Østergaard (2012)
Østergaard (2012)
Østergaard (2012)
Aalborg University
Special Report No. 80
Preface
This thesis has been submitted to the Faculty of Engineering, Science and Medicine at Aalborg
University in partial fulfillment of the requirements for the degree of Doctor of Philosophy in
Mechanical Engineering. The conducted work has been carried out at the Department of Mechanical
and Production Engineering at Aalborg University and at NKT-Flexibles in Brndby, Denmark, from
November 2008 to February 2012.
The work contained in the present thesis was carried out as part of an industrial PhD project involving
Aalborg University and NKT-Flexibles with financial support from the Danish Agency for Science,
Technology and Innovation. The project was supervised by Associate Professor, PhD Jens H.
Andreasen and Development Engineer, PhD Anders Lyckegaard, whom I thank for their support and
guidance.
I wish to thank my friend, PhD Alf Se-Knudsen for his support, optimistic mood and never failing
talent for expressing himself in differential equations rather than in words. Furthermore, a great
thanks to NKT-Flexibles Specialist Engineer Thomas Dettlaff for invaluable help and expertise in the
field of instrumentation, for ensuring that the numerous mechanical parts I constructed could be
assembled and for keeping me company watching a flexible pipe being bent repeatedly throughout
many a day and night. I would also like to thank the remaining crew involved in the experimental
work, especially Work Shop Technicians Jesper Nielsen and Tommy Harboe Friis for their work
related to dissections of failed flexible pipes.
I owe my gratitude to Lead Engineer Jan Rytter, who initiated and motivated the work with the lateral
buckling failure mode within NKT-Flexibles and to Pipe Design Engineers Lars Rude and Geir
Agustsson for their help with- and knowledge regarding well established design methods. I also wish
to thank Development Manager Niels Rishj and Lead Engineer Erik Bendiksen for encouraging the
present project and allowing the publication of obtained results.
Finally, I most of all wish to thank my family, Susi, Ronja and Malte for putting up with a boyfriend and
father who often while working on this project had his mind on wire mechanics rather than where it
should have been.
ii
iii
Abstract
The objective of the work documented in the present thesis has been by theoretical as well as
experimental means to study the physical behavior of the tensile armour layers of flexible pipes
related to a given failure mode involving lateral instability. This mode of failure, which is often
referred to as lateral wire buckling, is most common during flexible pipe laying in deep waters. In this
load scenario, a flexible pipe is subjected to compressive longitudinal loads and repeated bending
cycles.
Flexible pipes are usually designed as steel-polymer composite structures constituted by a number of
layers with different properties. Such structures have a wide range of applications in the offshore
industry. In the present work, the focus has mainly been on the mechanics of the two oppositely
wound layers of helical armour wires, since the investigated mode of failure occurs in these specific
pipe layers. Lateral wire buckling is a phenomenon characterized by the fact that it contrary to several
other flexible pipe failure modes is difficult to detect by visual inspection. However, it has on basis of
laboratory experiments been concluded, that lateral wire buckling causes a twist of a flexible pipe,
which can be quite severe. This twist occurs when the tensile armour layers no longer are torsionally
stable, which leads to a compression-twist coupling. Furthermore, lateral wire buckling is associated
with a shortening of a flexible pipe, while only small to moderate changes of circumference can be
detected.
As mentioned, lateral wire buckling of flexible pipe armouring layers can be reproduced
experimentally. In the present work, such experiments have been conducted by use of mechanical test
benches, in which a flexible pipe can be subjected to longitudinal compression and repeated bending.
It is, however, a widely accepted fact that results obtained by this experimental principle do not corelate to results obtained in the field, since failure during experiments occurs at lower compressive
load levels than encountered with offshore field conditions. The reason for this discrepancy is
unknown, but is possibly caused by differences in boundary conditions between the two scenarios.
However, experimental reconstruction of the lateral wire buckling failure mode remains a valuable
source of information related to failure in flexible pipes, since deformations and applied loads can be
measured, which is not the case in the field. Lateral wire buckling is characterized by large differences
in wire lay angles with respect to the initial helical state. However, the underlying mechanism does not
lead to failure of a flexible pipe, before repeated loading causes wire slippage towards geometrical
configurations in which the yield stress of the wire steel is exceeded.
In order to develop design methods for avoidance of failure in an unpressurized flexible pipe,
theoretical studies of armour wire mechanics have been conducted on basis of a formulation of the
mechanical equilibrium state of a beam embedded in a frictionless toroid. On this basis, the torsional
equilibrium state of all wires contained in the pipe wall is derived so that the compressive load
carrying ability can be calculated. The determination of this state rests on the assumption, that the
equilibrium states, which the wires will slip towards as cyclic bending is applied, coincide with the
equilibrium configurations determined directly, if friction is neglected. Effects due to friction and cyclic
loadings have partly been investigated without detection of significant impact.
The theoretical methods for analysis of armour wires that have been developed as part of the present
project are documented in four scientific journal papers. Furthermore, results obtained by these
methods are compared to experimental results in the present report as well as in two papers from
conference proceedings.
iv
Resume
Formlet med arbejdet, der er dokumenteret i nrvrende afhandling, har vret med svel teoretiske som
eksperimentelle metoder at studere den fysiske opfrsel af fleksible rrs trkarmering relateret til en given
svigtmekanisme involverende lateral instabilitet. Denne svigtmekanisme, der ofte benvnes lateral wire
buling, forekommer hyppigst under installation af fleksible rr p stor vanddybde. I dette belastningsscenarie
er et fleksibelt rr udsat for kompressive laster i lngderetningen og gentagne bjningscykler.
Fleksible rr er normalt designede som stl-polymer kompositte strukturer sammensat af en rkke lag med
forskellige egenskaber. Sdanne strukturer har et bredt spektrum af anvendelsesmuligheder i offshore
industrien. I det nrvrende arbejde er fokus hovedsageligt lagt p trkarmeringslagene, to lag af modsat
viklet helisk bndarmering, da den undersgte svigtmekanisme optrder i disse lag. Lateral wire buling er
som fnomen kendetegnet ved, at det i modstning til adskillige andre svigtmekanismer i fleksible rr kun
vanskeligt kan observeres ved visuel inspektion. Det er dog ved laboratorieeksperimenter, under hvilke
svigtmekanismen er blevet rekonstrueret under kontrollerede forhold, blevet fastslet, at lateral wire buling
forrsager et torsionelt vrid i fleksible rr, der kan vre ganske voldsomt. Dette er forrsaget af at
trkarmeringslagene ikke lngere stabiliserer hinanden torsionelt, sledes at en kobling mellem vrid og
forkortning opstr. Mens lateral wire buling er ledsaget af en forkortning af et fleksibelt rr, forekommer kun
sm til moderate ndringer i yderdiameter.
