on the liner wrinkling and collapse of bi-material pipe

The Dissertation Committee for Lin Yuan

Certifies that this is the approved version of the following dissertation:

On the Liner Wrinkling and Collapse of Bi-material Pipe under

Bending and Axial Compression

Committee:

Stelios Kyriakides, Supervisor

Michael Engelhardt

Kenneth M. Liechti

Krishnaswa Ravi-Chandar

Rui Huang



by

Lin Yuan, B.E.; M.E.

Dissertation

Presented to the Faculty of the Graduate School of

The University of Texas at Austin

in Partial Fulfillment

of the Requirements

for the Degree of

Doctor of Philosophy

The University of Texas at Austin

May, 2015

Dedication

To my parents.

iv

Acknowledgements

First of all, I would like to express my sincere gratitude to my advisor, Professor

Stelios Kyriakides, for his continuous guidance and support throughout my studies. His

enthusiasm, work ethic and commitment to excellence in the pursuit scientific research

are matchless. This unrivaled spirit and devotion towards research has significantly

influenced my growth as a researcher and an engineer. And I believe that now equipped

with such spirit, I will always be guided and inspired throughout my career.

I would like to also express my appreciation to the members of my dissertation

committee: Professors Kenneth M. Liechti, K. Ravi-Chandar, Rui Huang and Michael

Engelhardt for reviewing my dissertation and for their constructive comments. This

project was conducted with financial support from a consortium of industrial sponsors,

under the project Structural Integrity of Offshore Pipelines, which is acknowledged with

thanks. I also would like to thank to Butting management and engineers for their

cooperation throughout this study. Thanks also go to Benjamin Harrison for leading the

experimental effort in the axial compression of lined cylinders outlined in Chapter 6.

In the course of my studies in Engineering Mechanics, I was fortunate to receive a

lot of help from the senior fellow members of the group, they always inspired and

motivated me to go through the difficult times. This list must include Dr. Lianghai Lee,

Dr. Rong Jiao, Dr. Julian Hallai, Dr. Stavros Gaitanaros and Prof. Wen-Yea Jang. It is

also a blessing for me to know all my talented and friendly fellow graduate students: Nate

Bechle, Ben Harrison, Dongjie Jiang, Yafei Liu, Kelin Chen, Wei Gao, Martin Scales,

and Chenglin Yang.

Finally, I want to express my gratitude to my parents, my father Wenyi Yuan and

my mother Cuiqing Liu. I would not be able to finish this without your continuous love

v

and support. Your unconditional love is like the lighthouse, guiding me, encouraging me

and supporting me, to follow my heart, pursue my dream and career.

vi



Lin Yuan, Ph.D.

The University of Texas at Austin, 2015

Supervisor: Stelios Kyriakides

Pipelines and flowlines that carry corrosive hydrocarbons are often protected by

lining them internally with a thin layer of a corrosion resistant material. In a commonly

used fabrication method, the liner is brought in to contact with a carbon steel carrier pipe

by mechanical expansion. In applications involving severe plastic loading, such as the

reeling pipeline installation method, the liner can detach from the outer pipe and develop

large amplitude buckles that compromise the flow.

This work examines the mechanics of wrinkling and collapse of such a liner under

bending and axial compression. The modeling starts with the simulation of the expansion

process through which the two tubes develop interference contact pressure. Bending

induced ovalization causes separation of the liner from the outer pipe, which in turn leads

to wrinkling of the compressed side and at higher curvature collapse in shell-type mode.

The sensitivity of the collapse curvature to the various parameters is studied, and the

onset of collapse is shown to be very sensitive to small geometric imperfections in the

liner. The models developed are also used to demonstrate that modest amounts of internal

pressure can delay liner collapse to curvatures that make it reelable.

vii

This framework, suitably extended, is also used to examine the effect of girth

welds on liner collapse. It is found that a girth weld locally prevents this detachment

creating a local periodic disturbance. With increasing bending, the disturbance grows and

eventually yields to a shell-type collapse mode similar to the one that causes collapse

away from the weld.

The related problem of wrinkling and collapse of lined pipe under axial

compression is also studied using a second family of custom models. Following the

manufacturing expansion, such a model is compressed with the liner going through

axisymmetric wrinkling, followed by localization and collapse via a non-axisymmetric

buckling mode. Sensitivity studies show that the collapse strain exhibits a similarly

strong sensitivity to small geometric imperfections in the liner. As in bending, modest

amounts of internal pressure is demonstrated to delay liner collapse.

viii

Table of Contents

Nomenclature ......................................................................................................... xi

Chapter 1: INTRODUCTION ................................................................................ 1

1.1 Manufacture of Lined Pipe ..................................................................... 2

1.2 Liner Wrinkling and Collapse of Lined Pipe under Bending ................. 3

1.3 Liner Wrinkling and Collapse of Lined Pipe under Axial Compression .. ............................................................................................................... 4

1.4 Outline ..................................................................................................... 5

Chapter 2: MANUFACTURE OF LINED PIPE ................................................... 8

2.1 Manufacturing Process ............................................................................ 8

2.2 Simulations of Expansion Process .......................................................... 9

2.2.1 Analytical Model ........................................................................ 10

Stage I: Before liner reaches the carrier tube ............................. 11

Stage II: Expansion of both tubes .............................................. 11

Stage III: Unloading ................................................................... 13

2.2.2 Analytical Model Results ........................................................... 13

2.2.3 Finite Element Model ................................................................. 14

2.2.4 FE Models Results ..................................................................... 15

2.2.5 Comparisons ............................................................................... 15

2.3 Parametric Study ................................................................................... 16

2.3.1 Difference in Yield Stresses ....................................................... 16

2.3.2 Initial Gap between Carrier and Liner Tubes ............................. 18

Chapter 3: LINER WRINKLING AND COLLAPSE OF LINED PIPE UNDER BENDING .................................................................................................... 29

3.1 Finite Element Model ............................................................................ 30

3.2 Introduction of Initial State ................................................................... 31

3.3 Wrinkling of Perfect Structure .............................................................. 32

3.4 Wrinkling and Collapse of Imperfect Liner .......................................... 34

ix

3.5 Imperfection Sensitivity of Liner Collapse ........................................... 37


3.6.1 Pipe Diameter ............................................................................. 39


3.6.3 Liner Wall Thickness ................................................................. 42

3.6.4 Bending Under Internal Pressure ............................................... 42

Chapter 4: PLASTIC BIFURCATION BUCKLING OF LINED PIPE UNDER BENDING .................................................................................................... 65

4.1 Bifurcation Analysis ............................................................................. 66

4.2 Bifurcation Results ................................................................................ 69

4.2.1 Wrinkling Bifurcation Under Bending ....................................... 69

4.2.2 Parametric Study ........................................................................ 71

4.3 Imperfection Sensitivity ........................................................................ 73

Chapter 5: LINER WRINKLING AND COLLAPSE OF GIRTH-WELDED LINED PIPE UNDER BENDING ............................................................................ 93

5.1 Finite Element Model ............................................................................ 93

5.2 Wrinkling and Collapse of A Girth-Welded Pipe ................................. 95

5.3 Equivalent Imperfection of Unconstrained Lined Pipe ......................... 97



5.4.2 Pipe Diameter ........................................................................... 100

5.4.3 Bending Under Internal Pressure ............................................. 100

5.4.4 Liner Wall Thickness ............................................................... 101

5.4.5 Overlay Seal Weld ................................................................... 102

Chapter 6: LINER WRINKLING AND COLLAPSE OF LINED PIPE UNDER AXIAL COMPRESSION .......................................................................... 118

6.1 Demonstration Compression Experiments .......................................... 118

6.2 Finite Element Model .......................................................................... 120

6.3 Results ................................................................................................. 122

6.3.1 Wrinkling and Collapse of a Representative Lined Pipe ......... 122

x

6.3.2 Imperfection Sensitivity of Liner Collapse .............................. 125

6.3.3 Effect of Friction on Liner Collapse ........................................ 127

6.4 Parametric Study ................................................................................. 127

6.4.1 Initial Gap between Carrier and Liner Tubes ........................... 128

6.4.2 Pipe Diameter ........................................................................... 128

6.4.3 Liner Wall Thickness ............................................................... 130

6.4.4 Axial Compression Under Internal Pressure ............................ 130

Chapter 7: CONCLUSIONS .............................................................................. 146

7.1 Manufacture of Lined Pipe ................................................................. 146

7.2 Liner Wrinkling and Collapse of Lined Pipe under Bending ............. 147

7.3 Plastic Bifurcation Buckling of Lined Pipe under Bending ................ 148

7.4 Liner Wrinkling and Collapse of Girth-Welded Lined Pipe under Bending ........................................................................................................... 149

7.5 Liner Wrinkling and Collapse of Lined Pipe under Axial Compression .. ........................................................................................................... 150

APPENDIX A: ANALYTICAL MODELS OF LINED PIPE MANUFACTURING PROCESS .................................................................................................. 153

APPENDIX B: BIFURCATION BUCKLING UNDER AXIAL COMPRESSION .................................................................................................................... 158

APPENDIX C: NUMERICAL SCHEME OF BIFURCATION CHECK OF LINED PIPE UNDER BENDING .......................................................................... 161

APPENDIX D: DEMONSTRATION COMPRESSION EXPERIMENTS ON LINED CYLINDERS ............................................................................................. 165

References ........................................................................................................... 171

Vita .................................................................................................................... 175

xi

Nomenclature

D pipe outer diameter

oD mean diameter (= tD )

LD liner outer diameter

2L length of tube

M moment

oM fully plastic moment(= tDoo2 )

m number of circumferential waves

oP = oo Dt /2

R = 2/oD

LR = 2/)( LL tD

t pipe wall thickness

Lt liner wall thickness

w radial displacement

w liner imperfection

detachment of liner from outer pipe

curvature

1 = 2/ oDt

2 imperfection wavelength

stress

o nominal yield stress

o stress at a strain of 0.005

o amplitude of axisymmetric imperfection

m amplitude of non-axisymmetric imperfection

1

Chapter 1: INTRODUCTION

In many offshore applications, carbon steel pipe is lined internally with a thin

layer of a corrosion resistant material in order to protect it from corrosive ingredients in

hydrocarbons it carries during its operation. The most widely used product is assembled

by inserting a slightly undersized tubular liner inside the carbon steel pipe and then

mechanically expanding both so that the two tubes end up in interference contact with

each other (exact steps followed differ to some degree between manufacturers––e.g.,

Butting Brochure; Rommerskirchen et al., 2003; de Koning et al., 2003; Montague,

2004). In offshore operations, the carbon steel pipe carries most of the usual loads of

internal and external pressure, tension and bending while the thin liner (2-4 mm) protects

the line from corrosive ingredients in the hydrocarbons. However, in cases that involve

significant plastic loading of the composite structure, such as in the reeling installation

method or in lines susceptible to either lateral buckling or significant compression on the

sea floor, the liner can detach from the outer pipe and develop large wrinkles and buckles

that compromise the flow. An example of such a buckled failure following plastic

bending of 12-inch lined pipe is shown in Fig. 1.1 (from Hilberink, 2010). A viable

alternative is to use pipe with metallurgically “bonded” liner, commonly known as clad

pipe, however this product comes at a significantly higher cost.

The main objective of this dissertation is to use careful analysis to add clarity to

the sequence of events that lead to liner failure under bending and axial compression.

Furthermore, the study aims to understand the major factors that influence wrinkling and

collapse failure, and evaluate potential methods for delaying collapse. The problem is

directly influenced by the manufacturing processes of first the carbon steel carrier pipe,

second the forming of the noncorrosive alloy liner, and the process through which the

2

two are brought together. Thus below we first briefly introduce the manufacturing

process followed by one of the major producers of lined pipe (Butting). Subsequently, we

review the state of the art regarding the behavior of lined pipe under bending and under

axial compression.

1.1 MANUFACTURE OF LINED PIPE

Lined pipe consists of a carbon steel pipe with a thin inner layer of corrosion

resistant alloy liner. The two tubes are typically expanded together using one of several

methods currently in the market; they come into contact and remain so after unloading.

The objective is that the finished bi-layer composite ends with some interference contact

pressure between the two components (often called "mechanical bonding"). Different

manufacturers bring the two tubes together using some variation of mechanical expansion

(e.g., Butting Brochure; de Koning et al., 2003; Montague, 2004). In this study we will

concentrate on the expansion process followed by Butting.

In this process, the two tubes are brought into contact by hydraulic expansion.

The major steps are shown schematically in Fig. 1.2 (see Butting Brochure). For ease of

insertion, the diameter of the corrosion resistant tube is somewhat smaller than the inner

diameter of the outer tube leaving a small annular gap ( og ). The two tubes are enclosed

inside a die as shown in image , which leaves a gap between the outer surface of the

carrier tube and the die. The ends of the composite pipe are sealed and pressurization

commences. The liner expands and contacts the steel outer pipe (image ). The pressure

is further increased expanding both tubes until contact with the stiff die takes place

(image ). In the final step the pressure is gradually released (image ). The plastic

deformation induced by this process introduces changes to the mechanical properties of

both components and leaves behind residual stresses. Collectively these factors influence

3

the mechanical behavior of the composite pipe, and must be accounted for in any

subsequent mechanical loading of the composite pipe.

1.2 LINER WRINKLING AND COLLAPSE OF LINED PIPE UNDER BENDING

The first problem considered is pure bending of the lined pipe following the

manufacturing steps outlined above. Bending is of particular interest due to the desire to

install lined pipe using the reeling installation process. Here we review the state of the art

as it existed at the outset of this study. Several full-scale bending experimental programs

have been undertaken during the last several years. Those reported in the open literature

include a series of bending results by Gresnigt and co-workers (e.g., Focke, 2007;

Hilberink et al., 2010, 2011; Hilberink, 2011); bending of heated lined pipe by Cladtek

(Montague et al., 2010; Wilmot and Montague, 2011); repeated bending over circular

shoes (Tkaczyk et al., 2011); full-scale reeling simulations by Subsea7 and Butting (e.g.,

Toguyeni and Banse, 2012; Sriskandarajah et al., 2013), and others. Less developed are

complementary analytical/numerical efforts reported by the same teams apparently due to

the challenges of the problem. The most thorough study of the problem is due to Vasilikis

and Karamanos (2010, 2012) who used Finite Element models to analyze lined pipe

under pure bending.

Collectively the efforts listed above have contributed to the following state of

current understanding of the problem. Bending to curvature levels that correspond to

those seen by reeled pipe results in significant plastic deformation of both the carrier pipe

and liner. Concurrently, the composite structure develops Brazier-type (1927) ovalization

of its cross section. This in turn can result in loss of contact and partial separation of the

liner from the steel pipe. At some level of deformation, the separated section of the liner

buckles into a wrinkling mode, commonly seen in pure bending of single pipe (e.g., see

4

Ju and Kyriakides, 1991, 1992; Corona et al., 2006; Kyriakides and Corona, 2007;

Limam et al., 2010). We will demonstrate that, as is common to plastic buckling of

shells, wrinkling is followed by a second instability that leads to collapse of the liner in a

diamond-type buckling mode. As discussed above, the manufacturing process of lined

pipe introduces mechanical property changes and interference contact stresses.

Invariably, these changes influence the liner instabilities, but to date have been mostly

neglected as they tend to complicate the modeling. Inclusion of this prehistory will

constitute a significant first step in the modeling effort of this study.

1.3 LINER WRINKLING AND COLLAPSE OF LINED PIPE UNDER AXIAL COMPRESSION

The second problem considered is axial compression of the expanded lined pipe.

Compression severe enough to lead to plastic deformation and liner buckling occurs, for

example, in buried pipelines due to thermal loads from the passage of hot hydrocarbons

(e.g., see Jiao and Kyriakides, 2009, 2011). Depending on the extent of soil resistance to

sideways snaking, lines on the sea floor can also experience significant compression due

to thermal loads. Other causes of compression include fault movement, ground

subsidence, permafrost melting, etc. (e.g., see Ch. 11 and 12 in Kyriakides and Corona,

2007).

The problem of liner buckling and collapse due to axial compression has received

much less attention compared to bending. The first axisymmetric bifurcation of a liner

confined in an outer cylinder of the same properties was established in Peek and

Hilberink (2013) (see also Shrivastava, 2010). In addition, some compression

experiments on lined pipe have been reported in Focke et al. (2011). However, the results

were not sufficient to either demonstrate the problem challenges or to be used in direct

5

comparisons with analysis. For this reason, demonstration experiments on model lined

shell systems are conducted in support of this study.

1.4 OUTLINE

The present study uses careful modeling to study the sequence of events that lead

to liner buckling and collapse under bending and axial compression. This study starts

with analytical and numerical simulation of the expansion process through which lined

pipe is manufactured, described in Chapter 2. Chapter 3 presents a detailed model that is

used to simulate pure bending of lined pipes, that is capable of reproducing the initial

wrinkling and eventual collapse of the liner. The models include the prehistory and

residue stress fields induced by the manufacturing process. The models developed are

subsequently used to study the sensitivity of collapse to various problem parameters.

Chapter 4 outlines a numerical procedure for establishing the onset of the first bifurcation

bucking of such a lined pipe under bending. The critical strain at bifurcation and the

corresponding wavelength are compared to the corresponding values from the axially

loaded lined cylinder as well as with those of a liner shell alone under axial compression

and bending. In Chapter 5, the numerical framework of bending of Chapter 3 is suitably

extended to examine the effect of a girth weld on the bending capacity of lined pipe. The

extended model is used to conduct a parametric study of the factors that influence the

collapse of a girth-welded lined pipe. The problem of liner wrinkling and collapse under

axial compression is studied in Chapter 6 using an appropriate numerical model. Once

again the model includes the prehistory of the manufacturing process. The model is again

to conduct parametric study of liner collapse. Chapter 7 lists the main conclusions of the

study.

6

Fig. 1.1 Photograph of buckled lined pipe after bending.

(Hilberink, 2010)

7

Fig. 1.2 Schematic representation of the expansion process through which lined pipe

is manufactured (Butting Brochure).

8

Chapter 2: MANUFACTURE OF LINED PIPE

The manufacture of lined pipe is a cold mechanical process that plastically

deforms both the liner and the carrier pipe. This prehistory changes the mechanical

properties of both components and leaves behind residual stresses. Collectively these

factors influence the mechanical behavior of the composite pipe. In order to capture these

initial states, this chapter describes the expansion processes through which lined pipe is

manufactured (see Butting Brochure) and simulates it analytically and numerically. The

models developed are used to examine the effect of several parameters in this problem on

the induced material changes and residual stresses.

2.1 MANUFACTURING PROCESS

For 4-16-inch products, the carrier pipe is seamless produced by a piercing

process (e.g., see Kyriakides and Corona, 2007; Harrison et al., 2015). Most seamless

tubulars start as round billets produced by continuous casting. The billets are pierced

through the Mannesmann process at elevated temperature. In the plug mill, the round

billets get pierced and elongated simultaneously. Even though the process is operated

with precision by computers, some wall eccentricity and some internal surface

undulations are unavoidable. Thus the finished pipe typically has some eccentricity and

surface undulation.

The 2–4 mm thick corrosion resistant liner (e.g., SS-321, SS-316L, alloy-625,

alloy-825) is most often formed into a continuous longitudinally welded tube from coil.

Special care is given to the metallurgical quality and integrity of the weld while also

shaping its outer surface to conform to the circular shape of the steel pipe. The finished

tube, cut to approximately 12 m length, is placed inside the carrier pipe whose inner

9

surface is previously sandblasted and cleaned. The two tubes are then mechanically

expanded by internal pressurization. The amount of expansion is controlled so that the

tubes remain in contact after unloading.

Figure 2.1 shows schematically the hydraulic expansion as performed by Butting

(see Butting Brochure). For ease of insertion, the diameter of the corrosion resistant tube

is somewhat smaller than the inner diameter of the outer tube leaving a small annular gap

( og ). The two tubes are enclosed inside a die as shown in image of Fig. 2.1, which

leaves a gap between the outer surface of the carrier tube and the die. The ends of the

composite are sealed as shown schematically in Fig. 2.2 and pressurization commences.

The liner expands and contacts the steel outer pipe (image ). The pressure is further

increased, expansion of both tubes takes place until contact with the stiff die (image ).

In the final step the pressure is gradually released (image ).

The objective of the expansion process is to bring the two tubes together and

leave them in interference contact. This is achieved by using a liner material with a lower

yield stress than that of the carrier pipe. The effect of this difference is illustrated in

Appendix A by a simple exercise in which two thin-walled rings with elastic-perfectly

plastic materials are expanded a certain amount and unloaded. A contact stress develops

that is directly proportional to the difference in the two yield stresses. More complete

models are presented in the following section.

