experimental resultsdemonstrates that the numerical model can accurately predict the total buckle propagation process of

the tube under external pressure. To study the parametric dependence of the propagation pressure, the

modied RambergOsgood model is used to characterize the stressstrain behavior of the pipe

material, and then a series of parametric study on the propagation pressure of offshore pipelines with

the values ofD/tranging from 10 to 60 is conducted adopting the numerical technique developed. The

following conclusions can be drawn.

(1) If the material and geometric parameters of the tube are carefully established, the presented

numerical technique can offer an accurate prediction of the buckle propagation pressure.

(2) The material property has an important effect on the buckle propagation pressure. The paramount

inuencing factors are the strain hardening modulus E0 and the yield stress s0. The tubes withlarger strain hardening modulus and yield stress always possess larger propagation pressure. For

lower D/tvalues, the effects of strain hardening modulus and yield stress are more pronounced

compared with the case of larger values ofD/t. Therefore, higher strength and toughness steel

should be preferred to enhance the post-buckling behaviors in practical engineering, especially for

deepwater pipes.

(3) Based on experimental and extensive numerical results, an empirical formula for buckle propa-

gation pressurePpof offshore pipeline with various values ofD/tas well as different strain hard-

ening modulus and yield stress is proposed in the present study. The close agreement between

predicted and experimental results shows that the empirical formula can provide very reasonable

estimates of the propagation pressure.

Acknowledgments

The authors express their gratitude to the National Natural Science Foundation of China (Grant No.

51009122), the Fundamental Research Funds for the Central Universities (Grant No. 2010QNA4030),

and the National Science and Technology Major Project of China (Grant No. 2011ZX05027-002-005-

011) for the nancial support to this study.

References

[1] Hillenbrand HG, Graef MK Grob-Weege J, Knauf G, Marewski U. Development of linepipe for deepwater applications. In:Proceedings of the 12th international offshore and polar engineering conference, Kitakyushu, Japan; 2002, pp. 287294.

[2] Mesloh RE, Sorenson JE, Atterbury TJ. Buckling and offshore pipelines. Gas Mag 1973;7:403.[3] Palmer AC, Martin JH. Buckle propagation in submarine pipelines. Nature 1975;254:468.[4] Johns TG, Mesloh RE, Sorenson JE. Propagating buckle arrestors for offshore pipelines. In: Offshore technology conference,

Houston, Texas, USA, OTC2680; 1976, pp. 721730.



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