Som nvnt kan lateral wire buling i fleksible rrs trkarmeringslag reproduceres eksperimentelt. Indenfor
rammerne af det nrvrende projekt er dette foretaget under anvendelse af mekaniske testbnke, i hvilke et
fleksibelt rr udsttes for kompressiv last og gentagne bjningscykler. Det er et bredt accepteret faktum, at
dette eksperimentelle princip ikke rekonstruerer installationsbetingelserne p fyldestgrende vis, da svigt
forekommer ved lavere kompressive laster end observeret i felten. rsagen til denne diskrepans er ukendt,
men relaterer med stor sandsynlighed til forskelle i de randbetingelser, som et fleksibelt rr udsttes for i
felten og under laboratorieeksperimenter. P trods af dette forbliver laboratorieeksperimenter en vrdifuld
kilde mht. at studere svigt i fleksible rr, da deformationer og plagte belastninger kan mles, hvilket ikke er
tilfldet i felten. Lateral wire buling er karakteriseret ved store afvigelser i viklevinkel i forhold til den
oprindelige heliske tilstand. Der er dog frst tale om egentligt svigt, nr gentagne bjningscykler forrsager
at wirene slipper mod geometriske konfigurationer i hvilke flydespndingen er overskredet, da dette
muliggr dannelse af blivende plastiske deformationer.
For at vre i stand til at designe fleksible rr sledes at svigt ved lateral buling ikke forekommer, nr
et rr ikke et tryksat, er teoretiske studier i trkarmeringswires mekanik foretaget p grundlag af en
formulering for den mekaniske ligevgtstilstand for en bjlke p en friktionsls torusoverflade. P
grundlag af denne formulering er samtlige wires torsionelle ligevgtstilstand beskrevet, sledes at
den kompressive lastbreevne kan beregnes. Bestemmelse af denne tilstand hviler p en antagelse
om, at den ligevgtskonfiguration, som en armeringswire vil slippe mod efterhnden som
bjningscykler pfres, er sammenfaldende med den ligevgtskonfiguration, der bestemmes nr
friktion negligeres. Effekter forrsaget af friktion og cykliske belastninger har delvist vret berrt
uden at signifikant indflydelse kunne detekteres.
De teoretiske metoder til wireanalyse, der er udviklede som del af det udfrte arbejde, er
dokumenterede i fire videnskabelige artikler. Ydermere er sammenligninger med eksperimentelle
resultater dokumenteret i nrvrende rapport samt i to konferenceartikler.
vi
Dissertation
This dissertation is based on an introduction to the area of research and four papers submitted to
refereed scientific journals. Furthermore, two per-reviewed publications from conference proceedings
are included.
List of Publications
Journal Papers
1. Paper A
stergaard, N.H., Lyckegaard, A., Andreasen, J.H.
A method for prediction of the equilibrium state of a long and slender wire on a frictionless toroid
applied for analysis of flexible pipe structures
Engineering Structures, Vol. 34, pp. 391-399, 2012.
2. Paper B
stergaard, N.H., Lyckegaard, A., Andreasen, J.H.
Imperfection analysis of flexible pipe armour wires in compression and bending
Submitted to Applied Ocean Research, October 2011, under review.
3. Paper C
stergaard, N.H., Lyckegaard, A., Andreasen, J.H.
On modeling of lateral buckling failure in flexible pipe tensile armour layers
Submitted to Marine Structures, September 2011, under review.
4. Paper D
stergaard, N.H., Lyckegaard, A., Andreasen, J.H.
Simulation of frictional effects in models for calculation of the equilibrium state of flexible pipe
armouring wires in compression and bending
Rakenteiden Mekaniikka (Journal of Structural Mechanics), Vol. 44, No. 3, pp. 243-259, 2011,
Special Issue for the 24th Nordic Seminar on Computational Mechanics (NSCM-24).
Papers in proceedings
5. Paper E
stergaard, N.H., Lyckegaard, A., Andreasen, J.H.
On lateral buckling failure of armour wires in flexible pipes
Proceedings of the ASME 2011 30th International Conference on Ocean, Offshore and Arctic
Engineering, OMAE2011-49358.
6. Paper F
stergaard, N.H., Lyckegaard, A., Andreasen, J.H.
Simplified models for prediction of lateral buckling in flexible pipes armour wires
Accepted for presentation at the ASME2012 31th International Conference on Ocean, Offshore
and Arctic Engineering, OMAE2012-83080.
vii
Extended Abstract C
stergaard, N.H., Lyckegaard, A., Andreasen, J.H.
Lateral buckling of the tensile armor layers of flexible pipes
13th Internal Symposium of the Danish Center for Applied Mechanics and Mathematics
(DCAMM), 2011.
viii
Contents
Preface .................................................................................................................................................. i
Abstract ..............................................................................................................................................iii
Dissertation ........................................................................................................................................ vi
List of Publications ............................................................................................................................. vi
Contents ............................................................................................................................................ viii
1.
Introduction .................................................................................................................................. 1
1.1.
1.2.
1.3.
2.
Experimental setup............................................................................................................. 12
2.2.
3.
3.2.
3.3.
Test series II, 14 jumper test pipe samples in G2 bending rig ......................................... 21
3.4.
Test series III, 8 riser test pipe samples in G2 bending rig .............................................. 23
3.5.
4.
4.1.
4.2.
4.3.
Paper A ................................................................................................................................ 32
4.4.
Paper B ................................................................................................................................ 33
4.5.
Paper C ................................................................................................................................ 34
ix
5.
6.
4.6.
Paper D................................................................................................................................ 35
4.7.
Paper E ................................................................................................................................ 36
4.8.
Paper F ................................................................................................................................ 37
5.2.
Discussion of the definition of lateral wire buckling limit state design .............................. 40
5.3.
5.4.
5.5.
Modes of deformation.......................................................................................................... 45
5.6.
5.7.
5.8.
6.2.
References .......................................................................................................................................... 57
Internal NKT documents ................................................................................................................... 59
Appendix A: Paper A ... 61
Appendix B: Paper B ... 73
Appendix C: Paper C .... 87
Appendix D: Paper D .101
Appendix E: Paper E ......121
Appendix F: Paper F....133
1. Introduction
The present thesis deals with unbonded flexible pipes. This type of composite structures is usually
constituted by numerous steel and polymer layers. Flexible pipes have a wide range of applications in
the offshore industry related to drilling and extraction of oil and gas from subsea reservoirs. This type
of structure may be used as flowlines, running along the seabed, as conductors for water injection, or
as risers connecting a subsea reservoir to a floating platform at the sealevel. Flexible pipes are usually
designed in accordance with the API 17J code [1].