2.2 SIMULATIONS OF EXPANSION PROCESS

The mechanical property changes introduced by the process to the two

components are now established by simulating the process using two models. The first is

a semi-analytical model based on J2 incremental plasticity and second is an axisymmetric

finite element model that treats the two constituents as elastic-plastic.

10

2.2.1 Analytical Model

Here the two tubes are assumed to be thin-walled and they are taken through the

expansion process realistically as depicted in Figs. 2.1 and 2.2. The material of each tube

is modeled as an elastic-plastic solid that hardens isotropically. In both cases, the

structure is one-dimensional. However, both tubes experience biaxial states of stress

because, in addition to the pressure P, they are loaded axially by a compressive force

PA , 10 . The flow rule

ij

mnmn

pij

fd

f

Hd

1

(2.1)

is adopted where f is the current yield surface. Specializing (2.1) to plane stress and

adding the elastic strain increment, the incremental stress-strain relationships become

d

d

QQ

QQEd

d x

xxx

xxxx2

2

)2(1)2)(2(

)2)(2()2(11(2.2)

where

,14

12

ete E

EQ

and ),( x represent the axial and circumferential coordinates. tE is the tangent modulus

of the material stress-strain response. The two tubes are assumed to be axially connected

so xCxL . Axial equilibrium implies

2)1( CCxCLxL RPAA

or in incremental form,

dPBdAd xCxL (2.3)

where

11

,)1(

22C

LL

R

tRA

.)1(

2

C

C

R

tB

Stage I: Before liner reaches the carrier tube

Before the liner contacts the carrier tube, the hoop stress of the liner is

L

LL t

PR and for the outer tube 0C . Accordingly, the axial strains in Eq. (2.2)

become

)(1

)(1

1

11

xCxC

LxLxL

dcE

d

dbdaE

d

(2.4)

where 2

1 )2(1 LxLQa , )2)(2(1 xLLLxLQb , 21 )2(1 xCQc

By requiring xCxL dd , and substituting Eq. (2.4)

0111 xCL

LxL dcdP

t

Rbda (2.5)

Solving Eq. (2.3) and (2.5), the increments, xLd and xCd can be expressed in

terms of dP. Subsequently, the stresses of both tubes are updated and Q is evaluated. At

the end of each increment, xLd and xLd are evaluated, and the total strain is updated.

Stage II: Expansion of both tubes

After the liner reaches the carrier pipe, the incremental form of the equilibrium

equation in the hoop direction becomes

.dPDdCd CL (2.6)

where

.,C

C

C

LR

tD

R

tC

12

Accordingly, the axial strains in Eq. (2.2) become

)(1

)(1

22

22

CxCxC

LxLxL

dddcE

d

dbdaE

d

(2.7)

where 2

2 )2(1 LxLQa , )2)(2(2 xLLLxLQb ,

22 )2(1 CxCQc , ).2)(2(2 xCCCxCQd

By requiring xCxL dd once more

.02222 CxCLxL dddcdbda (2.8)

Once the liner contacts the carrier tube, the changes in the hoop strains of the two

tubes are equal, thus

CL dd or 02222 CxCLxL dhdgdfde , (2.9)

where

)2)(2(2 xLLLxLQe , 22 )2(1 xLLQf ,

)2)(2(2 xCCCxCQg , .)2(1 22 xCCQh

Equations (2.3), (2.6), (2.8) and (2.9) constitute the following system of linear

algebraic equations

0 A 0 BC 0 D 0b2 a2 d2 c2

f2 e2 h2 g2

dL

d xL

dC

d xC

dPdP00

. (2.10)

The stresses increments },,,{ CxCLxL dddd are solved from (2.10) at each

increment of dP. Subsequently, the stresses and tangent moduli of the tubes are updated

and evaluated. The incremental strain components of both tubes are then calculated, and

the total strains are updated.

13

Stage III: Unloading

After expanding the two tubes together a certain amount, the pressure is released

incrementally. Equations (2.10) still hold, except that 0Q . Accordingly, for every

increment of dP , the stress and strain components of both tubes are evaluated and

updated.

2.2.2 Analytical Model Results

The analytical model is now used to simulate the expansion process. Figure 2.3a

shows the calculated pressure-radial displacement (P-w) response for the pipe parameters

listed in Table 2.1. Here the pressure is normalized by the yield pressure of the steel pipe,

oP , based on its yield stress and final dimensions; the radial displacement of the liner, w

, is normalized by the initial gap og . The numbered points correspond to the images in

Fig. 2.1. Thus, between and the liner expands initially elastically and subsequently

plastically, and the expansion pressure remains small as the liner is relatively thin. At

the liner comes into contact with the carrier pipe, and consequently the response stiffens

significantly. The pressure increases sharply until the steel pipe yields. The two pipes are

then plastically expanded further until the outer one comes into contact with the stiff die

at . Subsequently the pressure is gradually removed ().

Figure 2.3b shows the hoop stresses, , developed in the two tubes during the

expansion (both normalized by the yield stress of the steel carrier tube o ). Between

and the liner is expanding freely. At the liner comes into contact with the carrier

pipe, and this is responsible for the small dip in the liner stress. Between and the

two are expanded together until the carrier pipe contacts the outer die. Finally, the

structures unload elastically to with both of them ending up with residual stresses due

to the interference contact. The stress is tensile in the steel pipe and compressive in the

liner. This is primarily due to the difference in the stress level that each component

14

unloads from, which is quite obvious in Fig. 2.3b. Other factors that affect the extent of

the interference stress will be discussed in the parametric study section. These residual

stresses result in an interference contact pressure of 265.7 psi between the two tubes. It

will be demonstrated in later chapters that the contact stress has a stabilizing effect on

liner collapse and thus it is an important parameter in the manufacturing process.

For comparison, the pressure-radial displacement response is compared with the

corresponding one from the elastic-perfectly plastic model outlined in Appendix A in Fig.

2.4 (the model in the Appendix is tailored slightly to take the initial gap between the liner

and carrier tube into account). Because of the absence of hardening for both materials, the

slope of the response predicted by the simpler model is smaller than that produced by the

present one. Despite this difference, the resultant contact pressure is 256 psi, which is

only 3.65% lower than the value of the more complete model.

Table 2.1 Main geometric and material parameters of lined pipe analyzed

D in† (mm)

t in† (mm)

EMsi* (GPa)

o ksi*

(MPa)

Steel Carrier

X65

12.75 (323.9)

0.705 (17.9)

30.0 (207)

65.0 (448)

Liner alloy 825

11.34 (288.0)

0.118 (3.0)

28.7 (198)

40.0 (276)

† Finish dimensions, *Nominal values

2.2.3 Finite Element Model

The inflation process is also simulated using an axisymmetric FE model

developed in ABAQUS 6.10 and shown in Fig. 2.5. The model involves a section of the

carrier pipe and the liner, as well as the outer die. The carrier pipe is meshed with 4-node

linear continuum elements (CAX4), and the liner is modeled by linear shell elements

15

(SAX1). The mesh adopted has four elements through the thickness of the carrier tube, 20

elements along a length of CR19.0 , which is sufficiently long for a uniform solution. For

numerical efficiency, the model is symmetric about the plane x 0 . The top edges of the

liner and outer pipe remain in the same plane perpendicular to x-axis.

Contact is modeled using the finite sliding option in ABAQUS. For a contact pair

between the liner and the carrier tube, the liner is assigned as the slave surface and the

inner surface of the carrier pipe as the master surface. As to the outer contact pair, the

outer surface of the carrier pipe is chosen as the slave surface and the inner surface of

stiff die as the master surface. The effect of friction during the expansion process is

assumed to be negligibly small, thus contact is assumed to be frictionless in the studies

(confirmed by parametric study). The materials of the two tubes are modeled as finitely

deforming solids that harden isotropically.

2.2.4 Finite Element Results

The pressure- and hoop stress-radial displacement responses calculated for the

system listed in Table 2.1 are shown in Fig. 2.6. As was the case for the analytical model,

the liner first expands on its own (-), and then both tubes are expanded together up to

point . At this point the pressure is gradually removed (). As a result, the steel pipe

ends up with tensile stress and the liner with compressive stress, with the two tubes being

in interference contact stress.

2.2.5 Comparisons

The pressure-radial displacement response calculated with this FE model is

compared to the corresponding one from the analytical model in Fig. 2.7a. Despite the

one-dimensional structural simplification made in the analytical model, good agreement

is observed before the liner contacts the carrier pipe. The expansion pressure is under

16

predicted by a small amount by the analytical model when the two tubes are deforming

together.

The hoop stresses of the carrier and liner tubes are plotted in Fig. 2.7b against the

radial displacement. A small difference between the two is again observed after the liner

contacts the carrier pipe. This is caused by the thin-walled assumption adopted for the

carrier pipe in the analytical model. Nevertheless, the resultant contact pressures are

found to be very close. The contact pressures is 272.9 psi from the FE model, which is

only 2.7% higher than the case of analytical model.

2.3 PARAMETRIC STUDY

In this section, we present results from a parametric study of the expansion

process using both the analytical and FE models. Two major factors are examined: the

difference in yield stresses of the two materials, and the initial gap between the carrier

and liner tubes.

2.3.1 Difference in Yield Stresses

The materials of the outer and the inner tubes of commercial lined pipe are

selected individually. Provided the corrosion resistance and strength properties are met

for the specific service conditions, several weldable material grades can be used for the

liner such as: SS-321, SS-316L, alloy-625, alloy-825 (see Butting Brochure). The same is

the case for the carbon steel pipe with a wide selection of material grades available, such

as X-52, X-60, X-70, X-80. Therefore, it is desirable to know the contact stress that will

be resulted from different combinations of the two materials.

Complete expansion simulations are conducted using the analytical model. The

materials are assumed to exhibit power law hardening as defined in the Ramberg-Osgood

stress-strain representation given by:

17

1

7

31

n

yE . (2.11)

In the simulations that follow, the carrier pipe is assumed to be of grade X-75 with the

material parameters listed in Table 2.2. The liner is assumed to be SS-304 with three

yield stresses also listed in Table 2.2; the four stress-strain responses are plotted in Fig.

2.8.

Table 2.2 Four stress-strain responses used in the parametric study.

E Msi (GPa)

o ksi

(MPa)

y ksi

(MPa)

n

X-75 30.0 (207)

75 (517.1)

69.89 (481.9)

13

SS-304 30.0 (207)

45

(310.3)

40.2 (277.2)

16

SS-304 30.0 (207)

55

(379.2)

50.1 (345.4)

16

SS-304 30.0 (207)

65

(448.2)

60.3 (415.8)

16

Three sets of pressure-radial displacement responses are presented in Fig. 2.9a.

The pressure is again normalized by the yield pressure of the carrier pipe. The radial

displacement of the liner, w , is normalized by the initial gap og . The pressure required

to expand the liner is seen to increase some amount as the liner yield stress increases.

Figure 2.9b shows the corresponding hoop stresses, , normalized by the yield stress of

of the carrier pipe. When the liner yield stress changes from 45 to 65 ksi, the stress in the

liner increases. This increase results in a smaller stress difference between the two

constituents on unloading. As a result, the corresponding residual contact stresses are

respectively 528.5, 330.1 and 130.4 psi. This sensitivity indicates that when practically

18

feasible, choosing material pairs with larger difference in yield stress will result in larger

contact stresses.

2.3.2 Initial Gap between Carrier and Liner Tubes

In the manufacturing process outlined in Section 2.1, the initial diameter of the

liner tube is chosen to be somewhat smaller than that of the outer pipe for ease of

insertion. In this section we will examine the effect of the initial annular gap between the

two pipes, og . To this end we simulate the manufacture of the composite system (Table

2.1) using the FE model, but start with somewhat different liner initial diameters so that

the initial gap varies. Figure 2.10 shows the normalized hoop stresses in the steel outer

pipe and in the liner plotted against the radial displacement, w gob , for four values of

og : {0.5, 1, 1.5, 2} obg , where obg is the value used in the calculations in Fig. 2.3. As

the gap increases, the liner has to deform more in order to come into contact with the

outer pipe, thus becoming increasingly more plasticized. The maximum stress in each

liner response corresponds to first contact with the outer pipe and the subsequent lower

stress section to simultaneous expansion of the two tubes. The residual hoop stresses left

in the two tubes on removal of the pressure are seen to decrease as og increases. As a

result, the corresponding contact stresses are 377.7, 272.9, 173.5 and 76.8 psi.

Apparently, this sensitivity indicates that the initial annular gap has a significant effect on

the resultant contact stress.

19

Fig. 2.1 Schematic representation of the expansion process through which lined pipe

is manufactured (Butting Brochure).

20

Fig. 2.2 Schematic representation of the expansion process with the end of the composite structure sealed and loaded by

compression.

21

(a)

(b)

Fig. 2.3 (a) Pressure-radial displacement response of the bi-material structure during

hydraulic expansion and (b) corresponding stresses-displacement responses

calculated by analytical model.

0

0.4

0.8

1.2

0 0.4 0.8 1.2 1.6

P

Po

w / go

Anal. Model

D = 12.750 in

tL= 3 mm

3

0

1

Composite2

Liner

-0.4

0

0.4

0.8

1.2

0 0.4 0.8 1.2 1.6

w / go

2

0

1

3

Steel Pipe

Liner

1

2

3

D = 12.750 in

tL= 3 mm

Anal. Model

22

Fig. 2.4 Comparison of pressure-radial displacement responses for simple model and

analytical model.

0

0.4

0.8

1.2

0 0.4 0.8 1.2 1.6

P

Po

w / go

Anal. Model

Simple Model

D = 12.750 in

tL= 3 mm

23

Fig. 2.5 Axisymmetric FE mesh of composite system for modeling the

manufacturing process.

24

(a)

(b)

Fig. 2.6 (a) Pressure-radial displacement response of bi-material structure during

hydraulic expansion and (b) corresponding stresses-displacement responses

calculated using the FE model.

3

0

1

0

0.4

0.8

1.2

0 0.4 0.8 1.2 1.6

P

Po

w / go

Composite

2

Liner

D = 12.750 in

tL= 3 mm

FE Model

-0.4

0

0.4

0.8

1.2

0 0.4 0.8 1.2 1.6

w / go

2

0

1

3

Steel Pipe

Liner

1

2

3

D = 12.750 in

tL= 3 mm

FE Model

25

(a)

(b)

Fig. 2.7 Comparison of (a) pressure-radial displacement response and (b)

corresponding stresses-displacement responses calculated using the

analytical and FE models.

0

0.4

0.8

1.2

0 0.4 0.8 1.2 1.6

P

Po

w / go

FE Model

D = 12.750 in

tL= 3 mm

Anal. Model

-0.4

0

0.4

0.8

1.2

0 0.4 0.8 1.2 1.6

w / go

FE Model

Anal. Model

D = 12.750 in

tL= 3 mm

26

Fig. 2.8 Four stress-strain responses for the carrier and liner tube.

0

20

40

60

80

100

0 2 4 60

100

200

300

400

500

600

700

(ksi)

(MPa)

X-75SS-304

Steel Pipe

Liner55

65

(ksi)

45

'

27

(a)

(b)

Fig. 2.9 (a) Pressure-radial displacement response and (b) corresponding stresses-

displacement responses for different values of liner yield stress.

0

0.4

0.8

1.2

0 0.4 0.8 1.2 1.6

P

Po

w / go

5565

(ksi)

45

D = 12.750 in

tL= 3 mm

'

-0.4

0

0.4

0.8

1.2

1.6

0 0.4 0.8 1.2 1.6

w / go

55

65

(ksi)

45

D = 12.750 in

tL= 3 mm

Steel Pipe

Liner

'

28

Fig. 2.10 Circumferential stress-displacement responses of bi-material structure

during hydraulic expansion for different values of initial annular gap.

-0.4

0

0.4

0.8

1.2

0 0.5 1 1.5 2 2.5

0.51

1.52

w / gob

go

gob

Steel Pipe Liners

D = 12.750 in

29

Chapter 3: LINER WRINKLING AND COLLAPSE OF LINED PIPE UNDER

BENDING

Offshore pipelines are in many cases designed to sustain severe enough bending

to plasticize the pipeline. For example, in the reeling installation process, the induced

bending strain can be as high as 1.5-3.0%. During operation, lines carrying hot

hydrocarbons are susceptible to lateral bending on the sea floor, which again can cause

plastic bending. Another scenario that can potentially lead to plastic bending and axial

compression is upheaval bucking of a pipeline buried in a trench. As mentioned in

Section 1.2, in the case of lined pipe, such bending levels can lead to wrinkling of the

liner that can result in large amplitude buckles that compromise the integrity of the

structure.

A review of the relevant literature on liner collapse has appeared in Chapter 1.

This chapter presents a numerical framework for establishing the extent to which lined

pipe can be bent before liner collapse. The first step involves introduction of the stress

history and residual stresses to the model induced by the manufacturing process as

developed in Chapter 2. The model is subsequently purely bent, leading to ovalization of

the composite pipe and some separation of the liner from the outer pipe. Loss of contact

by the liner leads to wrinkling. Wrinkles grow initially stably but at some stage a second

instability involving diamond-type shell buckling modes becomes energetically preferred.

This type of mode is responsible for large amplitude buckles in the liner that are

considered to be catastrophic. The model incorporates the geometric, material and contact

nonlinearities necessary for capturing the progressive evolution of these events up to

collapse. The models developed are subsequently used to study the influence of major

factors that govern liner collapse.

30

3.1 FINITE ELEMENT MODEL

The primary model involves a section of the composite pipe of length 2L, outer

diameter D and wall thickness t lined with a thin layer of non-corrosive material of

thickness Lt . For numerical efficiency, symmetry about the mid-span is assumed (plane

zy ) as well as about the plane of bending x z as shown in Fig. 3.1. The composite

structure is bent by prescribing the angle of rotation at Lx . The end plane is

constrained to remain plane, while the cross section is free to ovalize by imposing the

following multi-point constraint (MPC):

iref

irefL zz

xx

tan (3.1)

where ),( ii zx are the coordinates of the ith node in this plane and ),( refref zx are those

of a reference node (e.g., beam node at the center of the circle). The moment is calculated

at the plane of symmetry )0( x from:

N

iiiFzM 2 (3.2)

where iF is the axial force acting on the ith node of the cross section and iz is its distance

from the axis of the tube.

Unless otherwise stated, the half length of the model will be 20L , where is

the half wavelength of an initial axisymmetric geometric imperfection that will be

commonly introduced to the liner. Although actual wrinkle wavelengths under inelastic

bending differ to some degree, the value corresponding to the elastic buckling of a

circular cylindrical shell under uniform compression given below can be viewed as

representative (see Ju and Kyriakides, 1991, 1992; Corona et al., 2006; Kyriakides et al.,

2005; Kyriakides and Corona, 2007).

31

4/12)]1(12[

LLtR

, (3.3)

where LR is the mid-surface radius of the liner and is the liner Poisson’s ratio.

The steel carrier pipe is meshed with linear solid elements (C3D8) and the

contacting liner with linear shell elements (S4). The carrier pipe has four elements

through the thickness and both tubes are assigned 108 elements around the half

circumference. To accommodate the expected development of wrinkling, a finer mesh is

assigned to the compressed side of the cross section. The calculations will involve the

introduction of small initial geometric imperfections to the liner with a bias towards the

mid-span. The bias is introduced in anticipation of the expected localization of buckling

and collapse, and in order to accommodate the conduct of systematic parametric studies.

Consequently, a finer mesh is provided in the axial direction closer to the zy plane of

symmetry and coarser ones away from this zone as follows:

{ 40 x , 56 elements},

{ 144 x , 70 elements},

{ 2014 x , 30 elements},

Contact between the two tubes plays an important role in the problem. The finite

sliding option of ABAQUS is adopted with the carrier pipe as the master surface and the

liner as the slave surface. The effect of friction will be shown to be negligibly small and

thus contact is assumed to be frictionless unless otherwise stated.

3.2 INTRODUCTION OF INITIAL STATE

As demonstrated in Chapter 2, the initial mechanical expansion process that

brings the two pipes into contact introduces changes to the mechanical properties and

leaves behind residual stresses as well as a certain interference pressure. Collectively,

32

these initial conditions influence the mechanical behavior of the composite pipe and

consequently must be incorporated in the model.

Although the most direct approach is to simulate the manufacturing process using

the full FE model, the requirement to have liner geometric imperfections with

controllable shapes and amplitudes dictated an alternate approach. The manufacturing

process is analyzed separately using an axisymmetric model in which the liner is modeled

as a shell and the carrier pipe as a solid (see Fig. 2.5). Both are assigned a similar

through-thickness distribution of elements, or integration points, as those of the full

model, which is shown in Fig. 3.2a. At the end of the process (point in Fig. 2.1), the

state of stress and strain in each of the four solid elements are averaged and the state of

the stress, the plastic strains, and the equivalent plastic strains are transferred to the nodes

and the integration points to all through-thickness elements of the full model. The state of

the stress in the liner is essentially the same through all integration points and is

transferred directly to all elements of the liner in the full model. In the process, the two

pipes deform slightly and contact pressure develops between them.