During the past few decades, the direction taken by the world economy has made development of oil
and gas fields at large water depths feasible. This process has introduced a need for flexible pipes
capable of resisting the extreme loads induced by hydrostatic pressure occurring at water depths
larger than 1000 meters. Furthermore, the need for experimentally validated design tools for
prediction of the numerous different failure mechanisms has grown tremendously. Due to this
development, the complex mechanical behavior of flexible pipes arising from interaction between the
different pipe layers has been subject of both industrial and academic research. Among the failure
modes, which have been investigated most intensively by theoretical and experimental means, are
collapse resistance and fatigue failure. Despite research is still ongoing and computational models and
experimental principles are continuously being improved, the methods for prediction of these failure
modes have been developed to a stage, at which they can be applied successfully for engineering
analysis.
The work presented in this thesis is related to lateral wire buckling in the tensile armour layers of
flexible pipes, see Figure 1. This is one of a few failure mechanisms, which cannot yet be predicted
with sufficient accuracy. Hence, no computational tool for design against lateral wire buckling has
been proposed. In order to obtain a torsionally stable design, the tensile armour layers in flexible pipes
are usually constituted by two layers of initially helically wound steel wires
wires with opposite lay
directions. The specific failure mode is usually encountered during pipe laying. In this scenario, the
flexible pipe is in a free hanging configuration from an installation vessel to the seabed. During pipe
laying, the pipe is exposed to longitudinal compression due to hydrostatic pressure, since the pipe
bore is empty during installation. Furthermore, the pipe will be exposed to repeated bending cycles
due to natural loads and movements of the installation vessel. The lateral buckling failure mode was
first encountered in 1997 and is described by Braga and Kaleff [2], who also were the first to
reproduce it by experimental means in the laboratory.
Figure 1 Severe state of lateral wire buckling in the inner layer of armouing wires, triggered experimentally by
laboratory testing of 6 flexible pipe.
The theoretical work documented in this thesis constitutes studies of armour wire mechanics
investigating the mechanisms leading to lateral wire buckling. Furthermore, the industrial cooperation
enabled investigation of lateral buckling failure by reconstructing the failure mechanism in the
laboratory under controlled conditions. In order to do so, mechanical test rigs constructed specifically
for this purpose were used.
The thesis is outlined as an introduction to a number of peer-review papers enclosed in Appendix A-F.
In the present first chapter, the structures and failure mechanisms, which are subject of the
presented research, will be introduced.
In the second chapter, the principle used for laboratory experiments will be presented
In the third chapter, a summary of the experiments conducted by means described in chapter 2
will be presented.
In the fourth chapter, the theoretical approach to wire mechanics is addressed before a
summary of the publications related to the present project is included.
In the fifth chapter, additional experimental and simulated results are included for the sake of
completeness.
In the sixth and final chapter, the thesis will be finalized by concluding remarks. Furthermore,
the contributions from the present research to the field of flexible pipe mechanics will be
summarized and directions of future research will be discussed.
1.Carcass
2.Inner liner
3.Pressure armour
4.Tensile armour
5.Tensile armour
6. Outer sheath
Figure 2, left: Example of the most common flexible pipe design, right: High-strength tape for prevention of
wire birdcaging (ABC-layer).
Since flexible pipes are designed as composite structures with numerous layers, the global structural
behavior is complex and has been investigated intensively in numerous publications. Due to the
unbonded nature of the structures, layers may to some extend slip internally. Furthermore, couplings
between longitudinal strain, radial strain, and pipe twist may arise due to the internal helically wound
components. In general, a flexible pipe is designed in such a manner that coupling effects are
minimized. As a consequence of this, the number of wires in the outer layer is larger than in the inner
layer of tensile armour. The first commercially available design tool for flexible pipe design, CAFLEX
documented in [4], enabled analysis of straight flexible pipes based on radial, torsional, and
longitudinal equilibrium of internal components. The mathematical model applied considered layers
either as constituted by helical elements or as isotropic thin shells. The underlying theory was
partially published in 1987 by Fret and Bournazel [5].
Due to wire slippage in bending, the flexural moment-curvature responses of flexible pipes exhibit
hysteresic behavior. This behavior was described in the CAFLEX theory manual [4] and investigated
further by, among others, Witz and Tan [6], Tan, Quiggin and Sheldrake [7], Kraincanic. and Kabadze
[8], Alfano et. al. [9]-[10] and Dastous [11]. For low values of curvature, frictional resistance prohibits
wire slippage, and the wires contribute to the cross sectional moment of inertia around the global pipe
axis. When a critical curvature value s is exceeded, wire slippage occurs, and the wires bend solely
around their local axes. This causes the wire contributions to the pipe bending stiffness to decrease as
wire slippage occurs. This behavior has often been idealized as bilinear as shown in Figure 3, in which
S1 denotes the pipe bending stiffness prior to wire slippage, and S2 the bending stiffness after slippage
occurs. The transition from S1 to S2, which in Figure 3 is idealized as sudden, is in reality smooth, since
all wires do not slip exactly at the same level of curvature. However, for most practical purposes the
idealization is sufficient to model the global flexural behavior in a fulfilling manner.
M
s
S1
S2
Figure 3, left: Birdcaging of armouring wires reproduced experimentally by NKT-Flexibles, right: Idealized
flexural behavior of flexible pipe.
The consequence of failure in a flexible pipe applied as riser for extraction of oil and gas from a subsea
reservoir may be oil spill, possibly with tremendous environmental consequences. However,
installation of a replacement riser, which possibly has to be manufactured first, leads to production
down time. Furthermore, shut down of an oil well causes loss of the reservoir pressure, which must be
reestablished before the production can be continued. Due to these reasons, both oil companies and
flexible pipe manufactures are extremely cautious with respect to modifying a design, which in the
field already has shown potential as proven concept. Due to these commercial issues, the offshore
business is dominated by a conservative approach to the design process. Therefore, it is in general
considered feasible to accept the significant costs related to laboratory experiments for validation of
existing designs rather than to attempt to redesign the product in new and innovative ways. Tests are
used to reconstruct failure modes in flexible pipes, and obtained results are used for validation or
calibration of theoretical models. Examples of how the limit state design related to different failure
mechanisms may be defined are summarized by Jiao [12].
tests, it is under full-scale offshore conditions examined, if a flexible pipe can be installed without
occurrence of armour wire buckling, see Figure 4.