The veracity of this scheme was evaluated by comparing the stress and

deformation states induced by the expansion process using the axisymmetric model and

those of the full FE model. The two stress and strain distributions were found to be very

similar. In addition, the moment-curvature responses of the liner and carrier pipe

produced by the two initial state schemes for a particular case are compared in Fig. 3.2b.

The two sets of results are seen to be very close.

3.3 WRINKLING OF PERFECT STRUCTURE

It is well known that bending of thin-walled tubes leads to ovalization of the cross

section (Brazier, 1927). In the case of plastic bending the response eventually is

33

interrupted by buckling in the form of periodic wrinkling on the compressed side of the

structure (Ju and Kyriakides, 1992; Kyriakides and Corona, 2007). The wrinkles have

small amplitudes at first appearance, but their amplitudes gradually grow with curvature

contributing to some reduction in the stiffness of the response. At some higher level of

curvature, the structure develops a second instability that is usually catastrophic. For

higher D/t tubes, like those of the liners under consideration here, the second instability is

non-axisymmetric buckling (see Chapter 8 of Kyriakides and Corona (2007) for more

details).


D in† (mm)

t in† (mm)

EMsi* (GPa)

o ksi*

(MPa)

Steel Carrier

X65

12.75 (323.9)

0.705 (17.9)

30.0 (207)

65.0 (448)

Liner alloy 825

11.34 (288.0)

0.118 (3.0)

28.7 (198)

40.0 (276)


The onset of plastic bifurcations is best established by using deformation theory

instantaneous moduli. Indeed, the preferred procedure is to use flow theory for non-trivial

prebuckling calculations and deformation theory for bifurcation checks––using the state

of stress from the flow theory (see Chapter 13 Kyriakides and Corona (2007)).

Unfortunately, following this guideline in the present case is complicated by the

expansion prehistory and the other nonlinearities of the problem and thus will not be

followed. Consequently, we will first demonstrate the onset of wrinkling for the base case

(with its parameters listed in Table 3.1) using flow theory instead, realizing that the actual

bifurcation occurs at a lower curvature (see Chapter 4).

34

Figure 3.3 shows the calculated moment-curvature ( M ) responses of the

composite structure corresponding to the perfect base case where the normalizing

variables are based on the parameters of the outer pipe as follows:

tDM ooo2 , 2

1 / oDt , tDDo . (3.4)

Shown in the plot are the responses for the composite structure and of the individual steel

and liner pipes. Included is the ovalization induced to the liner represented by the change

in its diameter LΔD/D| . Figure 3.4 shows a set of liner deformed configurations with

color contours corresponding to the contact pressure between it and the outer tube. The

images correspond to the numbered points marked on the liner response in Fig. 3.3. In

this case, the expansion process resulted in a contact pressure of about 270 psi (1.86

MPa). Bending plasticizes and ovalizes both tubes and the combined effect leads to a

reduction in the contact pressure as illustrated in . At higher curvatures, the ovalization

of the liner overtakes that of the steel tube () eventually causing loss of contact at the

two extremes of the cross section as shown in . At a curvature of about 0.63 1 , the

long unsupported section of the liner that is under compression buckles into the periodic

wrinkling mode seen at a more developed stage in image (see also Vasilikis and

Karamanos, 2012). The amplitude of the wrinkles grows with curvature eventually

inducing a second diamond-type buckling mode not shown here. Although this sequence

of events is representative of the actual behavior, as pointed out above, the curvature at

the onset of wrinkling predicted with flow theory is artificially high. The subject of

plastic bifurcation under bending will be examined in detail in Chapter 4.

3.4 WRINKLING AND COLLAPSE OF IMPERFECT LINER

Manufactured lined pipe is characterized by small geometric imperfections (e.g.,

see §4.5 of Kyriakides and Corona, 2007 and Harrison et al., 2015). Thus, in the

35

remainder of the chapter we consider bending of lined pipe with initial liner

imperfections.

The first plastic instability that develops in circular cylindrical shells under

bending is wrinkling of the compressed side (Ju and Kyriakides, 1991). Depending on the

D/t of the shell, this is followed by a second bifurcation into a diamond-type buckling

mode that leads to localization and collapse (Ju and Kyriakides, 1992; Corona et al.,

2006; Kyriakides and Corona, 2007). Not surprisingly, initial wrinkling and diamond-

type buckling of the liner have been reported in the Delft experiments (e.g., Hilberink et

al., 2010, 2011). Motivated by this, we introduce to the liner two types of initial

imperfections, an axisymmetric one with half wavelength , as shown in Fig. 3.5a, and a

non-axisymmetric one with axial half wavelength 2 and m circumferential waves

shown in Fig. 3.5b (Koiter, 1963). The two are combined as shown in Eq. (3.5) and are

modulated by an axially decaying function in order to facilitate localization in the

neighborhood of the zy plane of symmetry.

2)/(01.0cos

2coscos

Nx

moL mxx

tw

(3.5)

In the process of transferring the initial state of stress to the full model, the initial

imperfection deforms and its amplitude is reduced. Figure 3.6 shows comparisons of the

initial and final imperfections for o m 0.05 and 4N . Figure 3.6a shows the

amplitude of the axisymmetric imperfection at mid-span to have been reduced by nearly

50% by the expansion process. Figure 3.6b shows the amplitude of the non-axisymmetric

imperfection at the mid-span for 8m to have been reduced by nearly 60% and the

contact with the outer pipe to have increased. In all calculations involving the base case

(Table 3.1), the models will be assigned the same prehistory due to the expansion. For

consistency, the imperfection amplitudes that will be quoted are the initial values.

36

We now consider a lined pipe with the same geometry and material properties as

in Section 3.3, manufactured in a similar manner but with a liner that has small initial

imperfections of the type described by Eq. (3.5). Here the value of is calculated as in

Eq. (3.3) and 8m (the effect of these choices will be discussed subsequently). The

amplitudes of the imperfections o and m used are listed in the figures; they represent

the values prior to expansion. Figure 3.7a shows the calculated moment–curvature

responses of the composite structure and the individual tubes. Figure 3.7b shows a

corresponding plot of the detachment, (0), of the compressed generator of the liner in

the plane of bending at 0x . Figure 3.8a shows two sets of deformed configurations

corresponding to the solid bullets marked on the liner responses in Fig. 3.7. The three

moment–curvature responses follow the same trends as those of the perfect geometry

case, but in the neighborhood of the axisymmetric imperfection is excited and small

amplitude wrinkles develop in the central part of liner (see corresponding images in

Fig. 3.8a where the color contours represent the magnitude of the separation of the liner

from the carrier pipe depicted as w .) The amplitude of the wrinkles grows as illustrated

in configuration and and so does the separation of the liner from the outer tube. This

reduces the bending rigidity of the liner causing the development of a moment maximum

at 0.6231 (marked in Fig. 3.7a with a caret "^"). This is a sign that wrinkling is

starting to localize while simultaneously the non-axisymmetric component of the

imperfection is excited. The switch to the diamond-type of mode, seen in configuration

, causes an abrupt increase in local separation of the liner from the outer pipe, (0). At

higher curvatures, the diamond buckles become more prominent as seen in configurations

and (note the different color scale). A three-dimensional rendering of the buckled

liner at a curvature of 1.0 1 is shown in Fig. 3.8b. The significant amplitude of the

buckles can render this structure non-operational.

37

Another view of the localization that takes place is presented in Fig. 3.7c, which

shows the compressed generators of the outer pipe and liner in the plane of bending at

different degrees of deformation. The separation of the liner from the outer pipe near the

center of the model in contours and is quite obvious. We will define the curvature at

the moment maximum and the sharp upswing in the separation between the two tubes as

the critical collapse curvature. It is reassuring that this sequence of events as well as the

collapse mode in images in Fig. 3.8 are qualitatively in good agreement with results

from full-scale bending experiments reported in Hilberink et al. (2010, 2011) and

Hilberink (2011).

3.5 IMPERFECTION SENSITIVITY OF LINER COLLAPSE

Information on actual liner imperfections introduced during the manufacture of

the two tubes and the composite structure at the present time are scarce. Collectively the

values of imperfection amplitude o and m used in the calculation described in Section

Section 3.4 are somewhat arbitrary. In order to better understand the effect of the

imperfections on the liner collapse, the two values of the imperfection amplitudes o and

and m are varied while keeping the outer pipe and liner geometry and material

properties the same as those in Table 3.1. Figure 3.9a shows sets of moment- and

maximum detachment-curvature responses for various values of o and fixed values of

m and m. Associating again the curvature at the moment maximum and the

corresponding point at which the liner detachment experiences significant sudden growth

with collapse, it is clear that collapse is extremely sensitive to this imperfection. This

point is further highlighted realizing that Lt03.0 , i.e., the axisymmetric imperfection

amplitude before expansion, corresponds to 0.09 mm, a value that is significantly smaller

than typical internal surface imperfections left behind by the manufacture of the seamless

38

carrier pipe. Furthermore, we reiterate that this amplitude is reduced by about 50% by the

expansion process.

The amplitude of m was also varied keeping o and m constant. Figure 3.10

shows similar sets of results for 0 m 0.06. Although these values are somewhat

larger than those of o in Fig. 3.9, it is clear that the liner collapse is sensitive to non-

axisymmetric imperfections also. The two sets of results in Figs. 3.9 and 3.10 are

summarized in Fig. 3.11 where the liner collapse curvature, CO, is plotted against the

two imperfection amplitudes. The results demonstrate that although the onset of collapse

is sensitive to both types of imperfections, it is much more sensitive to axisymmetric

ones.

The mode of the non-axisymmetric imperfection was also considered by varying

the value of m adopted in Eq. (3.5). Figure 3.12 shows moment- and maximum

detachment-curvature responses of the liner for three values of m from calculations based

on the base case parameters and for fixed values of imperfection amplitudes. The results

show that collapse is relatively insensitive to the value of m adopted. A careful evaluation

of this conclusion revealed that it is valid provided the imperfection amplitudes after

expansion have similar values, as was the case for m = 6, 8 and 10. It was observed that

for 6m the initial values of o and m had to be smaller in order to end up with

similar final imperfection amplitudes after expansion.

In the calculations thus far the value of the half wavelength of the axisymmetric

imperfections used corresponded to the elastic value, e , as defined in Eq. (3.3). A more

accurate value can only come from plastic bifurcation check of the composite pipe under

bending (see Chapter 4). Here, we vary within reasonable limits using the base case

parameters and constant values of imperfections amplitudes. Figure 3.13 shows the

39

collapse curvature of the liner to be quite insensitive to the value of adopted within the

the chosen range.

Thus far, contact between the liner and the carrier pipe is assumed to be

frictionless. We now examine the effect of friction on the results. In Fig. 3.14 we

compare moment- and maximum detachment-curvature responses of the liner for the

frictionless case and for Coulomb friction with coefficient 0.3 (based on the base

case parameters). Clearly friction has a negligibly small effect on the onset of collapse

and the post-collapse response of the liner. This is mainly because the evolution of the

liner collapse does not involve significant relative sliding between the two tubes. Based

on these observations friction is neglected in all subsequent calculations of lined pipe

bending.

Summarizing the results of this imperfection sensitivity study, it is clear that liner

collapse is very sensitive to small initial geometric imperfections left in the liner from the

manufacturing process. Furthermore, the outer pipe has been assumed to be perfectly

circular and to have uniform thickness, assumptions that require revisiting.


Thus far we have limited attention to a base case that involves a 12-inch outer

pipe with D/t 18 and a 3 mm thick corrosion resistant liner. In this section we present

results from a wider parametric study in which various additional factors that can

influence the collapse of liners are examined.

3.6.1 Pipe Diameter

We first consider composite systems of four different steel pipe diameters but

keep the D/t at approximately 18.0. In addition, the liner thickness is kept at 3 mm and

the material properties of both tubes are kept the same as those used in the base case.

40

Each composite system is assigned similar imperfections (Eq. (3.5)) and then

appropriately expanded as described in Section 2.1. In each case the imperfection half-

wavelength is determined from Eq. (3.3) while 8m . Due to the difference in pipe

diameter, the expansion process alters the initial imperfections to differing degrees. Thus,

for a more systematic comparison of their effect on liner collapse, the amplitudes of the

two imperfections are varied so that after expansion the maximum value of w / RL is

approximately the same for all four cases, 3100.778 .

The models are purely bent and the response of the two-pipe systems is recorded.

The results are summarized in Fig. 3.15, which shows plots of the liner moment- and

maximum detachment- curvature responses for outer pipes with diameters of 8.625,

10.75, 12.75 and 14.0 in (designated in the figure as 8, 10, 12, 14 in). In these plots the

normalizing variables are as follows bob Dt |21 , booob tDM |2 , where the

subscript ‘‘b’’ implies the variables of the base case, in other words those of the 12-inch

pipe system in Table 3.1. With this normalization the moment and curvature appear in

their natural order.

As expected, as the diameter of the pipe increases, the moment carried by the liner

increases. The behavior of the liner is similar to that described in Figs. 3.7 and 3.8:

bending causes the liner to separate from the outer pipe; it develops periodic wrinkles,

whose amplitude gradually grows, and at some point the non-axisymmetric imperfection

is excited enough to lead to the collapse of the liner. Collapse is associated with the

moment maxima in Fig. 3.15a and with the sharp upswing of the detachment variable

(0) in Fig. 3.15b. Clearly, as the pipe diameter decreases, the composite pipe can be

bent to a larger curvature before the liner collapses. This is caused by the fact that, as D

decreases, so does LL tR while the axial stress induced to the liner by bending

decreases.

41


In the manufacturing process used in the product analyzed, the liner tube initial

diameter is chosen to be somewhat smaller than that of the outer pipe for ease of

insertion. As demonstrated in Section 2.3.2, the initial annular gap between the two pipes,

og , has a significant effect on the resultant contact stress. In this section we examine the

effect of this gap on the collapse of the liner. To this end we simulate the manufacture of

the base case system (Table 3.1) again for four values of og : {0.5, 1, 1.5, 2} obg , where

obg is the value used in the base case (see Fig. 2.10). Because the annular gap influences

the contact stress that develops between the two tubes, the final value of a chosen initial

liner imperfection depends on og . Since it is desirable that the amplitudes of the

imperfections of the four cases studied be nearly the same, the initial values of o and

m are varied so that the final amplitude of the imperfections is Lt0255.0 for all four

cases. Figure 3.16 shows results from bending calculations on each of the four composite

tubes. Figure 3.16a shows the liner moment–curvature responses and Fig. 3.16b the

corresponding maximum separation-curvature results. The overall behavior of the liner is

similar in all cases, but clearly increasing og results in a decrease in the collapse

curvature of the liner. The importance of this parameter on the integrity of the liner under

bending is highlighted by the observation that the decrease in collapse curvature between

the smallest gap used and the largest is more than 50%. This sensitivity of the liner

collapse curvature to og indicates that, to the extent that is practically feasible, its value

should be minimized. This places tighter demands on the manufacture of the two tubes

for increased straightness and roundness.

It is also interesting to observe that increasing og has the effect of increasing the

moment carried by the liner, a direct consequence of the additional strain hardening

resulting from increased expansion undergone by the liner.

42

3.6.3 Liner Wall Thickness

As might be expected, the wall thickness of the liner plays a decisive role on its

stability under bending and deserves special attention (e.g., see Tkaczyk et al., 2011). We

thus consider a 12-inch composite system like the one in Table 3.1 but assign the liner

thickness six values between 2.0 and 4.5 mm. The annular gap is kept the same and so are

the mechanical properties. The liner is assigned initial geometric imperfections as defined

in Eq. (3.5) with the half-wavelength calculated for each value of Lt in accordance

with Eq. (3.3). Each composite system is expanded in the same way. The imperfection

amplitudes are chosen such that the post-expansion absolute values of the amplitudes are

similar for the six cases ( 310778.0/ LRw ).

Each composite system is purely bent, and the calculated liner moment- and

maximum detachment- curvature responses are shown in Fig. 3.17. Qualitatively the

behavior of the composite structures is similar to that described for the base case. The

results clearly show that increasing the liner thickness increases the moment carried by

the liner (Fig. 3.17a) and simultaneously delays the onset of liner collapse. It is important

to note however, that since the cost of lined pipe is significantly influenced by the

material cost of the non-corrosive liner, the improvement in collapse curvature resulting

from the increase in Lt demonstrated here must be weighed against the related increase

to the cost of the product. It is possible that calculations like the present ones can be used

to conduct a cost-performance analysis to select the optimal liner thickness for a given

outer pipe diameter.

3.6.4 Bending Under Internal Pressure

A practical method of delaying liner buckling and collapse during reeling that has

been proposed by industry is to internally pressurize the pipe (e.g., Endal et al., 2008;

Toguyeni and Banse, 2012; Montague et al., 2010). One method proposed is to isolate the

43

section of line that is to be reeled using moveable pigs and pressurize it to a few bars.

Once in contact with the reel one of the pigs is moved to include an adjacent stalk, which

is then pressurized and the reeling continues (Mair et al., 2013; see also Howard and

Hoss, 2011). Alternatively the whole line is pressurized (Mair et al., 2014).

As a way of evaluating the effectiveness of pressurization in delaying liner

collapse, the 12-inch base case is now bent under increasing values of internal pressure.

This is done following the initial expansion of the two-part system in accordance with the

steps described in Section 2.1. Figure 3.18 shows bending results for the base case and

for three levels of internal pressure: 30, 50 and 100 psi (2.07, 3.45 and 6.9 bar).

Qualitatively the behavior is similar to that of the unpressurized case. However, even

such modest levels of internal pressure have a stabilizing effect on the liner, causing a

delay in its collapse. It is interesting to observe that for the imperfection used at P = 100

psi (6.9 bar) the liner remained stable at curvatures of 12 and beyond (maximum

bending strain of about 5%).

Although the same exercise must be repeated for the more complex bending cycle

of spooling and unspooling a lined pipe on a reel, the results indicate that internal

pressurization during the process may indeed make otherwise non-reelable pipe systems

reelable. It is interesting to point out that previous studies have demonstrated that internal

pressure can delay buckling of shells under bending (e.g., Mathon and Limam, 2006;

Limam et al., 2010).

44

Fig. 3.1 FE mesh of composite pipe under bending.

x y

z

M

M

L

45

Fig. 3.2 (a) Schematic of elements distribution for the introduction of initial state.

46

Fig. 3.2 (b) Comparison of moment-curvature responses between the two initial state

schemes.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.2 0.4 0.6 0.8 1 1.2 1.4

M

Mo

Steel Pipe

Composite

Liner

D = 8.625 in

tL= 3 mm

= 18.1Dt

47

Fig. 3.3 Base case moment- and ovalization-curvature responses.

43210

0

0.4

0.8

1.2

0

1

2

3

0 0.2 0.4 0.6 0.8 1

M

Mo

D D

(%)

L

Steel Pipe

Composite

DD

Liner

48

Fig. 3.4 Liner deformed configurations with superimposed contours of liner contact

pressure; correspond to numbered bullets on liner response in Fig. 3.3.

49

Fig. 3.5 (a) axisymmetric and (b) non-axisymmetric imperfections adopted.

(b)

(a)

50

(a)

(b)

Fig. 3.6 Comparison of profiles of imperfections initially and after application of

manufacturing stress field: (a) axial and (b) circumferential ( 0x ) profile.

0

0.02

0.04

0.06

0 2 4 6

w

tL

x /

Initial: N = 4, = 0.05

Final

0

0.02

0.04

0.06

0 0.25 0.5 0.75 1

Initial: m = 8, m

=0.05

Final

w

tL

51

(a)

(b)

Fig. 3.7 Imperfect base case responses: (a) moment-curvature, (b) maximum

detachment-curvature.

41

2

3 657

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1

M

Mo

Steel Pipe

Composite

Liner

= 1%,

m = 6%

m = 8

4

321

0

0.04

0.08

0.12

0.16

0 0.2 0.4 0.6 0.8 1

RL

5

6

7 = 1%,

m = 6%

m = 8

52

Fig. 3.7 Imperfect base case responses: (c) axial profiles of compressed generators of outer pipe and liner.

-0.6

-0.4

-0.2

0

0 1 2 3 4 5

y

RL

x / RL

00.440.66 0.98

7

1

5

0

53

Fig. 3.8 (a) Sequences of liner deformed configurations showing evolution of

wrinkling corresponding to numbered bullets on response in Fig. 3.7a. On

the left are 3D renderings and on the right cross sectional views of

compressed side (for images and use Rw color scale).

54

Fig. 3.8 (b) Three-dimensional rendering of the buckled liner.

55

(a)

(b)

Fig. 3.9 Effect of axisymmetric imperfection amplitude on liner response. (a)

moment-curvature and (b) maximum detachment-curvature responses.

0

0.04

0.08

0.12

0.16

0 0.2 0.4 0.6 0.8 1

M

Mo

o(%)m = 8,

m = 6%

0.8 0.4 0.2 02.0 1.03.0

0

0.04

0.08

0.12

0.16

0 0.2 0.4 0.6 0.8 1

RL

o (%)m = 8,

m = 6%

0.8 0.4 0.2 02.0 1.03.0

56

(a)

(b)

Fig. 3.10 Effect of non-axisymmetric imperfection amplitude on liner response. (a)

moment-curvature and (b) maximum detachment-curvature responses.

0

0.04

0.08

0.12

0.16

0 0.2 0.4 0.6 0.8 1

M

Mo

m

(%)

4

m = 8, = 1%

2 1 06

0

0.04

0.08

0.12

0.16

0 0.2 0.4 0.6 0.8 1

m

(%)

m = 8, = 1%

421

0

6

RL

57

Fig. 3.11 Collapse curvature sensitivity to axisymmetric ( o ), and non-axisymmetric

( m ) imperfection amplitudes.

0

0.2

0.4

0.6

0.8

1

0 0.01 0.02 0.03

0 0.02 0.04 0.06

co

m

o

= 0.01, m = 8

m

= 0.06, m = 8

m

58

(a)

(b)

Fig. 3.12 Effect of circumferential wave number on liner response. (a) moment-

curvature and (b) maximum detachment-curvature responses.

0

0.04

0.08

0.12

0.16

0 0.2 0.4 0.6 0.8 1

M

Mo

m

610 8

= 1%,

m = 6%

0

0.04

0.08

0.12

0 0.2 0.4 0.6 0.8 1

610

8

m

= 1%,

m = 6%

RL

59

(a)

(b)

Fig. 3.13 Effect of axial wavelength of imperfections on liner response. (a) moment-


0

0.04

0.08

0.12

0.16

0 0.2 0.4 0.6 0.8 1

M

Mo

e

0.81.2 1.0

= 1%,

m = 6%

m = 8

0

0.04

0.08

0.12

0 0.2 0.4 0.6 0.8 1

0.8

1.21.0

= 1%,

m = 6%

m = 8

RL

e

60

(a)

(b)

Fig. 3.14 Effect of friction on liner response. (a) moment-curvature and (b) maximum

detachment-curvature responses.

0

0.04

0.08

0.12

0.16

0 0.2 0.4 0.6 0.8

M

Mo

0.3 0

= 1%,

m = 6%

0

0.04

0.08

0.12

0 0.2 0.4 0.6 0.8 1

0.30

= 1%,

m = 6%

RL

61

(a)

(b)

Fig. 3.15 Effect of pipe diameter on liner response for a constant liner wall thickness.

(a) moment-curvature and (b) maximum detachment-curvature responses.

0

0.04

0.08

0.12

0.16

0 0.4 0.8 1.2 1.6

M

Mob

1b

m = 8

14

10

12

8

D (in)

tL = 3 mm

= 0.778x10-3wR

L max

Dt

~18~

0

0.04

0.08

0.12

0.16

0 0.4 0.8 1.2 1.6

RL

1b

14

12

10

8

D (in)

62

(a)

(b)

Fig. 3.16 Effect of initial annular gap on liner response. (a) moment-curvature and (b)

maximum detachment-curvature responses.

0

0.04

0.08

0.12

0.16

0 0.2 0.4 0.6 0.8 1

M

Mo

D = 12.750 in

0.51

1.52

m = 8

go

gob

= 0.0255wtL max

0

0.04

0.08

0.12

0.16

0 0.2 0.4 0.6 0.8 1

D = 12.750 in

RL

0.511.52

go

gob

63

(a)

(b)

Fig. 3.17 Effect of liner wall thickness on liner response. (a) moment-curvature and

(b) maximum detachment-curvature responses.

0

0.04

0.08

0.12

0.16

0.2

0 0.4 0.8 1.2

M

Mo

D = 12.750 in

tL (mm)3

3.5

4

4.5

m = 8

2.5

2

0

0.04

0.08

0.12

0.16

0 0.4 0.8 1.2

D = 12.750 in

tL (mm)

3 3.5 4

4.5

RL

2.52

64

(a)

(b)

Fig. 3.18 Effect of internal pressure on liner response. (a) moment-curvature and (b)


0

0.04

0.08

0.12

0.16

0 0.4 0.8 1.2 1.6 2 2.4

M

Mo

D = 12.750 in

0

tL= 3 mm

= 1%,

m = 6%

m = 8

30 (2.07)50 (3.45)

100 (6.9)

P psi (bar)

0

0.04

0.08

0.12

0.16

0 0.4 0.8 1.2 1.6 2 2.4

D = 12.750 in

P psi (bar)

030 (2.07)

100 (6.9)

50 (3.45)

tL= 3 mm

RL

65

Chapter 4: PLASTIC BIFURCATION BUCKLING OF LINED PIPE UNDER

BENDING

In Chapter 3 we considered in detail the stability of the liner in a lined composite

pipe under bending. The nontrivial version of the problem was analyzed by considering

liners with small initial geometric imperfections. This chapter is concerned with the onset

of periodic wrinkling as a plastic bifurcation process. It is well established that plastic

bifurcations performed with incremental moduli using deformation theory are in much

better agreement than corresponding predictions yielded by flow theories (e.g., Batdorf,

1949; Hutchinson, 1975; Kyriakides and Corona, 2007). The success of such schemes in

pipeline and generally relatively thick-walled circular shell applications has been

demonstrated for axial compression (Peek, 2000; Kyriakides et al., 2005; Bardi and

Kyriakides, 2006), compression under internal pressure (Paquette and Kyriakides, 2006),

bending (Ju and Kyriakides, 1991; Peek, 2002; Corona et al., 2006) and bending under

internal pressure (Limam et al., 2010).

Peek and Hilberink (2013) recently developed an analytical expression for the

onset of axisymmetric wrinkling of the liner of a lined pipe under compression. The

results are along the lines of the axisymmetric plastic bifurcation check of Lee (1962) and

Batterman (1965) both of which are presented in summary form in Appendix B. The first

instability of a long circular cylinder under bending involves a similar periodic wrinkling

mode. The problem however is complicated by, among other factors, Barzier (1927)

ovalization induced to the cross section by bending. For this reason the bifurcation check

was performed numerically (Ju and Kyriakides 1991; Peek, 2002). Bending of a lined

cylinder is further complicated by contact nonlinearities making the bifurcation check

even more challenging.

66

This chapter presents a solution procedure for establishing the onset of the first

bifurcation buckling of such a lined pipe under bending. The critical strain at bifurcation

and the corresponding wavelength are compared to the corresponding values from the

axially loaded lined cylinder as well as with those of a liner shell alone under pure axial

compression and bending. The resultant bifurcation mode is subsequently used to

examine the imperfection sensitivity of the liner and the results are compared to those of

previous studies, which were based on idealized imperfections.

4.1 BIFURCATION ANALYSIS

We consider a long circular cylinder of line-grade carbon steel of diameter D and

wall thickness t lined with a thin layer of non-corrosive material thickness Lt (see Fig.

4.1a). The two tubes are assumed to be in perfect frictionless contact. We will consider a

model of length N2 that is under pure bending, where 2 is the wavelength of the

expected wrinkles. At the outset, will be assigned the value corresponding to elastic

buckling of a circular cylindrical shell with the geometry of the liner under axial

compression given by

4/12)]1(12[

LL

CetR

, (4.1)

where is the liner Poisson’s ratio. Symmetry about the mid-span (plane zy ) and

about the plane of bending zx is assumed. The steel carrier is meshed with linear solid

elements (C3D8) and the contacting liner with linear shell elements (S4). The carrier pipe

has two elements through the thickness and both tubes are assigned 14 elements per in

the axial direction. In the circumferential direction 36 elements are used for 4/0

and 72 for 4/ (see Fig. 4.1b). Unless otherwise stated the length of the model

will be defined by N = 8.

67

The bifurcation is expected to take place at a high enough curvature to plastically

deform the two tubes. To accommodate the preferred use of deformation theory of

plasticity for the bifurcation check, the material inelastic behavior will be modeled

through the J2 deformation theory of plasticity for the prebuckling solution also. This is

accomplished through a custom user-defined subroutine appended to the nonlinear code

ABAQUS. It is worth noting that, although under inelastic bending the stress-paths of, for

example, the intrados and extrados are somewhat non-proportional, the major aspects of

the bending response yielded by deformation theory are essentially identical to those

produced by J2 flow theory.

The nonlinear stress-strain relationships of J2 deformation theory are given by

ijjkiljlikklijs

s

s

sij

E

][2

1

)21()1(, (4.2a)

where Es(J2 ) is the secant modulus of the material and

s 12

EsE

12

. (4.2b)

The incremental version of (4.2a) required by the nonlinear solver is given by:

ijklij

klijjkiljlikij dJhh

sshh

h

Ed

221)21(3

3)(

2

1

1, (4.3)

where

1

2

3

sE

Eh ,

2dJ

dhh and 2/1

2

2/1

)3(3

2Jss ijije

.

The stress-strain responses of both tubes are represented by Ramberg-Osgood fits

given by:

1

7

31

n

yE . (4.4)

68

More details about the constitutive equations used in the bifurcation check are given in

Appendix C.

Table 4.1 Main geometric and material parameters of base case

D in (mm)

t in (mm) t

D E Msi

(GPa)

n y ksi

(MPa)

o ksi

(MPa)

Carrier X-65

12.75 (323.9)

0.705 (17.9)

17.75 30.3 (209)

0.3 52 72.5 (500)

73.5 (507)

Liner Alloy 825

11.34 (288.0)

0.118 (3.0)

99.4 30 (207)

0.3 17 41.0 (283)

44.0 (303)

The parameters },,{ nE y for the two tubes are listed in Table 4.1. For the carrier

pipe they were obtained from a fit of the measured tensile stress-strain response of a

nominally X65 line grade steel and for the liner from a fit of a measured response of

Alloy 825 ( o is the yield stress corresponding to a strain offset of 0.2%).

The model is bent by prescribing the rotation of the plane at Nx . The

increments are chosen to be small (~ 1000/1L ) and ABAQUS’s bifurcation check is

used to identify the critical eigenvalue (see Section 6.2.3 ABAQUS Analysis user manual

6.10). The bifurcation is in the form of periodic wrinkling of the liner most prominently

displayed on its compressed side. The initial value of used, i.e., Eq. (4.1), does not

necessarily agree with the actual bifurcation wrinkle half-wavelength. Thus, the complete

calculation is repeated for a number of different values of . The smallest bifurcation

curvature yielded is designated as the critical one, C , and the corresponding

wavelength as C2 .

69

4.2 BIFURCATION RESULTS

4.2.1 Wrinkling Bifurcation Under Bending

The bending response and the evolution of events that precede the bifurcation will

be demonstrated through results for the 12-inch composite pipe with the geometric and

material properties listed in Table 4.1. The calculated moment-curvature response (

M ) of the composite structure is plotted in Fig. 4.2a, where the normalizing

variables used are:

tDM ooo2 , 2

1 / oDt , tDDo . (4.5)

It is instructive to also include the corresponding moments carried by the

individual steel and liner pipes. As expected, the carrier pipe carries most of the moment

but both tubes are seen to have plasticized. Bending tends to ovalize both tubes but

included in the figure is the ovalization induced to the liner, represented here by the

change in diameter in the plane of bending, D / D |L . The ovalization is seen to grow in

the usual nonlinear manner with curvature, but more importantly the liner tends to ovalize

more than the carrier pipe. The set of deformed configurations of the liner shown in Fig.

4.2b illustrate the consequences of this differential ovalization (images correspond to

numbered bullets on the liner M response in Fig. 4.2a). Superposed on the

configurations are color contours that represent the magnitude of the separation between

the two tubes (radial separation w). Thus in image , at the relatively small curvature

of 1037.0 , the two tubes are essentially in contact; at at a curvature of 1066.0 the

ovalization of the liner clearly overtakes that of the steel tube causing measurable loss of

contact along two strips at the two extremes of the cross section. As the curvature is

further increased to 1144.0 at and 1170.0 at , the width of the separated liner

strips progressively grows. At at a curvature of 1185.0 , the compressed strip at the

top wrinkles. Marked on the response with an arrow () is the predicted bifurcation point

70

at a curvature of 1179.0 C . The corresponding wavelength is LC R246.0 . The

buckling mode yielded by the eigenvalue solver is shown in Fig. 4.3. A strip covering

approximately the top 60 degree sector of the liner has developed periodic wrinkles

whose amplitude is maximum at the plane of bending and gradually reduces to zero at

about 30o (see Fig. 4.2b). Beyond this angle the liner is in positive contact with the

carrier pipe. Needless to say that, at these small curvatures, the carrier pipe, although

plastically deformed and ovalized to a certain degree, is structurally in perfect condition.

The critical curvature and wrinkle wavelength given above were arrived at

following a series of calculations involving a short section of the model 2 long (see

Fig. 4.4a). The value of is varied calculating in each case the bifurcation curvature ( b

). Figure 4.4b shows the results of this process for the base case ( L1 is based on the

liner diameter and wall thickness). The results exhibit the usual behavior with the

bifurcation curvature increasing for both lower and higher values of than the critical

value.

In summary, the reported behavior is qualitatively similar to the one described in

Chapter 3, in which similar calculations were performed using the J2 flow theory of

plasticity. However, as is the case in other plastic buckling predictions, the bifurcation

curvature predicted by the present deformation theory analysis is significantly lower than

the value yielded by flow theory.

It is interesting to compare the moment-curvature response and bifurcation results

of the liner with corresponding ones for the liner bent alone (i.e., in the absence of the

steel carrier pipe). This has been performed with ABAQUS and confirmed with the

custom program BEPTICO (Kyriakides et al., 1994) and the associated bifurcation check

RIBIF described in Ju and Kyriakides (1991). Figure 4.5a shows a comparison of the

moment-curvature response of the liner in the composite pipe for the base case

71

parameters and the corresponding one for the liner shell alone. Figure 4.5b shows the

associated D responses. Interestingly, the moment-curvature of the single tube

response is only slightly lower than the one of the lined pipe. This, despite the fact that it

ovalizes significantly more (see Fig. 4.5b). Marked on the M response are the

calculated bifurcation points, which are also seen to be quite close, presumably because

the stress-states in the two shells do not differ significantly. Thus C for the single shell

is only 2% lower than that of the liner while RC is 4% higher.

4.2.2 Parametric Study

The critical state of the liner depends of course on its geometry and mechanical

properties. To explore this dependence we conduct a limited parametric study in which

the diameter of the carrier pipe is varied but the wall thickness and mechanical properties

of the liner are kept constant. Accordingly, we consider carrier pipes of 8.625, 10.75,

12.75, 14.0 and 16.0 inches, all of them having a 0.18/ tD and the X-65 mechanical

properties listed in Table 4.1. Since the liner thickness is kept constant at 3 mm, the

corresponding LtD / are respectively approximately 67.2, 83.8, 99.4, 108.6 and 125.0.

The mechanical properties of the liner are those of alloy 825 given also in Table 4.1.

Each of the five lined pipes was also purely bent and sets of bifurcation

calculations similar to the one described above were conducted for each. The calculations

yielded the liner critical strain, C , and the corresponding wrinkle half wavelength, C ,

each plotted against the liner D/t in Figs. 4.6a and 4.6b respectively (results identified by

“Lined Bending”). The critical strain varies from about 0.78% at the lowest LtD / to

0.40% at the highest. The corresponding C / R goes from 0.296 to 0.218. Included in

the figures are the corresponding critical quantities for bending of the liner shell alone

(designated as “Bending”). As was the case for the 12.75-inch system, the bifurcation

72

strains are very close to those of the liners in the corresponding lined pipes. The wrinkle

wavelengths on the other hand have somewhat higher values, by nearly 7% for the lowest

LtD / and about 3% for the highest.

For completeness we include in Fig. 4.6 the critical wrinkling variables (C,C )

under axial compression, first of the lined tubes and second those of liner shells alone,

identified by “Lined Axial” and “Axial” respectively. They were evaluated in the usual

way using Eqs. (B.6) and (B.9). Interestingly, of the four cases considered, the critical

strains of the compressed lined tubes are the highest, and those of the liner shells alone

are the lowest. Unlike bending, where differential ovalization causes some separation of

the two tubes, under compression, and if the tubes have the same Poisson’s ratio and

similar mechanical properties, they remain in contact until bucking, which has a delaying

effect on the instability. Thus, for the lowest LtD / considered, the critical strain is 11%

higher than that of the lined tube under bending and for the highest 22% higher. By

contrast, the compressed liner shell alone has the lowest wrinkling strain. Comparing

again the extreme values of LtD / , the values are 31% and 26% lower than those of the

lined tubes under bending. Clearly, uniform compression of the shell is the most

destabilizing loading condition of the four related cases.

The wavelengths follow the opposite trend with the compressed liners alone

having the longest wavelengths, which however are only slightly higher than those under

pure bending. The compressed lined tubes have the shortest wavelengths while those of

the lined tubes under bending fall between the two extreme sets of values. Overall, the

spread between the four sets of C is not that large which confirms that adoption of Ce

of the elastic compression problem (Eq. (4.1)) in non-trivial calculations can suffice as a

first step.

73

4.3 IMPERFECTION SENSITIVITY

As a way of analyzing the non-trivial response of the composite structure, the

calculated liner buckling modes are introduced as initial imperfections to the

corresponding structures, followed by bending. The FE model used is the one shown in

Fig. 4.1 with N = 8. The calculations that follow are similar to ones performed in Chapter

3 with the following differences: (a) the two shells are initially stress free; (b) the

imperfection corresponds to the calculated buckling mode (see Fig. 4.3) rather than the

axisymmetric one adopted in the preceding work; and (c) the imperfection is uniform

along the length (i.e., has no amplitude bias towards the mid-span). Here, the two

materials are modeled using the finite deformation J2 flow theory of plasticity, each

calibrated to the corresponding stress-strain responses in Table 4.1.

We use the same 12.75-inch composite pipe analyzed in Section 4.2.1 to describe

the ensuing sequence of events in some detail. The model half-length adopted is C8 ,

with the value of C established in Fig. 4.4. The buckling mode with an amplitude

o 0.01tL is introduced as an initial imperfection.

Figure 4.7a shows the calculated moment-curvature response of the composite

structure as well as those of the individual shells. Figure 4.7b shows the corresponding

detachment-curvature response, where )0( is the detachment of the compressed

generator of the liner in the plane of bending at the plane of symmetry (x = 0). Figure 4.8

in turn shows a set of deformed configurations of the liner corresponding to the numbered

bullets marked on the liner responses in Fig. 4.7. The color contours represent the extent

of local separation (w) from the outer pipe. Initially, the three moment-curvature

responses follow the same trends as those of the perfect geometry case. Image is well

past the bifurcation point ( 1179.0 C ) but no visible signs of wrinkling are observed

(due to the scale chosen). In the neighborhood of , the periodic imperfection is excited

74

and small amplitude wrinkles become visible in Fig. 4.8a, while simultaneously )0( is

seen to start to grow. At point at a somewhat higher curvature, the amplitude of the

wrinkles grows and so does the separation of the liner from the outer tube. The bending

rigidity of the liner is reduced resulting in the development of a moment maximum in the

liner response at 1736.0 (marked in Fig. 4.7a with a caret “^”). As a consequence,

wrinkling localizes as illustrated in images and at mid-span causing an abrupt

increase in )0( . In this neighborhood a diamond-type buckling mode becomes

energetically preferred and this switch starts to appear in image and is seen fully

developed in image . This buckling mode has a butterfly shape with a major wrinkle at

the center surrounded by four satellite ones. It can also be clearly seen in Fig. 4.8b that

shows the bent liner cut normal to the plane of bending. It is exactly the same mode

reported for the pre-deformed case of the same pipe system in Chapter 3, which however

was perturbed by an imperfection with an axisymmetric and a non-axisymmetric

component. Here, it has developed without any priming and presumably was triggered by

numerical noise. The amplitude of these wrinkles grows significantly with small

additional changes in curvature rendering the pipe quickly unserviceable. As in Chapter

3, we will designate the curvature at the liner moment maximum and the associated

upswing in liner separation and deepening of the wrinkles as the collapse curvature of the

liner, CO . In summary, although periodic wrinkling is the first instability, generally it is

is of small amplitude and is relatively benign. Collapse is caused by the diamond-type,

second instability that takes place at a higher curvature, as described in Chapter 3 (see

also corresponding results for axial compression in: Tvergaard, 1983; Yun and

Kyriakides, 1990; Bardi et al., 2006; Kyriakides and Corona, 2007).