During DIP-testing of flexible pipes with ABC-layers, lateral wire buckling was encountered in 1997. It
was observed, that buckling leads to disorder of the tensile armour wires in the circumferential pipe
direction, see Figure 5. The failure mode was discovered for flexible pipes subjected to compressive
loads and repeated bending cycles. Generally, the risk of lateral buckling is considered largest, when
the outer sheath of the pipe is damaged. This is due to the fact, that external pressure in the case of
flooded annulus to a much lower extend introduces contact stresses causing frictional effects to limit
wire slippage, than when the outer sheath is intact. Damaged sheaths are quite common during pipe
laying and wet annulus conditions must therefore be considered when designing flexible pipes against
lateral wire buckling. Due to this issue, DIP-tests are performed first with dry and afterwards flooded
annulus conditions.
Lateral wire buckling imposes a larger danger than birdcaging, since it often cannot be detected by
visual inspection of a test specimen without dissection. In order to study the failure mode under
controlled conditions, it was reconstructed experimentally in the laboratory. Braga and Kaleff [2],
developed the first principle for testing of flexible pipes in form of mechanical test benches. By this
experimental principle, a mounted flexible pipe could be subjected to repeated bending cycles.
Compression was applied by mounting a smaller flexible pipe inside the test sample pipe and
mounting those on the same base flange in one end of the setup. Furthermore, this end of the setup
was mounted in a manner enabling the test pipe sample to be torsionally free. As the inner pipe was
tensioned, the test sample pipe was compressed. Braga and Kaleff concluded that birdcaging could be
reproduced at compressive load levels corresponding well to the hydrostatic pressures causing failure
in the field. However, lateral wire buckling occurred at compressive load levels lower than detected
during field testing. The conclusion was furthermore drawn, that lateral buckling was associated with
shortening of the test sample and a pipe twist of the torsionally free pipe end. These observations
were confirmed during the experimental work included in the present project.
Similar experiments have been conducted by flexible pipe manufactures. Furthermore, the test
principle has been extended to include experiments conducted in mechanical bending frames mounted
inside pressure chambers. By this method, the stabilizing effect of hydrostatic pressure with dry
annulus conditions could be studied. Furthermore, compression could be applied directly to the pipe
end-cap without the necessity of a tension pipe in the bore of a test sample pipe. Experiments are
documented by Secher et. al. [3] and [17], Bectarte and Coutarel [18], and Tan et. al. [19]. The wire
deformation modes obtained experimentally in this project correspond well to the buckled wire
configurations, which were presented in [3], see Appendices E and G. Failure was during both
laboratory tests and DIP-tests detected in the inner layer of armouring wires. The reason for this is
deemed to be, that the inner layer of tensile armour experiences a larger compressive force than the
outer layer when the pipe is loaded. This is due to a minor radius of the layer, a lower number of wires
than in the outer layer, and complex interaction effects between the internal pipe components. The
outer layer may be prone to lateral buckling when the pipe is exposed to large externally applied
torsional moments. Such loads may occur during spooling of flexible pipes, but are not considered in
the present project.
While lateral wire buckling to a high extend has been investigated experimentally in the laboratory,
very few theoretical studies have been published. Custdio [20] formulated the equations of
equilibrium for the tensile armour wires analytically for a straight pipe. Buckling was considered a
bifurcation-problem formulated on basis of a radially elastic pipe wall. This effect was simulated by
use of elastic foundations. However, Custdio interpreted several of the obtained modes of wire
deformation as being unphysical and discarded those. Recently, further studies have been conducted
by Vaz and Ricci [13], using finite element modeling of a single armouring wire taking frictional effects
into calculation. Brack et. al. [21] also applied finite element methods for calculation of the load
carrying ability of an armouring wire. However, in none of the cases cyclic loadings were considered,
and obtained results were not compared to experimental data.
Installation vessel
Sea level
Flexible pipe in
free-hanging catenary
Touch-down
zone
Sea bed
Figure 4, left: Photo of flexible pipe touchdown zone during DIP-testing taken by ROV (underwater robot),
right: Principle sketch of flexible pipe during installation.
M1
M2
M3
M4
Figure 5 Modes of deformation in the inner layer of armouring wires detected during dissection of 6 flexible
pipe after it had been subjected to lateral buckling laboratory testing, M1: No sign of buckling, M2: Lateral wire
buckling detected as large localized gaps, M3: Lateral wire buckling detected as localized S-shaped mode of
deformation, possibly of a plastic nature M4: Severe state of lateral wire buckling, it is estimated that the yield
strength of the wires has been exceeded which has led to plastic deformations.
1.
2.
3.
Papp
Papp
Papp
Figure 6 Schematic drawing of the lateral buckling failure mechanism in flexible pipe tensile armour layers, Blue
wires: outer layer of armour wires, Red wires: inner layer of armour wires, 1, compression and cyclic bending:
Compression and repeated bending cycles are applied to a flexible pipe, 2, nonlinear geometrical softening of
inner layer: As bending cycles are applied, the wires contained in the inner layer slip towards a configuration in
which the load carrying ability is reduced with respect to both the initial configuration of the layer and to the outer
layer of tensile armour, 3, torsional coupling between layers: This causes a pipe twist in order to maintain
torsional equilibrium, which leads to relaxation of the compressive loads in the outer layer of armouring wires. This
twist adds further straining to the inner layer of armouring wires.
pu
B
R=
Flexible pipe
End fitting
pb
pu
Seabed
Figure 7, left: Schematic drawing of a flexible pipe idealized as having constant bending radius during laboratory
testing, right: Schematic drawing of the touchdown zone of a flexible pipe during DIP-testing.
The coupling between the layers is mainly due to the fact that all layers are fixed in end-fittings in both
ends of the pipe. Therefore, all wires are subjected to the same strain and twist on the boundaries.
While this relaxes the outer layer, the inner layer is stressed further by the pipe twist. The inner layer
is loaded further, since the wires, which are already compressed, are twisted against the winding
direction. This causes further shortening of the wires and may at some point lead to plasticity. This
may cause permanent deformations and, ultimately, failure of the pipe structure.