In the results shown in Fig. 4.7 the imperfection used corresponds to the actual

buckling mode shown in Fig. 4.3. It is worth comparing its response and collapse

75

curvature to that of the same composite pipe in which an axisymmetric imperfection of

the type given below is used instead (shown in Fig. 4.9):

2)100/(01.0cos Cx

CoL

xtw

, (4.6)

where C is the half wavelength of the critical bifurcation mode established in Fig. 4.4

(the multiplying function provides a small bias in amplitude towards the mid-span).

The calculated response is compared to the one using the bifurcation mode in Fig.

4.10; in both cases the imperfection amplitude is Lo t01.0 . The moment-curvature

response of the idealized imperfection is slightly below that of the bifurcation mode

imperfection while the corresponding detachment grows slightly faster with curvature.

However, the collapse curvatures of the two cases, represented by the moment maxima,

are very close indeed. Apparently, what influences the collapse curvature is the amplitude

and wavelength of the compressed side of the liner, which is common to both cases. The

results confirm that the adoption of axisymmetric imperfections in bending, as has been

done in Chapter 3 but also in Ju and Kyriakides (1992), Corona et al. (2006), etc., is

acceptable provided the correct wavelength is used.

The imperfection sensitivity of the base case is further examined by conducting

similar calculations for various values of initial imperfection. Figure 4.11 shows the

M and )0( responses for o {0, 0.002, 0.008, 0.03} Lt . Despite the

relatively small values of imperfections used, the curvatures at the moment maxima and

the corresponding upswings in the growth of )0( are seen to decrease rather

significantly as o increases. Thus, a drop of nearly 28% in the collapse curvature is

observed between Lo t002.0 and Lt03.0 . By contrast, in the absence of an

76

imperfection, the liner remains intact until the moment maximum of the composite

structure is reached, something that is unattainable in practice.

Similar collapse calculations were performed for several carrier pipe diameters

keeping their D/t at approximately 18.0 and the thickness of the liner constant at 3 mm.

Furthermore, the imperfection amplitude is kept at Lo R4102 for all cases. The

models were purely bent and the results are summarized in Fig. 4.12, which shows plots

of the liner moment- and maximum detachment-curvature responses for outer pipes with

diameters of 8.625, 10.75, 12.75, 14.0 and 16.0 in. (with corresponding LtD / of

approximately 67.2, 83.8, 99.4, 108.6 and 125.0). Here the normalizing variables obM

and b1 are based on the parameters of the 12.75 inch base case pipe in Table 4.1. The

behavior is similar to that in Figs. 4.7 and 4.8 for the 12.75-inch pipe. In other words,

bending causes separation of the liner from the carrier pipe, compression excites the

initial imperfection, which at some stage yields to the diamond shell-type buckling mode

that results in the collapse of the liner. As expected, as the diameter of the pipe increases,

the moment carried by the liner increases. However, the collapse curvature, represented

by the moment maxima in Fig. 4.12a and by the sharp upswing of the detachment

variable LR/)0( in Fig. 4.12b, decreases because LtD / increases (behavior similar to

one in Fig. 3.15 for pipe systems that had undergone the manufacturing pressurization

prehistory).

The strains at collapse ( CO ) for the five cases are plotted against the liner D/t in

Fig. 4.13. The collapse strains can be seen to decrease as the carrier pipe and LtD /

increases, going from approximately 3.1% at the lower to 1.55% at the higher ends.

Included in the figure are the corresponding bifurcation strains at the onset of wrinkling (

C ). They are seen to be significantly lower with values of 0.78% at the lower end and

0.40% at the higher end. Although this difference depends on the value of imperfection

77

used in the nontrivial calculations, it indicates that the onset of bifurcation is an overly

conservative criterion for the safe design of pipelines as collapse occurs much later and is

imperfection sensitive.

78

(a)

(b)

Fig. 4.1 (a) Cross section of a lined pipe. (b) Finite element mesh of lined pipe

domain analyzed under bending.

79

Fig. 4.2 (a) Base case moment- and ovalization-curvature responses.

4321

0

0.4

0.8

1.2

0

0.05

0.1

0.15

0.2

0.25

0 0.05 0.1 0.15 0.2

M

Mo

D D

(%)

L

DD

Steel Pipe

Composite

Liner

5

D = 12.75"

tL = 3 mm

80

Fig. 4.2 (b) Liner deformed configurations with superimposed contours of liner

separation from the outer pipe— correspond to numbered bullets on liner

response in Fig. 4.2(a).

81

Fig. 4.3 Liner bifurcation buckling mode with periodic wrinkles on compressed side.

82

(a)

(b)

Fig. 4.4 (a) Finite element model with the length of the model ( 2 ) varied (b) Liner

bifurcation curvature as a function of assumed wrinkle wavelength and

identification of the critical values.

0.96

0.98

1

1.02

1.04

0.2 0.22 0.24 0.26 0.28 0.3

b

1L

tL=3 mm

Dt

~18~

/ R L

D=12.75"

(c,

c)

83

(a)

(b)

Fig. 4.5 Comparison of (a) moment-curvature and (b) ovalization-curvature

responses of shell in a lined pipe and the same shell bent alone. Marked are

the calculated bifurcation points.

0

0.2

0.4

0.6

0.8

1

1.2

0 0.5 1 1.5 2

M

MoL

L

Single Tube

Liner

tL = 3 mm

Dt = 99.4

L

Alloy 825

0

2

4

6

8

10

0 0.5 1 1.5 2

D D

(%)

L

L

Liner

Single TubetL = 3 mm

Dt

= 99.4L

84

(a)

(b)

Fig. 4.6 (a) Critical bending strains as a function of liner shell D/t and (b)

corresponding critical wrinkle half-wavelengths. Included are results for

liner shell and shell alone under bending and axial compression.

0

0.2

0.4

0.6

0.8

1

60 80 100 120

8 10 12 14 16

tL= 3 mm

C

(%)

DL

/ tL

D (in)

Axial

Lined Axial

Bending

Lined Bending

Alloy 825Dt

~18~

0.1

0.15

0.2

0.25

0.3

0.35

60 80 100 120

8 10 12 14 16

R

tL= 3 mm

DL

/ tL

Axial

Lined Axial

Bending

Lined Bending

Alloy 825

D (in)

Dt

~18~

85

(a)

(b)

Fig. 4.7 Imperfect base case responses: (a) moment-curvature and (b) maximum

detachment-curvature.

41

23 65 7

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1 1.2 1.4

M

Mo

Steel Pipe

Composite

Liner

= 0.01

D = 12.75"

tL=3 mm

Dt =17.75

4

321

0

0.04

0.08

0.12

0.16

0 0.2 0.4 0.6 0.8 1 1.2 1.4

RL

5

6

7

= 0.01

Alloy 825

86

Fig. 4.8 (a) Sequences of liner deformed configurations showing evolution of

wrinkling corresponding to numbered bullets on responses in Fig. 4.7

87

Fig. 4.8 (b) Cross sectional view of compressed side of image that illustrates the

shell-type collapse mode.

88

Fig. 4.9 Liner axisymmetric imperfection with the same half wavelength of the

critical bifurcation mode in Fig.4.3.

89

(a)

(b)

Fig. 4.10 Comparison of (a) moment-curvature and (b) maximum detachment-

curvature responses for bifurcation mode and axisymmetric imperfections of

the same amplitude and wavelength.

0

0.02

0.04

0.06

0.08

0.1

0 0.2 0.4 0.6 0.8 1

M

Mo

o=0.01

Axisym. Imperf.

Bifurc. Mode

D = 12.75"

tL = 3 mm

Dt

~18~

0

0.02

0.04

0.06

0.08

0 0.2 0.4 0.6 0.8 1

RL

Axisym. Mode

Bif. Mode

90

(a)

(b)

Fig. 4.11 Effect of bifurcation mode imperfection amplitude on liner response and

stability: (a) Moment-curvature and (b) maximum detachment-curvature

responses.

0

0.02

0.04

0.06

0.08

0.1

0 0.2 0.4 0.6 0.8 1 1.2

M

Mo

o(%)

0.80.2

3.0

D = 12.75"

tL = 3 mm

Dt L

= 99.4

Perfect Liner

Dt

~18~

0

0.04

0.08

0.12

0.16

0 0.2 0.4 0.6 0.8 1 1.2

RL

o (%)

0.80.2

3.0

91

(a)

(b)

Fig. 4.12 Effect of pipe diameter on liner response for a constant liner wall thickness

and imperfection Lo R4102 . (a) Moment-curvature and (b) maximum

detachment-curvature responses.

0

0.04

0.08

0.12

0.16

0 0.4 0.8 1.2 1.6 2

M

Mob

1b

14

10

12

8

D (in)

tL = 3 mm

= 2 x10-4

RL

16

Dt

~18~

0

0.04

0.08

0.12

0.16

0 0.4 0.8 1.2 1.6 2

RL

1b

14

12

10

8

D (in)

16

Alloy 825

92

Fig. 4.13 Comparison of bifurcation and collapse strains as function of liner D/t.

0

0.4

0.8

1.2

1.6

2

2.4

2.8

3.2

60 80 100 120

8 10 12 14 16

(%)

CO

tL= 3 mm

C

Dt ~18~

DL

/ tL

=2 x 10-4

o

RL

Alloy 825

D (in)

93

Chapter 5: LINER WRINKLING AND COLLAPSE OF GIRTH-WELDED

LINED PIPE UNDER BENDING

A pipeline usually consists of 12 m-long length sections, which are girth welded

together. Figure 5.1a shows a photograph and Fig. 5.1b a schematic of a pipe longitudinal

cross section in the neighborhood of a girth weld joining two BuBi® pipes. The ends of

the liners are terminated with a seal weld and a 50 mm overlay weld (see Toguyeni and

Banse, 2012; Shriskandarajan et al., 2013). The ends of the pipes are then beveled to

accommodate the girth weld between two joints (for more on current welding procedures

see Jones et al., 2013). In this set up, the edge of the liner is connected to the outer pipe.

As demonstrated in Chapters 3 and 4, under bending the liner tends to ovalize more than

the thicker carrier pipe and partially detach from it. The girth and overlay welds prevent

the separation of the liner end and in the process, create a local periodic disturbance to

the liner. In this Chapter it will be demonstrated that this disturbance causes wrinkling

that eventually leads to a local collapse of the liner. The numerical framework established

in Chapter 3 is suitably extended and used here to examine the effect of girth welds on

liner collapse and to study its sensitivity to several problem parameters.


A section of expanded lined pipe containing the girth weld is modeled with FEs in

ABAQUS 6.10. The model has overall length 2L, outer diameter D, and carrier and liner

thicknesses t and Lt . The problem is simplified slightly by neglecting the presence of the

overlay weld. Thus the liner is fixed to the outer pipe at x = 0, which in turn becomes a

plane of symmetry (Fig. 5.2). In addition, the problem is symmetric about the plane of

bending, which allows consideration of one-quarter of the structure as shown in the

figure.

94

The steel carrier pipe is meshed with linear 3D elements (C3D8) and the

contacting liner with linear shell elements (S4). Unless otherwise stated, the half-length

of the model will be DL 30.2 . The carrier pipe has four elements through the thickness

and both tubes are assigned 108 elements around the half circumference. The radial

constraint provided by the girth weld is expected to result in a local disturbance.

Consequently, a finer mesh is provided in the axial direction closer to the zy plane of

symmetry and coarser ones away from this zone as follows:

{ Dx 46.00 , 56 elements},

{ DxD 61.146.0 , 70 elements},

{ DxD 30.261.1 , 30 elements},

The girth weld is modeled by tying the nodes of the shell at 0x to the

corresponding nodes of the innermost solid elements. Contact between the two layers

plays an important role in the problem so it is modeled using the finite sliding option of

ABAQUS with no friction, and the carrier pipe as the master surface and the liner as the

slave surface (The effect of friction was considered and found to be small). The model is

bent by prescribing incrementally the rotation of the plane at Lx . As in Chapter 3, the

end plane is constrained to remain plane, while the cross section is free to ovalize. The

moment induced at 0x is evaluated.

The simulation of bending starts with the mechanical property changes and

residual stresses induced by the expansion process in place (see Chapter 2). The bending

is performed using the same isotropic hardening model used to expand the composite

structure.

95

5.2 WRINKLING AND COLLAPSE OF A GIRTH-WELDED PIPE

The main characteristics of the problem will now be illustrated using the same 12-

inch lined pipe in Chapter 3 designated as the base case. The pipe has the geometric and

material parameters listed in Table 3.1. Figure 5.3a shows the moment-curvature ( M

) response of the composite pipe together with those of the steel carrier pipe and the liner

individually ( tDM ooo2 and 2

1 / oDt , tDDo are based on the carrier pipe

parameters). Figure 5.5 shows selected deformed configurations of the liner with

superimposed color contours that represent the magnitude of the separation of the liner

from the carrier pipe (for clarity the full model is shown). Bending plasticizes both tubes

as evidenced by the responses. Simultaneously, both tubes ovalize with the liner

ovalizing more. Thus, the initial contact stress between the two tubes gets gradually

reduced and eventually the liner partially separates from the steel pipe (as presented in

Chapter 3). The girth weld prevents this differential deformation of the liner and the

constraint causes an axially periodic disturbance at the upper and lower ends of the liner.

The evolution of the disturbance caused by the weld is illustrated in Fig. 5.4 that

shows plots of the separation, w, of the most compressed generator of the liner (at the top

of the model in Fig. 5.2, i.e., at 0 ) from the steel pipe at different values of curvature

(girth weld at 0x , RL mid-surface radius of liner). The disturbance takes the form of

of periodic axial wrinkles with exponentially decaying amplitude, which is reminiscent of

other similar boundary effects caused in thin shells by constraints or point loads (see

Yuan, 1957). For convenience, the axial distance x is also normalized by the wavelength

characteristic variable RLtL . It is noteworthy that the wavelength differs from that of

bifurcation wrinkling mode of the liner away from the constraint. Thus here the

wavelength is 1.93 LLtR and for the bifurcation wavelength calculated is 1.73 LLtR ,

(see Section 4.2.1). For comparison the multiplier of LLtR for pure bending of the liner

96

liner alone is 1.80, and for axial compression of the elastic liner 1.73. The largest

amplitude, max , occurs adjacent to the weld and is also plotted vs. curvature in Fig.

5.3b.

The evolution of the wrinkles is also depicted in the color contours on the

deformed configurations in Fig. 5.5. Thus, in image based on the scale used, only the

most deformed wrinkles next to the “weld” are visible. In image the major wrinkles

deepen (see also max in Fig. 5.3b) and the ones next to it become discernible. In image

, at a somewhat higher curvature, separation increases and a number of additional

wrinkles become evident. Soon after image , the liner reaches a moment maximum at a

curvature of 1716.0 . Beyond this point, deformation localizes (image ) and the

growth of the most deformed wrinkles accelerates as evidenced by the upswing in max

in Fig. 5.3b. Simultaneously, a diamond-type buckling mode characterized by a number

of circumferential waves appears for the first time. It is seen initially in image and

more prominently in , where the detachment is plotted with different color scales due to

the significant increase in amplitude. A three-dimensional rendering of the buckled liner

at a curvature of 103.1 is shown in Fig. 5.6. Remarkably, the collapse mode is very

similar to the one calculated for the mother lined pipe (see Fig. 3.8), and is also similar to

liner buckle images recorded in full-scale bending experiments reported in Hilberink et

al. (2010, 2011) and Hilberink (2011). The significant amplitude of such liner buckles

can render the structure non-operational and the sharp curvatures can be sources of

failure or fatigue fractures. As was the case for the mother lined pipe, we will define the

curvature at the moment maximum and the associated sharp upswing in the separation

between the two tubes as the critical collapse curvature designated by CO . It is

important to note that, at this curvature, the outer pipe although plastically deformed is

free of buckles and in perfect operational condition.

97

In summary, the events reported here are similar to those observed in Chapter 3

and 4 for the mother structure free of welds. There the wrinkling in the ideal geometry

appears through a bifurcation, and in the actual structure, is excited by small initial

geometric imperfections. By contrast, in the neighborhood of a girth weld, wrinkling is

excited by the constraint provided by the weld. In both situations, the amplitude of

wrinkles grows and at some point deformation localizes, the growth of the local

amplitude of wrinkles accelerates and, simultaneously, a diamond-type buckling mode

develops that leads to catastrophic collapse of the structure.

5.3 EQUIVALENT IMPERFECTION OF UNCONSTRAINED LINED PIPE

In our study of lined pipe free of the constraining effect of welds, wrinkling and

collapse were established for liner imperfections combining an axisymmetric mode and a

shell-type mode with m circumferential waves as follows:

2)/(01.0cos

2coscos

Nx

moL mxx

tw

(5.1)

where the variables take the meaning defined under Eq. (3.5). It was earlier demonstrated

that the collapse curvature of the liner is significantly dependent on both imperfection

amplitudes and much less on m. Furthermore, comparison of collapse curvatures

calculated using the actual buckling mode and imperfections like the one in Eq. (5.1)

found them to be very similar when the same amplitude and wavelength are used. With

this as background, we will use this type of imperfection and the FE model of Chapter 3

to explore the combination of amplitude levels required to collapse the liner at the same

curvature as in the welded system. The reader is reminded that in the case of the girth

welded lined pipe the disturbance is provided by the weld and is fixed. In both models the

98

structure is assigned the prehistory and residual stresses introduced by the manufacturing

process (see Section 2.1).

Table 5.1 Collapse curvatures for various combinations of imperfection amplitudes o

and m .

1/CO

o

m 0.006 0.01 0.02

0.01 - - 0.727 0.02 - - 0.704 0.03 - 0.719 - 0.04 0.719 0.704 - 0.05 0.692 - -

One set of comparisons appears in Fig. 5.7 that shows the moment-curvature

response of the welded case (the moment is truncated). The collapse curvature, marked

on the response with a “^”, occurs at 1716.0 . Included are the responses of imperfect

liners with 02.0o and two values of m : 0.01 and 0.02. Their collapse curvatures

marked on the responses with “”, are seen to span the value of the welded case (values

listed in Table 5.1). These combinations of imperfection amplitudes were chosen from a

wider imperfection sensitivity study of collapse curvatures to serve the purpose of this

comparison. Of course these values are not unique and this is demonstrated in Table 5.1

that lists additional combinations of imperfection amplitudes that produce collapse

curvatures that straddle the 1716.0 value of the girth-welded case. The results in Fig.

5.7 and Table 5.1 provide measures of the severity of the disturbance provided by the

girth weld.

99


In this section we examine the effect of additional problem parameters on the

response and collapse of a liner in the neighborhood of a girth weld.


In Chapter 2 it was reported that the initial annular gap, og , between the

undeformed liner and outer pipe (see image in Fig. 2.1a), influences the residual

contact stress between the two pipes following the expansion. This in turn affects the

response and stability of the liner under bending (see Fig. 3.16).

In this section, expansion simulations were conducted again for four values of

obo gg / ( obg is the value used for the base case simulation in Section 5.2). The stress

histories were introduced to the girth-welded model in Fig. 5.2 and subsequently the

composite pipe models were bent. The calculated M responses are shown in Fig.

5.8a and the associated max responses in Fig. 5.8b. Qualitatively the results are

similar to those of the base case. However, as in Chapter 3, increasing the annular gap

increases the moment carried by the liner and simultaneously decreases the curvature at

the moment maximum, i.e., CO . These trends are directly related to the additional

deformation and strain hardening induced by the expansion process. The results

demonstrate that keeping the size of the annular gap og as small as possible can result in

direct increase in the curvature to which the girth-welded lined pipe can be bent. This

conclusion is similar to the one drawn for lined pipe free of welds. It is put forward

realizing that physical and manufacturing limitations exist on the extent to which this

guideline can be followed.

100

5.4.2 Pipe Diameter

In the case of seamless pipes of up to a diameter of 16 inch, it is an industry

standard not to increase the thickness of the liner with outer pipe diameter. Hence, the D/t

of the liner tends to increase as the pipe diameter increases. For this reason, pipe diameter

can influence the collapse of the liner and should be examined. To evaluate this effect for

girth-welded pipe we consider five pipe diameters: 8.625, 10.75, 12.75. 14, 16 inches.

The D/t of the five pipes is kept at approximately 18.1 and the liner thickness at 3 mm.

Each system is appropriately expanded, a weld is introduced to the model as in Fig. 5.2,

and the FE model is subsequently purely bent.