The lacking one-to-one correspondence between results obtained from experimental lateral buckling
experiments in mechanical rigs and DIP-tests is a complex issue, for which no absolutely fulfilling
explanation has yet been given. No measurements of the pipe responses during pipe laying are
available, since it is a tremendous challenge to solve the problems related to instrumentation of a
flexible during a DIP-test. Information regarding the touchdown zone of a flexible pipe is therefore
only available from images transmitted by ROVs (small subsea robotsystems) during the tests. An
example of such an image is presented in Figure 4. However, the view of an ROV is quite limited, and it
is often difficult to identify possible 3D-effects. It is therefore difficult to know the exact circumstances
under which lateral buckling occurs with respect to seabed interaction and possibly out-of-plane
motion of the test pipe. The reasons for the lacking correspondence are therefore of a highly
speculative nature. However, the boundary conditions used for laboratory experiments may be
considered and compared to the installation scenario, see Figure 7. It is clear that the pipe end denoted
B in the experimental setup is torsionally fixed, while this end during DIP-tests may only be partially
fixed. This effect may be one reason for the lacking correspondence. Furthermore, the pipe may not
necessarily be subjected solely to in-plane bending. NKT-Flexibles has as the present project was
finalized never encountered lateral wire buckling in the field. Therefore, no exact knowledge regarding
in-field wire buckling has been established.
Another issue regarding the lacking one-to-one correspondence between the two test principles is the
difficulties related to determination of the limit compression causing failure during a DIP-test. In order
to determine the limit compressive load, the test had to be performed several times at varying water
depths until failure was detected. The necessary process would be extremely time consuming and
expensive. However, an example of such a campaign is documented by Secher et. al. [3] and [17].
Furthermore, for obvious commercial reasons, flexible pipe manufactures, oil and gas companies and
offshore contractors wish to validate the installability of a given flexible pipe. Therefore, the main
objective of a DIP-test is usually to conclude the test without failure rather than to determine limit
compressive loads. Furthermore, a flexible pipe may be installed at water depths at which the limit
compressive load is exceeded, if only a limited number of bending cycles insufficient to trigger failure
is applied. Secher et al. [3] and [17] claimed to have obtained correspondence between DIP-tests and
laboratory tests conducted in pressure chambers, however, using very short test pipe samples. In
order to draw conclusions regarding the validity of this issue, experiments with varied length of test
pipe sample must be conducted in order to examine the influence of test sample length on the
obtained results.
The experimental results obtained in the present work suggest, that buckling may develop in the
elastic regime. Furthermore, in some cases a tremendous number of bending cycles must be applied,
before the yielding limit of the wire steal is exceeded. These observations may alter the understanding
of the phenomenon. Presently, the only criteria for when lateral buckling is considered as failure
available are contained in certain flexible pipe specifications provided by oil and gas companies. The
API 17J code [1] only states that wire buckling should be considered, but does not prescribe by which
methods. This demonstrates the need for clarification of the difference between the distinct terms
lateral wire buckling and failure by lateral wire buckling. From a design perspective, the limit state
may be taken as the conditions, at which compressive loads cause torsional imbalance of the pipe
structure. However, yielding of the wires has not necessarily occurred in this state. Therefore, the limit
state may alternatively be considered as the conditions in which compressive loads and torsional
imbalance have led to permanent deformation of the pipe structure due to plastic behavior of the
wires. The limit state design issue is elaborated further in section 5.8.
Experimental reconstruction of the lateral wire buckling failure mechanism in eight available
test sample pipes. This part of the project has included construction of the necessary
mechanical gear, execution of bending tests, and pipe dissections.
Derivation of the equations governing the equilibrium of an armouring wire within the wall of
a flexible pipe bent to a constant radius of curvature.
Investigation of lateral wire buckling of a single armouring wire with small imperfections.
These studies are based on full non-linear analyses for calculation of the wire equilibrium
paths.
Modeling of the mechanical behavior of both layers of armouring wires arising from lateral
wire buckling. Analyses will be based on multiple single wire analyses and proper
idealizations.
Estimation of how effects such as friction, boundary effects, and adjacent pipe layers influence
the load carrying ability.
The ultimate goal would be to deliver a set of experimentally validated design rules for flexible pipes.
These could then be implemented directly in the pipe design process and in the installation limitations
for a given flexible pipe design. However, due to lack of information regarding the exact conditions
encountered during pipe installation, the scope of work has been limited to prediction of the limit
compressive load causing lateral buckling during a laboratory experiment. The methods may enable
prediction of the outcome of a DIP-test. This may be addressed either on basis of calibration against
DIP-test results or on basis of future measurements taken during DIP-tests, which reveal the true
conditions encountered during pipe installation.
A visualization of the scope of work is presented in Figure 8.
Global analysis of
armouring layers
Experimentally validated
Design tool
Simplified analysis
with adjacent layers
Single wire
imperfection analysis
DIP-test results /
measurements
Present project
Frictionless single
wire equilibrium
Laboratory
experiments
Preliminar understanding of
the lateral wire buckling problem
Figure 8 Schematic overview of the work contained in and related to the present project.
10
11
a.
P
L
e
b.
c.
P
e
dy
M
P
dS
Figure 9 Schematic drawings of test setup, a) Test principle used for laboratory experiments, Pipe end A
longitudinally and torsionally fixed, Pipe end B longitudinally and torsionally free, b) Pipe geometry, c) Pipe
equilibrium.
12
However, industrial experiments conducted for validation of computational models applied in flexible
pipe design are extremely time consuming and expensive. The scale of the experiments conducted in
order to qualify flexible pipe design methods can with respect to execution time and cost often be
compared to, for example, fatigue tests of wind turbine blades, see for example Kensche [23]. Another
example of similar experiments are those for determination of pile-soil interaction effects used to
validate computational methods in civil engineering, see for example Imamura et. al. [24]. Therefore, it
is in general necessary to assume the outcome of such experiments to be reasonably deterministic.
This corresponds to neglecting effects caused by statistical variance of the experimental outcome.
Despite the fact that it from a statistical point of view is difficult to justify this approach to
experimental work, it is crucial to assume the outcome of the experiments deterministic, when
experiments like the present are carried out with limitations in cost and time. In the present project, at
least half of the conducted experiments would have been used only to obtain a rough measure for the
statistical variance, if a classical approach to design and analysis of experiments had been followed. It
was therefore chosen not to consider these issues when planning the tests.
Figure 10 Overview of upgraded test rig based on CAD model, 1) Test pipe sample, 2) Static frame end A
with hydraulics for tensioning of the inner pipe, 3) Moving frame end B, test pipe supported by system of
bearings allowing a test pipe twist, 4) Idealized bending device, 5) Supporting frame (idealized model).
13
Figure 11 GI Lateral buckling test rig during the initial trial test of a 6 riser pipe sample, bending applied using
electrical drives.
Figure 12 G2 lateral buckling test rig during the test of the first 14 jumper, bending applied by a hydraulical
system.
14
The end-fittings were mounted on base-flanges connected to pinned H-formed frames. These were
above the test sample pipe connected by a bar, which could be shortened. This system constitutes the
bending device of the test rig, since shortening of the top bar causes the test pipe sample to bend
downwards as the H-formed frames rotate. The key difference between the initial and the upgraded
test rig is, that larger and more robust mechanical components were used for the upgrade. This
allowed larger pipes to be tested. Furthermore, the bending device in the initial setup was constituted
by a system of electrical screwdrives. This was replaced with a more robust hydraulic system in the
upgraded rig.
A maximum compressive load of 1000 kN could be applied in both bending rigs. In order to protect the
crew operating the rig in case of failure of the setup or rupture of the test pipe sample, a number of
two meter high concrete blocks shielded the position in which the computers used for datalogging and
control were placed. Furthermore, it could during the planning of a few experiments with extreme
load inputs not be ensured that the test pipe sample behaved in the desired manner. These
experiments were therefore conducted at times, when the test facilities and workshops were empty.
Base flange
Flange
Test end-fitting
Test pipe (flexible)
Test end-fitting
Tension pipe (flexible)
Base flange
Tension pipe endbody
Compression hydraulics
Threaded bar
v+
Figure 13 Sectional drawing of assembled test pipe sample, compression applied to test pipe sample when the
compression hydraulics moved in the direction marked v+, means for pipe centralization not included.
Figure 14, left: Assembled bearing arrangement for torsionallly free pipe end, right: Plastic centralizers for test
of the 8 risers and 14 jumpers constructed as two half-parts assembled by two pins and a band-it metal strip.
15
Figure 15, left: Unmounting of tension pipe with plastic centralizers during teardown after test of 14 jumper,
right: Dissected end-fitting from 6 riser pipe, wire fixated with weldings to a weld-ring.
Compression was applied by tensioning the inner pipe, which was centralized in the bore of the test
sample pipe by appropriate means. A sketch demonstrating this principle is shown in Figure 13.
During the execution of the tests of the 6 riser, polymeric insulation materials were used, since the
difference between the bore diameter of the test sample pipe and tension pipe was small. For the tests
of the remaining pipes, custom built plastic centralizers were used, see Figure 14 and Figure 15. The
torsionally free end of the test pipe sample was supported by a system of bearings allowing this end to
twist. The bearing system was constituted by two axial bearings transferring the longitudinal forces
from the tensionpipe to the surrounding structure for each direction of motion in the compression
hydraulics. Furthermore, two axial bearings enabling transferring of bending were applied. The
bearing arrangement is shown in assembled form in Figure 14.
Measurement
Compressive load on test pipe sample
Device
Load cell (max. 1000 kN)
Bending load
Extensometers
Inclinometers
Inclinometer
Extensometer
Position sensor
Temperature sensor
16
hg
4
(1)
The system for compressing the test sample pipe was controlled directly by adjusting the hydraulic
station, until the desired compressive load was measured by the load cell mounted for this purpose.
The system used to bend the test pipe sample was in the G1 test rig controlled from a control box
placed directly behind the bending rig. This control box interfaced directly to the PLC controlling the
electrical drives. In the G2 test rig, in which bending was applied hydraulically, the bending system
control was programmed in a labview application. This could run parallel to the interface used for
datalogging on the computer integrated in the setup. The maximum radius of curvature was calculated
on basis of the response from two inclinometers placed on each side of the pipe midpoint. With the
rotation responses i+1 and i from transducers placed in Si+1 and Si, the pipe curvature is approximated
as a finite difference by
=
i +1 - i
S i +1 - S i
(2)
As described in section 1.2, lateral wire buckling is associated with a twist of end B, a shortening, but
only a small change of pipe circumference. These values were all measured during the experiments. All
instrumentation was by appropriate means connected to a digital interface that enabled the
measurements to be shown online during the experiments. Furthermore, the logged responses were
saved in an ascii-file for further studies and postprocessing. The radius of curvature of the pipe
midpoint was processed online and written to the computer interface during the experiments. This
allowed small adjustments necessary due to changes of the flexural behavior of the test pipe sample.
17
Initially, a test of anti-birdcaging performance was conducted. During this part of the
experiment, the compressive load was applied to a straight test pipe sample. The objective was
to ensure, that the compressive loads applied during the actual lateral buckling test would not
cause birdcaging. Static compression was applied with a holding time of 20 minutes.
The test pipe sample was tensioned by a load of 50 kN and bent a number of times (usually
between 10 and 20) in order to ensure, that the wires contained in the tensile armour layers
were as close to the initial helical configuration as possible. Handling prior to the test may have
caused wire slippage.
Finally, the actual lateral buckling experiment was performed by applying a prescribed load
program. During most experiments, a maximum of 1200 cycles of repeated bending from
straight configuration was chosen as a reasonable estimate for a DIP-test. In similar
experimental studies, a total of 800 bending cycles was chosen, see [3].
18
15
Upper measuring limit of instrument
Instrument
torn off
10
Twist (deg)
48 bending
cycles
-5
Instrument
unmounted
-10
-15
500
1000
1500
2000
time (s)
2500
3000
3500
300
Upper measuring limit of instrument
250
48 bending
cycles
200
180 bending
cycles
Experiment stopped,
tests bends performed
150
100
50
1000
2000
3000
4000
5000 6000
time (s)
7000
8000
9000 10000
Figure 17 Compressive stroke measured during the initial trial test of a 6 riser.
19
OD1
OD2
OD3
OD4
OD5
30
25
180 bending
cycles
15
10
1000
2000
3000
4000 5000
time (s)
6000
7000
8000
9000
Figure 18 Change of circumference measured during the initial trial test of a 6 riser.
During dissection of the pipe sample, the following conclusions were drawn:
A severe remaining twist of approximately 90 degrees of pipe end B was measured. This was
the only sign of failure within the pipe wall, which could be observed by naked eye.
The anti-birdcaging tape of the test pipe sample had a highly irregular pitch. This may have
influenced the test result.
No signs of failure could be detected in the outer layer of tensile armour wires.
The inner layer of armoring wires was found in a severely buckled state, see Figure 1. It was
estimated, that plasticity in the wires had occurred during the test.
The radius of curvature was during the initial trial test estimated on basis of measured deflections.
These responses were logged by extensometers placed between the top-bar and the test pipe sample.