Figures 5.9a and 5.9b show the calculated M and corresponding max

responses respectively (the normalizing variables obM and b1 are based on the

parameters of the base case listed in Table 3.1). As the diameter of the composite

structure increases, the basic behavior remains the same: the constraint provided by the

weld results in wrinkling adjacent to it, the wrinkles grow and lead to a moment

maximum. Close to the moment maximum, the shell mode of buckling is excited and the

liner collapses in the manner illustrated in the configurations of Fig. 5.5. Increasing the

diameter of the composite structure increases the diameter of the liner and, since Lt is

fixed, its diameter-to-thickness ratio increases. Consequently, the moment carried by the

pipe increases. However, as evidenced by the results in Fig. 5.9, the collapse curvature

decreases primarily because of the increase in LL tD / .

5.4.3 Bending Under Internal Pressure

In Chapter 3 it was demonstrated that under pure bending even small values of

internal pressure reduce the ovalization of the liner and delay its separation from the

carrier pipe, which has a corresponding increase in the collapse curvature of the liner (see

also Endal et al., 2008; Toguyeni and Banse, 2012; Mair et al., 2013; Howard and Hoss,

101

2011). A similar study was performed here for lined pipe with a girth weld. The

composite system analyzed corresponds to the 12-inch pipe in Table 3.1. The composite

structure is again first expanded and then purely bent under internal pressure levels of: 0,

50, 75, 100, 150 psi (0, 3.45, 5.2, 6.9 and 10.35 bar). Figures 5.10a and 5.10b show the

calculated M and corresponding max responses respectively. The behavior is

qualitatively the same as that of the unpressurized pipe in Section 5.2. However, the

results in Fig. 5.10b clearly show that even such modest levels of internal pressure delay

the separation of the liner from the outer pipe. Figure 5.11 shows plots of the separation,

w, of the most compressed generator of the liner from the steel pipe in the neighborhood

of the weld at a curvature of 168.0 for three levels of pressure: 50, 75 and 100 psi (3.45,

5.2 and 6.9 bar). The corresponding plot for zero pressure appears in Fig. 5.3, however

notice the significantly smaller scale of Ltw / adopted in Fig. 5.11. Clearly, pressure

suppresses the growth of the periodic disturbance induced by the weld. At the curvature

considered, at 50 psi the disturbance is significantly smaller than in image in Fig. 5.3

at zero pressure. At 75 psi the disturbance has reduced significantly and at 100 psi it is

barely discernible. As a result of this suppression of the weld-induced disturbance, and

generally of liner separation from the outer pipe, delays significantly the moment

maximum and the onset of collapse. In summary, internal pressure delays the collapse of

the liner in the neighborhood of a girth weld.


As in all instabilities of thin-walled structures, the wall thickness of the liner plays

a decisive role on its stability under bending and deserves special consideration (Tkaczyk

et al., 2011). In Section 3.6, it was demonstrated that, as expected, increasing the wall

thickness delays the onset of liner collapse for the mother pipe. The effect of liner wall

102

thickness on the stability of a lined pipe with a girth weld has also been examined in the

present study using the basic parameters of the 12-inch composite system in Table 3.1.

The liner thickness is varied between 2.0 and 4.5 mm. The composite system is first

expanded and then purely bent. The calculated liner moment- and maximum detachment-

curvature responses for six wall thicknesses are shown in Figs. 5.12a and 5.12b

respectively. The behavior of the composite structures is qualitatively similar to that of

the base case in Fig. 5.3. The constraint of the girth weld causes a periodic disturbance

with the largest amplitude adjacent to it, and the amplitude decaying away from it. Figure

5.13 shows axial plots of the disturbance for three of the liner thicknesses, with the

distance x being measured from the weld. The wavelength is proportional to LLtR and

thus when the ordinate is normalized by LR , which is essentially constant, the

wavelength is seen to increase. Increasing the liner thickness increases the moment

carried by the liner and simultaneously delays the onset of liner collapse. In other words,

results are as expected and in line with those for lined pipe free of girth welds. However,

since the cost of lined pipe is significantly influenced by the material cost of the non-

corrosive liner, the increase in collapse curvature caused by an increase in Lt

demonstrated here must be weighed against the associated increase in the cost of the

product. Calculations like the present ones and the ones in Chapter 3 can help develop a

cost-performance analysis to select the optimal liner thickness for a given application.

5.4.5 Overlay Seal Weld

The ends of the liners are terminated with a seal weld and a 50 mm overlay weld

(see Fig.5.1). To evaluate the effect of such welds on the collapse of liner, we consider a

section of expanded lined pipe that includes an overlay seal weld. The evolution of the

disturbance is found to be the same as the case without the overlay weld: the weld creates

103

an axially periodic disturbance to the liner; the wrinkling grows with increasing bending

and the liner eventually collapses with a diamond-shaped mode. The calculated M

and corresponding max responses are compared with the corresponding ones from

the case without the overlay in Fig. 5.14. Clearly, the inclusion of overlay seal weld has a

negligibly small effect on the onset of collapse and the subsequent response of the liner.

104

(a)

(b)

Fig. 5.1 Typical girth weld of lined pipe with 50 mm overlay seal weld.

(a) Photograph of an individual weld and (b) schematic of pipe section.

105

Fig. 5.2 Finite element mesh of composite pipe under bending.

106

(a)

(b)

Fig. 5.3 (a) Moment-curvature of composite pipe and of individual tubes. (b)

maximum detachment-curvature response.

5 64321 7

0

0.2

0.4

0.6

0.8

1

1.2

0 0.4 0.8 1.2

M

Mo

Girth Welded

D = 12.75 in

tL = 3 mm

= 18.1Dt

Steel Pipe

Composite

Liner

432

5

6

7

0

0.04

0.08

0 0.2 0.4 0.6 0.8 1 1.2

max

RL

1

107

Fig. 5.4 Liner detachment along compressed pipe generator close to the weld (x = 0).

0

0.1

0.2

0.3

0 1 2 3

0 4 8 12 16 20

w(0)

tL

x / RL

1 0.34

0.51

0.60

0.68

2

3

4#

tL = 3 mm

= 18.1Dt

D = 12.75 in

x / (RLtL)0.5

108

Fig. 5.5 Sequence of liner configurations corresponding to responses in Fig. 5.4

(contours detachment).

109

Fig. 5.6 Three-dimensional rendering of the buckled liner.

110

Fig. 5.7 Moment-curvature responses of girth-welded liner and ones with imperfections that yield approximately similar

collapse curvatures.

0.04

0.08

0.12

0 0.2 0.4 0.6 0.8 1

M

Mo

D = 12.75 in

tL = 3 mm

= 18.1Dt

Girth Weldedm = 8, o = 0.02

m

0.02 0.01

111

(a)

(b)

Fig. 5.8 Effect of initial annular gap on welded liner response: (a) moment-curvature

and (b) maximum detachment-curvature responses.

0

0.04

0.08

0.12

0.16

0 0.2 0.4 0.6 0.8 1 1.2

M

Mo

0.51

1.52

go

gob

D = 12.75 in

tL = 3 mm

= 18.1Dt

Girth Welded

0

0.04

0.08

0 0.2 0.4 0.6 0.8 1

0.511.52

go

gob

max

RL

112

(a)

(b)

Fig. 5.9 Effect of pipe diameter on liner response for constant Lt : (a) moment-


0

0.04

0.08

0.12

0.16

0.2

0 0.4 0.8 1.2 1.6 2

M

Mob

1b

14

10

12

8

D (in)

tL = 3 mm

Dt ~18~16

0

0.04

0.08

0 0.4 0.8 1.2 1.6 2

max

RL

1b

14

12

10

8

D (in)

16

113

(a)

(b)

Fig. 5.10 Effect of internal pressure on welded liner response:(a) moment-curvature

and (b) maximum detachment-curvature responses.

0

0.04

0.08

0.12

0.16

0 0.4 0.8 1.2 1.6

M

Mo

D = 12.75 in

tL = 3 mm

= 18.1Dt

Girth Welded150(10.35)

100(6.9)

0P psi (bar)

50 (3.45)

75 (5.18)

0

0.04

0.08

0 0.4 0.8 1.2 1.6

max

RL

150(10.35)

100(6.9)

050 (3.45)

P psi (bar)

75 (5.18)

114

Fig. 5.11 Liner detachment along compressed pipe generator close to the weld for different values of internal pressure.

0

0.01

0.02

0 1 2 3

0 4 8 12 16 20

w(0)

tL

x / RL

=0.68 t

L = 3 mm

= 18.1Dt

D = 12.75 in

x / (RLtL)0.5

50 (3.45)

75 (5.18)

100(6.9)

P psi (bar)

115

(a)

(b)

Fig. 5.12 Effect of liner wall thickness on welded liner response: (a) moment-


0

0.04

0.08

0.12

0.16

0.2

0 0.4 0.8 1.2

M

Mo

D = 12.750 in

tL (mm)

3

3.5

4

4.5

Girth Welded

2.5

2

= 18.1Dt

0

0.04

0.08

0 0.4 0.8 1.2

tL (mm)

3 3.5

4 4.5

max

RL 2.5

2

116

Fig. 5.13 Normalized liner detachment along compressed pipe generator close to the weld for different liner thicknesses.

0

0.5

1

0 0.4 0.8 1.2 1.6 2 2.4

w(0)

maxw

x / RL

= 18.1Dt

D = 12.75 in

3 4.52

tL (mm)

Girth Welded

117

(a)

(b)

Fig. 5.14 Effect of overlay weld on liner response: (a) moment-curvature and (b)


0

0.04

0.08

0.12

0.16

0 0.2 0.4 0.6 0.8

M

Mo

Overlay Weld

Girth Weld

D = 12.75 in

tL = 3 mm

= 18.1Dt

0

0.01

0.02

0.03

0 0.2 0.4 0.6 0.8 1

RL

Overlay Weld

Girth Weld

118

Chapter 6: LINER WRINKLING AND COLLAPSE OF LINED PIPE UNDER

AXIAL COMPRESSION

Axial compression is another loading that can lead to liner wrinkling and collapse.

As outlined in Section 1.3, compression severe enough to plasticize the line can develop

in buried pipelines by the passage of hot hydrocarbons. Pipelines are also compressed

when crossing an active fault, by ground subsidence, by foundation liquefaction in

earthquake prone areas, etc. (e.g., see Chapters 11 and 12 in Kyriakides and Corona,

2007). Under high enough compressive strain, the liner can wrinkle and eventually

collapse, causing similar operational disruptions as the related bending failures described

in the previous chapters. Compression of lined pipe, although of equal practical

importance, has received much less attention in the literature than bending.

This chapter investigates the extent to which typical lined pipe can be axially

compressed before liner collapse. Demonstration experiments on model lined systems are

first used to illustrate the wrinkling and collapse of the liner. The problem is then

modeled numerically starting with the introduction of the prehistory induced by the

manufacturing process described and analyzed in Chapter 2. As is the case in the bending

problem, compression of the composite model pipe leads to stable growth of the wrinkles

at first, and then to catastrophic diamond-type shell buckling modes at much higher strain

levels. The evolution of these events up to collapse are carefully monitored and reported.

The sensitivity of the collapse strain to various parameters of the problem is studied, and

several methods for delaying failure are evaluated.

6.1 DEMONSTRATION COMPRESSION EXPERIMENTS

Some compression experiments on lined pipe have been reported in Focke et al.

(2011). In the most relevant test to our study, a composite pipe consisting of a 10-inch X-

119

65 carbon steel pipe with a D/t = 29.4, lined with a 2.45 mm SS-304L liner was

compressed under displacement control between stiff platens. The composite structure

was compressed until the outer tube buckled. Diamond buckling patterns were reported to

have developed in the liner, but the evolution of events leading to the liner collapse was

not delineated.

The absence of dependable experimental data from compression experiments is

partly due to the large load required to test actual lined cylinders. To enhance the

understanding of the problem, we conducted a series of demonstration experiments on

model composite cylinders that could be fabricated and tested in a laboratory

environment. The model structures consist of a relatively thick epoxy shell cured around

a 2.5 in (63.5 mm) stainless steel liner shell with a wall thickness of about 0.020 in (0.5

mm). The composite pipe is compressed between platens introducing the same strain to

both components. More details about the liner and epoxy shell, their material properties,

and the fabrication process are given in Appendix D, where the results of one of the

experiments conducted are also outlined.

It is important to observe that because of the combination of their geometric and

material parameters, shells used to line typical pipelines buckle in the plastic range. This

was indeed the case for the demonstration experiment described in Appendix D. Thus,

the sequence of events that were observed is in essence along the lines of that followed

by plastic buckling of a circular cylindrical shell alone (e.g., see Tvegaard, 1983; Yun

and Kyriakides, 1990; Kyriakides et al., 2005; Bardi and Kyriakides, 2006; Bardi et al.,

2006; Kyriakides and Corona, 2007). The liner shell buckles first into an axisymmetric

mode at an increasing load. The amplitude of the axisymmetric wrinkles is initially very

small but grows with increasing compression. At a higher strain, the liner buckles a

second time into a non-axisymmetric diamond-type mode, which leads to collapse of the

120

liner inside a usually intact outer tube. Both instabilities are sensitive to small initial

geometric imperfections in the liner. Figure. 6.1 shows a photograph of a 225-degree

sector of the composite cylinder at the end of the experiment after undergoing a

shortening of 2.4%. Protruding inwards are diamond-shaped buckling patterns in the

stainless steel liner with 5 circumferential waves (m = 5) while the outer epoxy shell

remained essentially intact.

The first axisymmetric bifurcation of a liner confined in an outer cylinder of the

same properties was established in Peek and Hilberink (2013) (see also Shrivastva, 2010).

Their developments are outlined in Appendix B and the critical stress C and half-

wavelength of the mode are given in Eq. (B.9). The demonstration experiments have

shown that as was the case for bending (see Chapter 3), the two governing instabilities of

the problem are separated by significant strain. Thus the present study uses a more

elaborate numerical model to understand the postbuckling behavior from the first

bifurcation to final collapse of the liner and the factors that influence collapse.


The problem is analyzed using a FE model developed in ABAQUS 6.10 shown in

Fig. 6.2. The expansion is first simulated as in Chapter 2, thus capturing the induced

changes in the mechanical properties and residual stresses. Subsequently, these are

introduced as initial conditions to the structural model. The model involves a section of

the composite pipe of length L2 , outer diameter D , and carrier and liner wall thickness

of t and Lt respectively. For numerical efficiency, symmetry about the mid-span is

assumed (plane 0x ). The steel carrier is meshed with linear 3D elements (C3D8) and

the contacting liner with linear shell element (S4). Unless otherwise stated, the half-

length of the model will be 12L , where is the half wavelength of the characteristic

121

characteristic axisymmetric geometric imperfections that are introduced to the liner.

Motivated by the experimental observations, the imperfection used consists of an

axisymmetric and a non-axisymmetric component with m circumferential waves as

follows: 2)/(01.0cos

2coscos

Nx

moL mxx

tw

(6.1)

where corresponds to the critical state of the perfect lined structure in Eq. (B.9). Here a

purely sinusoidal axisymmetric component is chosen over the actual bifurcation mode for

easier definition of the imperfection. It turns out that this difference does not affect either

the response or the collapse strain in any significant way.

The carrier pipe has four elements through the thickness and both tubes are

assigned 240 elements around the half circumference. Imperfection (6.1) has a bias

towards the plane of symmetry. So in anticipation of the expected localization of

buckling and collapse in this neighborhood, this area has a finer mesh as follows:



{ 128 x , 20 elements},

Contact between the two layers plays an important role in the problem, so it is

modeled using the finite sliding option of ABAQUS with the carrier pipe as the master

surface and the liner as the slave surface. The effect of friction is given special

consideration, but for a significant part of the study contact is assumed to be frictionless.

The model is compressed by prescribing incrementally the displacement of the plane

Lx while constraining this end to remain in a plane normal to x. The force at 0x is

evaluated by summing the nodal forces on this plane.

122

The geometric imperfections are introduced to the liner in the initial stress free

state. As reported in Chapter 3, the expansion process has the result of altering the shape

and reducing the amplitude. The resultant changes are illustrated in Fig. 6.3, which shows

comparisons of the initial and final imperfections geometries for 05.0 mo and

8N . In Fig. 6.3a the expansion is seen to have reduced the amplitude of the

axisymmetric imperfection at mid-span by nearly 40%. Figure 6.3b shows the amplitude

of the non-axisymmetric imperfection at the mid-span for 8m to have been reduced by

nearly 60% and the contact with the outer pipe to have increased. Naturally, the changes

induced to the imperfection geometry by the expansion process depend on the

imperfection itself but also the geometric and material properties of the two tubes. Thus

for consistency, unless otherwise stated, the imperfection amplitudes that will be reported

will be the initial values.

6.3 RESULTS

6.3.1 Wrinkling and Collapse of a Representative Lined Pipe

The response and stability of a 12-inch system to axial compression are now

examined in some detail. The outer pipe is an X65 line-grade steel with a nominal yield

stress ( o ) of 65 ksi (448 MPa) and 09.18/ tD . The contacting liner is made out of

alloy 825 with a nominal yield stress of 40 ksi (276MPa) and a 10.96/ tD . A

Poisson's ratio ( ) of 0.3 is assigned to both tubes. The dimensions of the two

components after expansion are listed in Table 6.1. It is worth noting that since both

tubes are plastically deformed in the expansion process their apparent properties at

compression are altered to some extent.

123


D in† (mm)

t in† (mm) t

D

E Msi* (GPa)

o ksi*

(MPa)

Steel Carrier X65

12.75 (323.9)

0.705 (17.9)

18.09 30.0 (207)

65.0 (448)

0.3

Liner Alloy 825

11.34 (288.0)

0.118 (3.0)

96.10 28.7 (198)

40.0 (276)

0.3


The liner was assigned an initial imperfection with 04.0o , 008.0m and

8N . A parametric study demonstrated that for this combination of geometric and

material properties the preferred value of m is 2, which was adopted. The critical value of

the half wavelength yielded by (B.9) for this case is LR23.0 . Figure 6.4a shows the

calculated compressive force-shortening response ( xF ) of the composite structure as

well as those of the individual tubes. Here the force is normalized by the yield force of

the steel carrier pipe alone oF ( Ao ) and the shortening by the initial length of the

model, L; thus Lx / also represents the average compressive strain. During compression

the axisymmetric imperfection is at some point excited, developing wrinkles that separate

from the outer pipe. Figure 6.4b shows the detachment, 0 , of the most deformed

wrinkle at 0x vs. the shortening ( LR is the radius of the liner). The evolution of the

wrinkles is portrayed in Fig. 6.5, which shows nine deformed configurations

corresponding to the solid bullets marked on the liner responses in Fig. 6.4. with

superimposed color contours representing the liner separation, w . For clarity, the

images are grouped under three different scales of the separation.

Following an initial stiff elastic response, the composite structure yields at a strain

of about 0.25%, with both shells starting to deform plastically. Image just after

yielding shows the liner to be free of wrinkles (based on the color scale used). The

124

response of the composite is dominated by that of the carbon steel, which continues to

harden and thus the force continues to increase up to a strain higher than 1.6%. By

contrast, the response of the liner traces a relatively flat load plateau. The perfect liner

bifurcates into the axisymmetric mode at a strain of 0.60%. Thus for the imperfect case

examined here, a wrinkle has appeared at the plane of symmetry in image at a strain of

0.57%. It is interesting that on the scale of this image, this first wrinkle only covers part

of the circumference. This is because the local amplitude of the imperfection is reinforced

by the 2m component. At higher strains of 0.72% for image , 1.0% for image

and 1.30% for image , the number of axisymmetric wrinkles increases covering now

the full circumference. Their amplitude also increases and this is also reflected in the

gradual increase of the amplitude of the central wrinkle in Fig. 6.4b. In image and

the non-axisymmetric component of the imperfection gets excited and deformation

localizes in the neighborhood of the plane of symmetry. This causes additional loss of

stiffness of the liner and at an average strain of 1.75%, its force reaches a maximum value

(marked in Fig. 6.4a with a caret "^"). Beyond this point the deformation is localizing in

the neighborhood of the plane of symmetry resulting in the sharp upswing in the value of

0w observed in Fig. 6.4b. Beyond the load maximum the non-axisymmetric mode

becomes dominant and the liner starts to collapse. The collapse mode has a butterfly

shape with a major wrinkle at the plane of symmetry, surrounded by four satellite ones

(see image and ); this collapse pattern repeats diametrically opposite to the viewing

plane. By point at a strain of 2.22%, the maximum inward deflection approaches 7%

of the liner radius, a value high enough to render the structure out of service, even though

the carrier pipe remains intact.

We will define the average strain corresponding to load maximum and the sharp

upswing in the separation between the two tubes as the critical collapse strain. It is

125

reassuring that this sequence of events are qualitatively in quite good agreement with the

observation made in the demonstration experiments outlined in Section 6.1 and in

Appendix D.

6.3.2 Imperfection Sensitivity of Liner Collapse

Seamless pipe produced either by the plug or mandrel mill process (e.g., see

Kyriakides and Corona, 2007) leaves behind an internal surface relief that is related to the

piercing, rolling and external finishing of the product. When the thin liner is plastically

expanded against this surface, the relief comes through and acts as initial imperfection.

Scanning of the internal surface of the liner reported in Harrison et al. (2015), has

revealed that this surface relief has a Fourier content with characteristic circumferential

and axial waves specific to the manufacturing process of the seamless outer pipe. Here

we perform a limited evaluation of the effect of the imperfection variable o , m and m

on the collapse strain for the base case lined pipe variables in Table 6.1. The axial

wavelength is the one corresponding to the critical one from Eq. (B.9), the length of

model continues to be 12 , and 8N in Eq. (6.1).

In this spirit, Fig. 6.6 shows sets of axial force ( oL FF / )- and maximum

detachment ( LR/)0( )- average axial strain responses for several values of o for fixed

fixed values of m and m. In all cases, the liner yields rather abruptly and follows a

rather flat stress plateau up to the point of collapse. The detachment of the most deformed

wrinkle at 0x initially grows gradually with strain and picks up abruptly when the

mode-two collapse is approached. Associating again the strain at the load maximum and

the corresponding point at which the liner detachment experiences significant sudden

growth with collapse, it is clear that collapse is extremely sensitive to this imperfection,

as indeed was the case for bending (see Fig. 3.9). This point is further emphasized

126

realizing that Lt06.0 , i.e. the axisymmetric imperfection amplitude before expansion,

corresponds to 0.18 mm, a value that is within the range of measured internal

imperfections in liners. For the combination of variables of this parametric study this

imperfection resulted in 32% reduction in collapse strain. It is interesting to point out that

the small initial drop in force recorded in all xF responses is purely a plastic effect

related to the liner prehistory.

The amplitude of m was varied in a similar manner keeping o and m constant.

Figure 6.7 shows the corresponding sets of results for 04.00 m . The behavior is of

course very similar to that in Fig. 6.6, as the collapse strain exhibits a similar sensitivity

to the amplitude of the non-axisymmetric imperfection as that of o . Quantitative plots

of the average strain at collapse, CO , vs. the two imperfection amplitudes appears in

Fig. 6.8. The sensitivity to both is about the same. Although here the collapse strain

appears somewhat more sensitive to the non-axisymmetric imperfection, and the opposite

was the case for bending (see Fig. 3.11), overall the sensitivity to imperfections in the

two problems is similar, and more importantly very significant. Included in the figure is

the critical bifurcation strain for the perfect structure ( %60.0C ) evaluated via Eq.

(B.9). Its value can be seen to be significantly lower than the collapse strain values for all

imperfections amplitudes considered.

The wave number of the non-axisymmetric imperfection is considered next,

which entails varying the value of m adopted in Eq. (6.1). Figure 6.9 shows force- and

maximum detachment-average axial strain responses of the liner for five values of m

from calculations based on the base case parameters (Table 6.1) and for fixed values of

o and m . The results show that collapse is mildly influenced by the number of

circumferential waves in the imperfection, with 2m resulting in the lowest collapse

strain. This prompted adoption of 2m in the calculations performed for the base case.

127

6.3.3 Effect of Friction on Liner Collapse

For the cases shown thus far friction was not considered. The effect of friction on

the problem was considered in a separate study, which started with the expansion process

and was followed by compression of expanded lined tubes with various geometric

imperfections. The main conclusions of this study can be summarized as follows:

a. Friction does not play a significant role in the mechanical expansion of the two tubes.

The main influence of the expansion on the liner collapse under compression is

through the changes it introduces to the mechanical properties of the liner.

b. Friction has some effect on the stability of the liner of expanded pipe under

compression. This effect is illustrated in Fig. 6.10, which shows calculated liner

force- and maximum detachment-average axial strain responses for the base case

parameters for four values of Coulomb friction. Collapse is seen to be somewhat

delayed by friction. Furthermore, friction can influence the extent of collapse and

make it more localized. However, the overall conclusion is that, at least for the

idealized way that liner imperfections are introduced in this study, the effect of

friction is modest, does not change the behavior sufficiently and can be neglected.


In the preceding section liner buckling and collapse under compression was

demonstrated through a composite tube consisting of a 12-inch outer pipe with 18tD

and a 3 mm thick liner. In this section we present results from a wider parametric study

that considers other factors that can influence the collapse of the liner. This includes

aspects of the manufacturing, consideration of other composite system diameters, the

effect of liner wall thickness, and the effect of internal pressure during compression.

128


The expansion process through which the liner and the carbon steel pipe are

brought into contact was shown in Section 3.6.2 to influence the curvature at which the

liner collapses under bending. In particular, the initial annular gap between the two tubes,

og , was shown to either delay or accelerate collapse. We thus start by simulating once

more the expansion process of the base case system (Table 6.1) but vary the magnitude of

the initial annular gap to four values of og : {0.5, 1, 1.5, 2} obg , where obg is the gap

used in the simulation of the base case (see Table 6.1).

We subsequently compress each expanded system and monitor the response of the

liner. As noted earlier, the expansion process also influences the final amplitude of the

geometric imperfection. For a more objective comparison, the initial values of o and

m used in each calculation were varied so that the final maximum amplitude of the

imperfection was Ltw 0245.0 . Figure 6.11 shows the liner force-average axial strain

and the corresponding maximum detachment-average axial strain responses for the four

values of og . Because of the additional hardening of the liner by the increased straining

with og , plastic deformation occurs at a higher stress, but the collapse of the liner occurs

earlier. It is interesting to observe that the collapse strain of the largest gap is about 50%

lower than that of the smallest gap, a result that was similar for the bending problem. This

sensitivity of liner collapse to og for both compression and bending suggests that, to the

extent that is practically feasible, the initial gap between the two tubes should be

minimized. For this to be achieved tighter tolerances on tube straightness and roundness

are required.

6.4.2 Pipe Diameter

We next consider lined pipe systems of four different diameters, while keeping

the D/t at approximately 18.0. Furthermore, as is mainly the practice, the liner thickness

129

is kept constant at 3 mm. Compressive responses from outer pipes with diameters of

8.625, 10.75, 14.0 and 16 in. (designated as 8, 10, 14, 16 in) are compared to those of the

12-inch pipe analyzed in Section 6.3.1. The mechanical properties assigned to the two

tubes are those in Table 6.1. Changing D has a corresponding change of the liner LL tR .

Each composite system is assigned similar imperfections (Eq. (6.1)) but with the value of

yielded by the bifurcation analysis for the new liner dimensions (Eq. (B.9)). It turns

out that 2m remains the critical circumferential wave for all pipe dimensions

considered and so it is adopted in this set of calculations. Each system is appropriately

expanded as described in Chapter 2. The expansion process alters the initial geometric

imperfections to differing degrees for each D so the amplitudes of o and m were

varied so that after expansion the maximum value of the imperfection, max|LRw , was

the same for all cases, 310516.0 .

Figure 6.12 shows axial force and maximum detachment in the liner vs. average

strain results for the five pipe diameters. In order for the axial forces to appear in their

natural order, they are all normalized by obF , the yield force of the 12-inch outer pipe

base case. The overall behavior of the liners is similar to that of the 12-inch base case but

with some important differences. First, as expected, as the diameter of the composite pipe

increases the force carried by the liner increases. Second, and more importantly, the strain

at collapse decreases. This more unstable nature of the liner is the direct result of the

increase of LL tR with D. The collapse strain decreases by about 50% when the

diameter goes from 8.625 in. to 16.0 in. The corresponding drop in the collapse curvature

(or bending strain) under bending for approximately similar levels of imperfections is

about 40%.

130


In Chapter 3 we showed that, as expected, the wall thickness of the liner plays a

decisive role on its stability under bending (see also Tkaczyk et al., 2011). Here we

investigate its role on the axial compression problem using 12-inch composite pipe like

the one in Table 6.1 but assign the liner thickness six values between 2.0 mm and 4.5

mm. The annular gap is kept the same and so are the mechanical properties. The liner is

assigned initial geometric imperfections as defined in Eq. (6.1) with the wavelength

yielded by the bifurcation analysis in (B.9). The circumferential wave number 2m

proved again to yield the lowest collapse strains. Each composite system was expanded in

the same way. The imperfection amplitudes were chosen so that the post-expansion

values of the amplitudes were similar for all the six cases ( 3max 10516.0| LRw ).

Each composite system was then compressed and Fig. 6.13 shows the resultant force and

maximum detachment vs. average axial strain responses. Qualitatively the general

behavior of each system is similar to that of the 3 mm base case. However, as the wall

thickness increases, the force carried by the liner increases and collapse is delayed. In

other words, increase in liner thickness has a similar stabilizing effect as it has for

bending. On the other hand, since the cost of the product is significantly dependent on the

material cost of the non-corrosive liner, the improvement in collapse strains resulting

from the increase in Lt demonstrated for both loadings must be weighed against the

related increase to the cost of the product.

6.4.4 Axial Compression Under Internal Pressure

Motivated by the proposal from industry to reel and unreel lined pipelines

internally pressurized in order to avoid buckling and collapse of the liner (e.g., Endal et

al., 2008; Toguyeni and Banse, 2012; Montague et al., 2010), Chapter 3 demonstrated

even modest amounts of internal pressure can stabilize the liner under bending. It is thus

131

imperative that the effect of internal pressure be examined here for the axial loading

problem also. To this end, the 12-inch base case in Table 6.1 is now compressed under

increasing values of internal pressure. Figure 6.14 shows liner force and maximum

detachment vs. average axial strain responses for internal pressure levels of 10, 20, 25, 30

psi (0.69, 1.38, 1.72 and 2.07 bar) along with those of unpressurized case. The pressure

produces qualitatively similar behavior to that of the unpressurized case. However, even

such modest pressure levels as those considered have a significant stabilizing effect as

collapse is progressively delayed with increasing pressure. This is illustrated by the

observation that for the particular imperfection amplitudes chosen, for pressure of 30 psi

(2.07 bar) the liner did not collapse even at the relatively high compressive strain 3%.

This is an encouraging result since most of the applications where lined pipes may see

compressions outlined at the beginning of this chapter involve pipes in operation where

invariably they tend to carry hydrocarbons at some level of internal pressure. The

stabilizing effect of internal pressure on elastic buckling of cylindrical shells was reported

among others by Weingarten et al. (1965) and for plastic buckling by Paquette and

Kyriakides (2006). In both cases the induced hoop stress levels were higher than those

induced to the liner at the pressure levels of the present applications. Relatively high

pressure levels tend to delay buckling by reducing the amplitude of the imperfections.

The present problem involves unilateral buckling and it appears that even modest

pressure levels resist the inward growth of wrinkles.

132

Fig. 6.1 Liner diamond-type buckling from an axial compression test on a polymeric

outer cylinder lined with a stainless steel.

133

Fig. 6.2 Finite element mesh of the model composite cylinder loaded in

compression.

134

(a)

(b)

Fig. 6.3 Comparison of profiles of imperfections initially and after application of

manufacturing stress field: (a) axial and (b) circumferential ( 0x ) profile.

0

0.02

0.04

0.06

0 2 4 6 8 10 12

w t

L

x /

Initial: N = 8, = 0.05

Final

0

0.02

0.04

0.06

0 0.5 1 1.5 2

Initial: m = 4, m

=0.05

Final

w t

L

135

(a)

(b)

Fig. 6.4 Imperfect base case responses: (a) Force-displacement, (b) maximum

detachment- displacement.

321 65 7

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.4 0.8 1.2 1.6 2 2.4

F

Fo

x / L (%)

Steel Pipe

Composite

Liner4 8

= 4%

9

m

= 0.8%

m = 2

0

0.04

0.08

0.12

0 0.4 0.8 1.2 1.6 2 2.4

5 67

= 4%,

m = 0.8%

m = 2

8

x / L (%)

9

321 4

RL

136

Fig. 6.5 Sequences of liner deformed configurations showing evolution of wrinkling

corresponding to numbered bullets on response in Fig. 6.4a.

137

(a)

(b)

Fig. 6.6 Effect of axisymmetric imperfection amplitude on liner response. (a) Force-

displacement, (b) maximum detachment-displacement responses.

0

0.04

0.08

0.12

0.16

0 0.4 0.8 1.2 1.6 2 2.4

o(%)

0.4 01.04.0

FL

Fo

m = 2, m

= 0.8%

x / L (%)

0.26.0 2.0

0

0.02

0.04

0.06

0.08

0 0.4 0.8 1.2 1.6 2 2.4

RL

o (%)m = 2,

m = 0.8%

0.20.46.0 02.0 1.04.0

x / L (%)

138

(a)

(b)

Fig. 6.7 Effect of non-axisymmetric imperfection amplitude on liner response. (a)

Force-displacement, (b) maximum detachment-displacement responses.

0

0.04

0.08

0.12

0.16

0 0.4 0.8 1.2 1.6 2 2.4

m

(%)

0.4 0.2 0.1 01.02.04.0

m = 2, = 4%

x / L (%)

FL

Fo

0

0.04

0.08

0 0.4 0.8 1.2 1.6 2 2.4

m

(%)

m = 2, = 4%

0.40.20.1

0

1.0

RL

2.0

4.0

x / L (%)

139

Fig. 6.8 Collapse strain sensitivity to axisymmetric ( o ) and non-axisymmetric

( m ) imperfection amplitudes.

0

0.5

1

1.5

2

2.5

0 0.02 0.04 0.06

0 0.02 0.04 0.06

CO

(%)

m

o

= 0.04, m = 2

m

= 0.008, m = 2

m

C

140

(a)

(b)

Fig. 6.9 Effect of circumferential wave number on liner response. (a) Force-

displacement, (b) maximum detachment-displacement responses.

0

0.04

0.08

0.12

0.16

0 0.4 0.8 1.2 1.6 2 2.4

m

42 3

= 4%,

m = 0.8%F

L

Fo

x / L (%)

6 8

0

0.02

0.04

0.06

0.08

0 0.4 0.8 1.2 1.6 2 2.4

m

RL

x / L (%)

= 4%,

m = 0.8%

4

23

6 8

141

(a)

(b)

Fig. 6.10 Effect of friction on liner response. (a) Force-displacement, (b) maximum

detachment-displacement responses.

0

0.04

0.08

0.12

0.16

0 0.4 0.8 1.2 1.6 2 2.4

FL

Fo

x / L (%)

0.30 0.2 0.4

= 4%,

m = 0.8%

m=2

0

0.02

0.04

0.06

0 0.4 0.8 1.2 1.6 2 2.4

RL

x / L (%)

0.3

00.2

0.4

= 4%,

m = 0.8%

m=2

142

(a)

(b)

Fig. 6.11 Effect of initial annular gap on liner response. (a) Force-displacement, (b)

maximum detachment-displacement responses.

0

0.04

0.08

0.12

0.16

0 0.4 0.8 1.2 1.6 2 2.4

D = 12.750 in

0.51

1.52

m = 2

go

gob

= 0.0245wtL max

FL

Fo

x / L (%)

0

0.02

0.04

0.06

0.08

0 0.4 0.8 1.2 1.6 2 2.4

D = 12.750 in

RL

0.511.52

go

gob

x / L (%)

143

(a)

(b)

Fig. 6.12 Effect of pipe diameter on liner response for a constant liner wall thickness.

(a) Force-displacement, (b) maximum detachment-displacement responses.

0

0.04

0.08

0.12

0.16

0.2

0 1 2 3

m = 2

14

10

12

8

D (in)

tL = 3 mm

Dt

~18~

= 5.16x10-4wR

L max

16

x / L (%)

FL

Fob

0

0.02

0.04

0.06

0.08

0 1 2 3

RL

1412

10

8

D (in)16

x / L (%)

144

(a)

(b)

Fig. 6.13 Effect of liner wall thickness on its response. (a) Force-displacement, (b)


0

0.08

0.16

0.24

0 1 2 3

D=12.750 in

x / L (%)

FL

Fo

tL(mm)

3.5

3

2.5

2

4

4.5

m = 2

= 5.16x10-4wR

L max

0

0.02

0.04

0.06

0.08

0.1

0 1 2 3

RL

3.53

2.52 t

L(mm)

4

x / L (%)

4.5

D=12.750 in

145

(a)

(b)

Fig. 6.14 Effect of internal pressure on liner response. (a) Force-displacement, (b)


0

0.04

0.08

0.12

0.16

0 0.8 1.6 2.4 3.2

D = 12.750 in

0

tL= 3 mm

= 4%,

m = 0.8%

m = 2

P psi (bar)

10 (0.69)

25 (1.72)

20 (1.38)

30 (2.07)FL

Fo

x / L (%)

0

0.02

0.04

0.06

0.08

0 0.8 1.6 2.4 3.2

D = 12.750 in

P psi (bar)0

10 (0.69)

25 (1.72)

20 (1.38)

tL= 3 mm

RL

30 (2.07)

x / L (%)

146

Chapter 7: CONCLUSIONS

Low-carbon steel linepipe used in offshore and other operations is often lined

internally with a thin layer of corrosion resistant material in order to protect it from

corrosive contents. In applications where such bi-material pipe is loaded plastically, as

for example in the installation of a pipeline using the reeling method, the liner can detach

from the outer pipe and collapse forming large amplitude buckles that compromise the

flow and generally the integrity of the structure. This study presented a numerical

framework for establishing the extent to which lined pipe can be bent or axially

compressed before liner collapse. For both loadings the onset of wrinkling can be

idealized as a plastic bifurcation problem. This aspect was examined independently in

Chapter 4. Another aspect of the problem considered is the effect of girth welds on the

wrinkling and collapse of the liner. Following are major conclusions drawn from each

part of this study.

7.1 MANUFACTURE OF LINED PIPE

The manufacture of the lined pipe considered in this study involves mechanical

expansion of the liner and the steel outer pipe. Expansion alters the mechanical properties

of the two pipes and results in interference contact pressure between the two. The

manufacturing process of lined pipe was simulated using analytical and numerical

models. Comparisons of the results from both models show good agreement between

them, despite the thin-walled assumption made in the analytical model.

The following observations can be made for a parametric study of the

manufacturing process.

147

a. In this specific manufacturing process, for practical reasons, the two tubes start with

an annular gap between them. It was shown that reducing the initial gap between the

liner and carrier tube can increase the resultant contact stress significantly.

b. Increasing the difference between the yield stress of the two tubes can also increase

the resultant contact stress.

7.2 LINER WRINKLING AND COLLAPSE OF LINED PIPE UNDER BENDING

Pure bending of the composite structure leads to the usual Brazier ovalization of

its cross section. Differential ovalization leads to gradual reduction of the contact stress

between the two components and to the eventual separation of the liner from the steel

tube. Without the support of the substrate, the liner sector on the compressed side in turn

develops periodic wrinkles. The wrinkles initially grow stably, but as is common to shell

plastic buckling problems, at some point yield to a diamond-type shell buckling mode

that involves several circumferential waves. This second instability is associated with a

drop in the load carried by the liner, is local in nature, and results in collapse of the liner.

The collapse curvature was found to be very sensitive to initial geometric

imperfections corresponding to the two modes: that is, the axisymmetric periodic

wrinkling mode with wavelength 2 , as well as to non-axisymmetric mode with m

circumferential waves. This sensitivity was studied by adopting an imperfection that

additively combines the two modes. Within the range of parameters considered, collapse

was relatively insensitive to and m. Also, the effect of friction was found to be

negligibly small. Other highlights of the results are as follows:

148

a. Reducing the annular gap between the liner and the carrier tube increases the collapse

curvature of the liner, i.e., it has a stabilizing effect on the liner.

b. Increasing the diameter of the composite structure, but keeping the liner thickness

constant, reduces the collapse curvature of the liner.

c. Increasing the wall thickness of the liner of a given system has the intuitively

expected effect of delaying liner collapse. However, this benefit has to be considered

vis-à-vis the resultant additional cost of the product.

d. Bending lined pipe in the presence of relatively modest levels of internal pressure was

shown to delay liner collapse. Internal pressure tends to delay separation of the liner

from the outer pipe with corresponding delay in the wrinkling and non-axisymmetric

buckling instabilities.

7.3 PLASTIC BIFURCATION BUCKLING OF LINED PIPE UNDER BENDING

For lined pipe parameters of practical interest, the onset of periodic wrinkling on

the compressed side of the liner is a plastic bifurcation process. As is customary in plastic

bifurcation problems, the bifurcation check was performed using the J2-deformation

theory of plasticity. The solution procedure established identified the bifurcation mode to

consist of periodic wrinkling of the compressed side of the liner. The critical strain and

wavelength were studied parametrically and compared first with corresponding results for

the liner shell bent alone using the bifurcation check of Ju and Kyriakides (1991).