It was after the experiment found difficult to estimate the radius of curvature. By spline-interpolations
and regressions it was determined that at least 5 meter of bending radius had been applied during the
experiment, but the value at the pipe midpoint may have been lower. The difficulties were mainly
imposed by the fact, that estimating curvature included differentiating the deflection response twice,
which amplified noise in the responses significantly. The problem was during the remaining
experiments solved by replacing the extensometers with inclinometers mounted directly on the pipe
sample. The consequence is that the radius of curvature of pipe midpoint during the remaining tests
was estimated on basis of the rotation of the test pipe sample rather than on basis of deflections, see
equation (2).
As the electrical drives were turned on during the initial trial test, significant influence on the
responses from the remaining transducers was detected. Furthermore, a few transistor radios located
in the work shop began transmitting noise. The effect was found to be caused by the frequency
transformers in the electrical drives. In order to minimize noise in the logged measurements from the
remaining transducers, these were shielded electrically.
It was on basis of the detected behavior of the test pipe sample chosen to set a pipe twist of 45 degrees
as the stop criteria for future experiments.
20
Pipe ID
Experiment ID
Load
cycle
ID
Number of
bending
cycles
Applied
compression (kN)
Bending
radius (m)
*4
Result
obtained by
dissection *3
Pipe twist
before
unloading (deg)
Initial
trial test
0 *2
180 / 48*1
265
Failure
>90
Test
series 1:
204
265
11
Failure
45 (increasing)
800
80
11
No failure
<1 (stable)
II
392
210
11
Failure
45 (increasing)
1200
160
11
No failure
3 (slowly
increasing)
II
151
265
Failure
45 (slowly
increasing)
6000
277
18 to 24
No failure
<1 (stable)
II
1200
269
7 to 9
No failure
6.5
(increasing)
III
1200
411
9 to11
Failure
27
I*2
18
950
12
Failure
10 (rapidly
increasing)
II
1200
950
12
Failure
17 (increasing)
6 riser
L= 5m
Test
series 2:
14
jumper
L= 7.5m
Test
series 1:
1 *2
1200
700
12
Failure
27 (slowly
increasing)
8 riser
II
1200
300
12
No failure
<1
III
2400
400
12
No failure
15 (slowly
increasing)
L= 5m
* 1) On basis of the datalog it was after the test estimated that failure by lateral wire buckling occurred after 48
bending cycles,
*2) Experiment / load cycle was due to extreme load input conducted at an appropriate time when the test
facilities were empty (for example at night or started in the afternoons before holidays/weekends).
*3) Failure should in this context be interpreted as detection of lateral wire buckling during dissection of the test
pipe sample. The deformation patterns detected in the inner layer of armouring wires are included in section 5.5.
*4) Bending applied from straight configuration to prescribed value unless stated in the table.
Table 2 Summary of conducted experiments.
21
50
80
Test I, LC 1, 265 kN, 11 m bending
Test II, LC 1, 80 kN, 11 m bending
Test II, LC 2, 210 kN, 11 m bending
Test III, LC 1, 160 kN, 11 m bending
Test III, LC 2, 265 kN, 8 m bending
30
Pipe twist (deg)
60
Compressive stroke (mm)
40
20
10
70
50
40
30
20
10
0
-10
200
400
600
800
Number of bending cycles
1000
1200
-10
200
400
600
800
Number of bending cycles
1000
1200
Figure 19, left: Pipe twist of the 6 risers applied bending cycles, right: Compressive stroke of the 6 risers.
criteria of 45 degrees was reached after 204 bending cycles. Failure was detected in the inner layer of
armoring wires during dissection. The modes of deformation were partly S-shaped, partly large gaps
also with an S-like shape (of the types M2 and M3, see Figure 5). The result showed, that reducing the
maximum bending radius of the pipe midpoint for a fixed compressive level had as consequence, that a
higher number of bending cycles had to be applied in order to trigger failure.
The second experiment in the present test series was constituted by two load cycles. The first load
cycle showed, that 800 bending cycles with 80 kN of compressive load could be applied without
causing failure to occur. This load cycle was conducted in order to ensure, that experiments could be
performed without leading to lateral buckling failure. The second load cycle was performed with a
compressive load of 210 kN but still with 11 m of bending radius. This caused failure in the test pipe
sample. Lateral buckling was during dissection of the test pipe sample detected as large gaps in the
inner layer of tensile armour wires.
The third experiment in the test series was also constituted by two load cycles. The first of those had
as objective to decrease the load with respect to the second experiment. A load level of 160 kN was
chosen. After 1200 bending cycles, the twist was approximately 3 degrees and progressing very
slowly. It was concluded, that if the experiment simulates installation, the limit compressive load of the
test pipe sample is between 160 and 210 kN for an 11 meter bending radius. The second load cycle
was performed with 265 kN compression and the maximum bending radius of the pipe midpoint
increased to 8 meters. This caused failure to occur. The progression in the twist response was during
this load cycle very rapid compared to what had been observed during previous experiments during
the last few bending cycles. The experiment was stopped with a twist of approximately 45 degrees, but
due to the very fast progression of the response, this was mainly due to coincidence. The last load cycle
reveled, that lateral wire buckling may lead to instability progressing at a quite high speed as cyclic
bending is applied.
3.3. Test series II, 14 jumper test pipe samples in G2 bending rig
After the first series of lateral buckling experiments, an 8 riser test pipe sample from the third series
of experiments was attempted tested in the original lateral buckling rig. However, this test pipe sample
had a bending stiffness so large, that it caused damage in the test rig due to misalignment of the screw
drives in the bending arrangement. After tear down of the test pipe sample without any sign of failure,
the test rig was upgraded as part of a NKT-Flexibles commercial project. The upgrade, including
assembly of mechanical parts and construction of a new tension pipe, was of approximately two
months duration.
22
The second experimental series using two 14 test pipe samples was started in June and July 2009
with the first test pipe sample. The test pipe sample used was not designed as a riser connecting a
reservoir at the seabed to a floating platform, but as a jumper connecting the top of a tower of rigid
steel risers to a floating platform. The load program, which is summarized in Table 2, was defined on
basis of client specifications as simulation of field conditions. Initially, 6000 bending cycles were
applied simulating shut down periods expected to occur in the pipe service-lifetime. For each 1200
bending cycles, the test was interrupted and the pipe sample was tensioned and bent repeatedly. This
procedure was followed in order to attempt to draw the armoring wires back towards the helical
configurations, before the experiment was continued. This simulates, that the test pipe is
repressurized and tensioned. No signs of failure could be detected during this part of the experiment.