Interestingly, the critical strains of the lined structure and of the single shell were found

to have very similar values while the critical wavelengths of the lined pipe were

somewhat smaller than those of the shell bent alone. The results were also compared with

the critical variables of lined pipe under pure compression (Peek and Hilberink, 2013).

The critical strains under bending were consistently lower than those under axial

149

compression while the wavelengths were somewhat longer. This is because unlike

bending that leads to early separation of the liner from the outer pipe, under compression

the liner remains in close contact with the outer pipe until buckling.

The post-bifurcation of the lined pipe under bending was subsequently studied by

introducing to the liner an initial imperfection in the form of the wrinkling buckling

mode. Again, bending causes the liner to separate from the outer pipe inducing initially a

gradual growth of the periodic wrinkles. At higher curvatures, the wrinkles were shown

to yield to a diamond-type buckling mode whose amplitude grows with decreasing local

liner moment. In other words, the liner collapses while the carrier pipe remains intact.

Collapse, while imperfection sensitive, occurs at a much higher curvature and bending

strain than the critical wrinkling bifurcation values. Collapse is thus designated as the

critical design variable. Interestingly, liners assigned axisymmetric initial imperfections,

such as those adopted in Chapter 3 were found to collapse at very similar values of

curvature. This supports the notion that collapse is mainly influenced by the amplitude

and wavelength of the imperfection on the compressed side of the liner.

7.4 LINER WRINKLING AND COLLAPSE OF GIRTH-WELDED LINED PIPE UNDER

BENDING

A pipeline usually consists of 12 m-long length sections, which are girth welded

together. The edge of the liner is connected to the carrier pipe, which locally prevents the

detachment creating an axially periodic disturbance to the liner. This local disturbance

plays the same role as geometric imperfections in the main body of the lined pipe. The

periodic deformation of the liner grows with increasing bending and eventually leads to a

diamond-shaped collapse mode. A study of the effect of imperfections on liner collapse

in the absence of a girth weld, as in Chapter 3, identified several imperfection pairs that

lead to liner collapse at the same curvature as that of the girth-welded case. From this

150

comparison it was concluded that the disturbance provided by the weld is rather severe,

making a girth weld a "weak" spot in the liner.

Several factors that influence the onset of liner collapse in the neighborhood of a

girth weld were examined and the following trends were established:

a. Minimizing the initial annular gap between the liner and the carrier pipe can delay

liner collapse.

b. Bending lined pipe in the presence of even modest levels internal pressure delays

liner collapse.

c. Increasing the diameter of the composite structure while keeping the liner thickness

constant leads to earlier liner collapse.

d. As in most structural stability problems, increasing the liner wall thickness delays

liner collapse but adds to the cost of the product.

7.5 LINER WRINKLING AND COLLAPSE OF LINED PIPE UNDER AXIAL COMPRESSION

This part of the study considered the extent to which a lined pipe can be

compressed before the liner buckles and collapses. Demonstration experiments reported

on model lined systems show that the liner, while supported by contact with the outer

pipe, first buckles unilaterally into an axisymmetric wrinkling mode at a relatively low

strain. The wrinkles grow stably with compression but yield to a non-axisymmetric

diamond-type mode at a higher strain that causes uncontrolled growth of the diamond

buckles, in other words the liner collapses.

The problem has been modeled with nonlinear finite elements that incorporate the

initial mechanical properties of the liner and carbon steel outer pipe. The modeling again

starts with the simulation of the expansion manufacturing step through which the two

tubes are brought into contact. With residual stresses and changes in mechanical

151

properties locked in, the model is then axially compressed monitoring the deformation of

the two tubes. The initial axisymmetric wrinkling, the growth of the wrinkles, the switch

to the non-axisymmetric mode and the ensuing collapse of the liner have been confirmed.

As was the case for the bending problem, these events including the collapse strain are

sensitive to small initial geometric imperfections in the liner. The model is thus endowed

with geometric imperfections with axisymmetric and non-axisymmetric components and

the axial wavelength yielded by plastic bifurcation analysis. The numerical model was

subsequently used to examine the sensitivity of the collapse strain to the main parameters

of the problem that has led to the following conclusions:

a. The collapse strain is sensitive to both the axisymmetric and non-axisymmetric

imperfections considered. It is less sensitive to the circumferential wave number

adopted in the non-axisymmetric component and not very sensitive to the axial

wavelength of the imperfections. The main source of such imperfections in actual

lined pipes is the internal surface roughness of the seamless outer pipe. It is thus

imperative that imperfections in manufactured pipes be quantified and to the extent

possible reduced.

b. In the manufacture of lined seamless outer pipe, the two tubes start with an annular

gap between them. It was shown that reducing this gap can delay liner collapse.

c. Increasing the diameter of the outer pipe, while keeping the liner wall thickness

constant, increases the diameter-to-thickness ratio of the liner and reduces the

collapse strain. On the other hand, increasing the liner wall thickness on any

composite pipe increases the collapse strain.

d. Modest amounts of internal pressure can delay liner collapse up to strains at which

the outer pipe collapses.

152

Finally it is worthwhile comparing the axial compression results to corresponding

ones for lined pipe under bending (Chapter 3). The overall behavior is similar to that

described for axial compression. The liner develops periodic wrinkles at relatively low

strain, which grow and lead, at a much higher strain, to the collapse by shell-type modes.

Unlike axial compression, under bending the wrinkling is limited to the compressed side

of the liner that separates from the carrier pipe due to differential ovalization. Collapse

exhibits a similar sensitivity to imperfections with the collapse strains being of the same

order of magnitude as the ones reported in axial compression.

153

APPENDIX A: ANALYTICAL MODELS OF LINED PIPE MANUFACTURING

PROCESS

An analytical model has been developed for the manufacturing process of lined

pipe based on thin-walled shell assumptions. The two materials are idealized as elastic-

perfect plastic, and the structure is assumed to be one-dimensional.

We consider two thin-walled rings of radius R in contact and loaded by internal

pressure, P (see Fig. A.1) The outer one has wall thickness t and the inner one t . The

two materials are assumed to be elastic-perfectly plastic with elastic moduli E and E ,

and yield stresses y and y (where 1},,{ ). The equilibrium of the system is

given by

PRtt CL (A.1)

where C and L are the hoop stresses of the outer ring and the liner ring respectively.

When both rings are in the elastic range, the pressure is related to the hoop strain as

follows

.)1( R

tEP (A.2)

Since 1 , the inner ring will yield first (see Fig. A.2). The corresponding yield

pressure can be calculated as R

tP y

)1(1

. Beyond this point, the relation between

the pressure and the strain is

.R

tE

R

tP y (A.3)

The second ring yields at a pressure of

R

tP yo )1( (A.4)

154

and subsequently the structure expands freely without additional effort. When the

pressure is gradually removed the stresses in the two rings take the form:

yC

1

)(

, .1 yL

(A.5)

Accordingly, there is interference contact stress developed between the two rings, and it

can be found to be

.1 R

tP yc

(A.6)

It can thus be observed that reducing leads to an increasing contact stress between the

two rings. To evaluate this effect on contact stress, expansions are conducted for three

liner yield stresses: 45, 55 and 65 ksi, while keeping the carrier pipe yield stress 75 ksi

constant. The pipe parameters are chosen as 51.8tR , 1675.0 , 1 . Figure A.3

shows the stress-strain response of the two rings during the expansion. As the liner yield

stress increases, the difference between the stress levels at unloading is reducing. As a

result, the residual hoop stresses left in both tubes on the removal of the pressure are seen

to decrease, and the interference contact stresses are 516.0, 348.2,180.4 psi respectively.

In other word, the larger the yield stress difference, the larger the resultant contact stress

will be.

155

Fig. A.1 Schematic representation of two thin-walled rings of radius R in contact and

loaded by internal pressure.

156

Fig. A.2 Stress-strain responses of two rings during the expansion process.

0

0

Inner

Outer

y

y

E

E

157

Fig. A.3 Stress-strain responses for different values of liner yield stresses .

0

0

Inner

Outer

E

55

65

ksi

45

75

y

y

ksi

158

APPENDIX B: BIFURCATION BUCKLING UNDER AXIAL COMPRESSION

Unlike buckling of elastic shells where the critical buckling stress corresponds to

a multitude of buckling modes, axisymmetric and non-axisymmetric, the first buckling

mode of thicker shells that enter the plastic range is associated with the periodic

axisymmetric buckling mode of Lee (1962) and Batterman (1965) (see experiments in

Bardi and Kyriakides, 2006; Kyriakides et al., 2005; and analyses in Peek, 2000;

Kyriakides and Corona, 2007). The linearized incremental buckling equations for such

modes are:

0xxN , (B.1)

.wtwNR

NM xxoxx

Here ),( x represent the axial and circumferential coordinates, ),( MN are the stress

and moment intensities, ),( wu are the axial and radial displacements, and is the

applied axial compressive stress. The corresponding kinematical relations are given

uoxx ,

R

wo , wxx and 0 (B.2)

where ),( ooxx are the membrane strains and ),( xx are the curvatures. The

instantaneous stress-strain relations are given by

xx

CC

CC

2212

1211 , (B.3)

and the instantaneous stress and moment intensities are given by

and . (B.4)

159

(a)

(b)

Fig. B.1 Axisymmetric plastic bifurcation modes under axial compression: (a) shell alone

and (b) liner shell inside carrier pipe.

160

As is customary, for plastic buckling ]C[ are chosen to be the incremental

deformation theory moduli. It can be easily shown that the buckling mode is

x

aw cos~ and ,sin~x

bu (B.5)

(see Fig. B.1a) the critical stress and half wavelength are then given by

R

tCCCC

2/12122211

3 , .

)(12

2/14/1

2122211

211 Rt

CCC

CC

(B.6)

Buckling of the liner in a lined pipe under compression is again axisymmetric, but

now because of the contact with the outer pipe it is constrained to buckle inwards as

shown in Fig. B.1b (Shrivastava, 2010; Peek and Hilberink, 2013; for an example of

unilateral buckling see Chai, 2008). Thus the radial displacement must satisfy the

following conditions at the contact points:

0 www at x . (B.7)

The buckling mode can be shown to be

2

3cos

2cos3~ xx

aw (B.8)

and the critical stress and half wavelength become:

R

tCCCC

2/12122211

33

5 , 2/14/1

2122211

211

)(122

3Rt

CCC

CC

.(B.9)

161

A comparison of bifurcation strains of lined cylinders under axial compression

and bending as a function of D/t has been presented in Chapter 4 (see also Yuan and

Kyriakides, 2014b). Included are corresponding results for the liner shells alone under the

same loadings.

162

APPENDIX C: NUMERICAL SCHEME OF BIFURCATION CHECK OF LINED

PIPE UNDER BENDING

A numerical scheme has been developed for the plastic bifurcation check of lined

pipe under bending. To accommodate the preferred use of deformation theory of

plasticity for the bifurcation check, the material inelastic behavior will be modeled

through the J2 deformation theory of plasticity for both the prebuckling solution and the

bifurcation check. Bending of a lined cylinder is complicated by, among other factors,

Barzier (1927) ovalization induced to the cross section and also the severe contact

nonlinearities between the two tubes, making the bifurcation check even more

challenging. For this reason, it is accomplished through a custom user-defined material

subroutine (UMAT) appended to the nonlinear code ABAQUS.

The nonlinear stress-strain relationships of J2 deformation theory are given by

ijjkiljlikklijs

s

s

sij

E

][2

1

)21()1(, (C.1a)

where )( 2JEs is the secant modulus of the material uniaxial stress-strain response and

2

12

1

EsE

s . (C.1b)

Here the liner is modeled by linear shell element (S4) in the finite element model,

which requires specialized plane stress formulation. And thus, explicit, incremental

version of strain-stress relationships for plane stress is written as follows (see Kyriakides

and Corona, 2007):

163

x

x

x

x

d

d

d

d

d

d

dD , (C.2)

where

,

181)2(3)2(3

)2(6)2(1)2)(2(

)2(6)2)(2()2(11

2

2

2

xsxxxx

xxxxxs

xxxxsx

s qvqq

qqq

qqq

EdD

and

1

4

12

et

s

e E

Eq

.

The inverted version

,

is passed to the nonlinear solver as the Jacobian matrix for shell element bifurcation

checks.

As described in Chapter 4, the carrier pipe is meshed using linear continuum

elements (C3D8). Accordingly, the incremental version of (C.1) required by the nonlinear

solver is given by:

12

13

23

33

22

11

12

13

23

33

22

11

d

d

d

d

d

d

d

d

d

d

d

d

dC , (C.3)

where

.21)21(3

3)(

2

1

1 2

Jhh

sshh

h

EC klij

klijjkiljlikijkl

Because of preexisting symmetries, it can be written in compact notation as

d i Cijd j for the convenience of coding in UMAT,

164

where

1212

31123131

231223312323

3312333133233333

22122231222322332222

111211311123113311221111

C

CC

CCC

CCCC

CCCCC

CCCCCC

Cij . (C.4)

The stress-strain responses of both tubes are represented by Ramberg-Osgood fits

given by:

E

1 3

7

y

n1

. (C.5)

The parameters },,{ nE y for the two tubes are from a fit of the measured tensile stress-

strain response of a nominally X65 line grade steel and Alloy 825 (see Table 4.1).

In order to accurately identify the critical curvature, rotation, instead of the

moment, is prescribed at the 2x plane (see Fig. 4.4a). In addition, the increments are

chosen to be small (~ 1000/1L , L1 is based on the liner diameter and wall thickness).

Subsequently, ABAQUS' perturbation analysis is conducted for every increment of the

prebuckling solution, which in essence a plastically bent and ovalized composite pipe.

After identifying the critical eigenvalue (see Section 6.2.3 ABAQUS Analysis user

manual 6.10), the bifurcation curvature ( b ) is calculated afterwards.

165

APPENDIX D: DEMONSTRATION COMPRESSION EXPERIMENTS ON

LINED CYLINDERS

Lined specimens tested consisted of a thin stainless steel circular cylindrical shell

around which a relatively thick epoxy cylinder was cast as shown in Fig. D.1. The epoxy

used was Araldite GY502 with 35% Aradur 955-2 curing agent. It was selected because

of its relatively high ultimate strain and its good machinability. Figure D.2a shows the

stress-strain response measured in a compression test on a specimen cast from the same

batch as the lined test specimen. The elastic modulus and yield stress are given in Table

D.1. For this application, it is important to note that the material, although rate dependent,

retained a positive tangent modulus up to relatively high strains. The liner was a seamless

SS-304L in annealed condition with the stress-strain response shown in Fig. D.2b.

The epoxy was cast in a custom Teflon mold arranged to be concentric with the

liner shell. After curing and removal from the mold, the outer surface of the epoxy was

machined ensuring uniform thickness and concentricity with the liner. The ends where

then faced off producing a composite specimen of length L with nearly parallel ends. The

dimensions of the liner and epoxy are listed in Table D.1.

Table D.1 Main geometric and material parameters of lined cylinder tested

D in (mm)

t in (mm)

L in (mm)

E ksi (GPa)

o ksi

(MPa)

SS-304 Liner

2.494 (63.35)

0.0197 (0.500)

2.377 (60.38)

28,230 (195)

34.39 (237)

Epoxy Outer Shell

3.165 (80.4)

0.336 (8.53)

2.377 (60.38)

174 (1.20)

5.36 (37.0)

166

The composite cylinder was subsequently compressed between hardened steel

platens under displacement control that corresponds to strain rate of 410 s-1. The

specimen was unloaded periodically for visual inspection of the liner and reloaded.

Figure D.3 shows the recorded load-unload-reload axial force-displacement response.

The loading part of the response exhibits an initial essentially linear trajectory that

terminates into a knee caused by first yielding of the metal liner. At higher “strains,” the

upper loading trajectories can be seen to form a nearly linear locus up to a strain of just

under 2%, which primarily reflects the nearly linear hardening of the SS-304. At even

higher strains, the response exhibits some reduction in stiffness caused by the gradual

nonlinearity of the epoxy. The test was terminated at a strain of approximately 2.4%.

The visual inspections revealed the following. The first appearance of wrinkles

affecting only part of the circumference, occurred on the forth unloading from a strain of

about 1.01%. During subsequent unloadings the wrinkle amplitudes and angular spans

grew covering more of the circumference. The wrinkles were axially periodic protruding

away from the constraining epoxy. Measurements performed after similar tests found the

axial wrinkle wavelength to be close to those of the bifurcation analysis in Eq. (B.9) in

Appendix B. For example, in an experiment on a lined system with liner dimensions

close to the ones reported above, the measured value of was LR201.0 which compared

with the bifurcation value from (B.9) of LR206.0 (the standard deviation of

measurements was 0.24%).

The switch to the diamond mode of buckling, shown in Fig. 6.1, was first

observed after the 8th unloading from a strain of about 1.86%. It is interesting to observe

that this occurred at an increasing overall load of the composite specimen. The amplitude

of these wrinkles grew during subsequent compression making them more distinct. Thus

for example, the wrinkles shown in Fig. 6.1 were developed at a strain of 2.7%.

167

We close this section with a couple of experimental details. The specimen whose

response is shown in Fig. D.3, was compressed between parallel platens. The non-

linearity in the initial part of the response indicates that the specimen ends were slightly

out of parallel. Finally, because the test was conducted for demonstration purposes, the

displacement recorded was the “machine” displacement, which differs somewhat from

the actual shortening of the specimen.

168

Fig. D.1 Geometry of the lined cylinder tested.

169

(a)

(b)

Fig. D.2 (a) Load-unload compressive stress-strain response of araldite

GY502/Aradur epoxy used for the outer cylinders. (b) The stress-strain

response of the SS-304L liner shell used in the demonstration experiments.

0

2

4

6

8

0 1 2 3 4 50

10

20

30

40

50

(ksi)

= 5.355 ksi

(MPa)

Epoxy: Araldite GY502/Aradur 955-2

E = 174.1 ksi

= 10-4.

0

20

40

60

0 2 4 6 80

100

200

300

400

(ksi)

= 34.39 ksi

(MPa)

SS-304L Annealed

E = 28.23 x 103 ksi

= 10-4.

170

Fig. D.3 Load-unload response of the lined cylinder tested (see Table D.1).

Axisymmetric wrinkling was observed after a strain of 1.01%. The switch to

the diamond mode was first observed at a strain of 1.86%.

0

4

8

12

16

20

0 0.4 0.8 1.2 1.6 2 2.40

20

40

60

80F(kips)

/ L (%)

SS304/Epoxy

Exp. LIAX14

F(kN)

L

= 10-4

.

171

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175

Vita

Lin Yuan entered Zhejiang University in 2003 and graduated in 2007 with a

Bachelor's degree in Civil Engineering; he graduated in the top 5% of his class and

received many awards including the "Excellent Student in Zhejiang University" and

"First-grade Scholarship" awards. Recommended for admission, he subsequently earned

a Master's degree in 2009 in Structural Engineering at the same university. In August

2009, he entered the Graduate School of The University of Texas at Austin to pursue a

Ph.D. degree in Engineering Mechanics. In the course of his studies he made several

presentations at national meetings and co-authored the following conference and journal

publications:

Yuan, L., Kyriakides, S., 2013. Wrinkling failure of lined pipe under bending. Proc. 32nd Int’l Conf. Ocean, Offshore & Arctic Eng., OMAE2013-11139, June 2013, Nantes, France.

Yuan, L., Kyriakides, S., 2014. Liner wrinkling and collapse of bi-material pipe under bending. Int. J. Solids Struct. 51, 599-611.

Yuan, L., Kyriakides, S., 2014. Plastic bifurcation buckling of lined pipe under bending. Europ. J. Mech.-A/Solids 47, 288-297.

Yuan, L. and Kyriakides, S., 2014. Wrinkling and collapse of girth-welded lined pipe under bending.” Proc., 33rd Int’l Conf. Ocean, Offshore and Arctic Eng., OMAE2014-23577. June 2014, San Francisco, California.

Yuan, L., Kyriakides, S., 2015. Liner wrinkling and collapse of girth-welded bi-material pipe under bending. Appl. Ocean Res. 50, 209-216.

Yuan, L., Kyriakides, S., 2015. Liner wrinkling and collapse of bi-material pipe under axial compression. Int. J. Solids Struct. 60-61, 48-59.

Harrison, B., Yuan, L. and Kyriakides, S., 2015. Measurement of lined pipe liner imperfections and the effect on wrinkling under bending. Proc. 34th Int’l Conf. Ocean, Offshore and Arctic Eng., St John’s, NL, Canada, May 31-June 5, 2015, Paper OMAE2015-41228.

Permanent address: [email protected]

This dissertation was typed by the author.

on the liner wrinkling and collapse of bi-material pipe

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