After this load cycle, which was of a duration of a few days, two load cycles simulating the installation
process were conducted. The second of these load cycles led to severe pipe twist and shortening of the
test pipe sample, see Figure 20. However, the progression of these responses was, as cyclic bending
was applied, significantly slower than detected during the first test series, see Figure 19. During
dissection of the test pipe sample, lateral buckling was detected in the inner layer of tensile armour as
localized gaps of moderate size. However, it was unclear if plasticity had occurred in the wires.
The second lateral buckling experiment of the present series was conducted in December 2010. The
test objective was, motivated by the NKT-Flexibles qualification, to examine the response of a flexible
pipe to extreme loads. Therefore, a load scenario close to maximum rig capacity was chosen.
Performing 20 bending cycles with this load input and afterwards applying 1200 bending cycles with
decreased loads simulating service should constitute an experimental simulation of the long term
effects of extreme loads. During the first load cycle simulating extreme loads, the pipe twist increased
in a stable manner until 18 bending cycles had been applied. Then a very rapid twist occurred and the
experiment was stopped, see Figure 21. No reason for this mechanical behavior could be detected in
the measured responses or by visual inspection of the test pipe sample. It was chosen to continue the
experiment and load cycle II was initiated. The pipe sample was left for approximately 48 hours
between the two load cycles. Furthermore, tension and repeated bending was applied before load
cycle II was initiated. The twist increased in a more stable manner during the last part of the
experiment. During dissection of the test pipe sample it was determined, that gaps of moderate size
occurred throughout the inner layer of armouring wires without signs of localization. No other reasons
for the rapid twist during load cycle I could be found. It therefore seems, that some global mode of
buckling has been triggered by the extreme loads applied.
30
100
Logged response
LC 2 started
LC 3 started
25
80
Compressive stroke (mm)
20
Pipe twist (deg)
Logged response
LC 2 started
LC 3 started
90
15
10
70
60
50
40
30
20
10
-5
1000
2000
3000
4000
5000
6000
Number of bending cycles
7000
8000
9000
1000
2000
3000
4000
5000
6000
Number of bending cycles
Figure 20, left: Pipe twist of the first 14 jumper, right: Compressive stroke of the first 14 jumper.
7000
8000
9000
23
10
18
16
14
5
4
12
10
8
6
2
1
0
500
1000
1500
2000
2500 3000
time (s)
3500
4000
4500
5000
200
400
600
800
1000
number of bending cycles
1200
1400
Figure 21, left: Pipe twist of the second 14 jumper, load cycle I, right: Pipe twist of the second 14 jumper, load
cycle II.
200
30
180
25
160
20
15
10
140
120
Test I, LC 1, 700 kN, 12 m bending
Test II, LC 1, 300 kN, 12 m bending
Test II, LC 2, 400 kN, 12 m bending
100
80
60
40
500
1000
1500
Number of bending cycles
2000
20
0
500
2500
1000
1500
Number of bending cycles
2000
2500
Figure 22, left: Pipe twist of 8 risers, right: Compressive stroke of 8 risers.
3.4. Test series III, 8 riser test pipe samples in G2 bending rig
The third test series was conducted from February to March 2011 using two 8 riser test pipe samples
in the upgraded test rig shown in Figure 10. The first experiment was conducted with load inputs
corresponding to DIP-test conditions encountered for a pipe sample with the same pipedesign at 1500
meter of water depth in the Brazilian Campos Basin. Considering the pipe twist and compressive
stroke in Figure 22, which were measured during the experiment, the responses can be observed to
remain stable and increase slowly until a certain number of bending cycles have been applied. Then
the measured values can be observed to increase in a more progressive manner throughout a number
of bending cycles and afterwards soften and progress at lower speed as cyclic bending is applied. This
behavior differs from the responses measured during the first test series in which the responses
continuously progressed until the experiment was stopped, see section 3.2. During dissection of the
first 8 test pipe sample, lateral buckling was detected as large localized gaps in the inner layer of
tensile armour.
The second experiment of this test series had as scope to estimate the compressive limit depth leading
to lateral buckling for a fixed bending radius of 12 meters. Initially, 1200 bending cycles were
24
performed with a compressive load of 300 kN without signs of lateral buckling. During the second load
cycle, the compressive load was increased to 400 kN, which caused the pipe twist and compressive
stroke to increase in a manner similar to the first experiment. Since the responses had not reached a
constant level after 1200 bending cycles had been applied, it was chosen to apply additional 1200
bending cycles. Since this did not cause the responses to stabilize entirely, it was estimated, that lateral
buckling had occurred and the experiment was stopped. However, during dissection of the test pipe
sample, no signs of lateral buckling could be observed and only small variations in pitch length could
be detected in the inner layer of armouring wires.
25
but has due to the very limited tension capacity of the mechanical rigs not been examined
experimentally yet.
The conducted experiments have for flexible pipes subjected to the chosen boundary conditions
enabled rough estimates of the compressive load carrying ability of the 6 and 8 risers for fixed
bending radii, while the load carrying ability of the 14 riser is more unclear, since the experiments
were not conducted for fixed pipe curvature.
The deformation patterns observed during the dissections are contained in section 5.5 along with
further elaborations on how the limit state design against lateral buckling may be chosen.
26
27
4. Theoretical work
In this section, an overview of the theoretical work conducted in the present project will be presented.
Initially, elaborations regarding the choice of approach to wire mechanics will be included. Afterwards,
the content of the appended papers will be summarized one by one.
The mechanics of beams have been subject of research for several centuries. The well-known fourth
order differential equation usually referred to as the Bernoulli-Euler equation (3), which is derived
assuming deformations and rotations small, is probably among the theoretical results which to the
widest extend have been applied for engineering analysis
(3)
(4)
Equation (3) relates the fourth order derivative of deflection w as function of longitudinal position x to
the applied distributed load q with EI as measure for the flexural stiffness. The basic underlying
assumption, which is considered valid for long and slender members, is, that shear strains can be
neglected when calculating the deflection. As a consequence, cross sections plane before deformation
are assumed to remain plane in bending. For short members, this cannot be considered to be the case
and an extended formulation developed by Timoshenko can be applied in order to account for shear
effects. Large deflections and rotations were taken into calculation in an approach usually referred to
as the elastica equation, often by formulating the problem in rotations as function of wire arclength s
rather than in deflections, see Bazant and Cedolin [25]
in which P is the longitudinal load applied in a distance e, see Figure 9. The solution to equation (4) is
given by an elliptic integral of first kind. Similar means for formulation of the equilibrium equations
for offshore cables were followed by Terndrup-Pedersen, [26]. The extension of these approaches
necessary to describe the mechanical behavior of long and slender space curved beams has been
investigated for more than a hundred years. Despite of the fact that very few analytical solutions have
been obtained, a set of equations governing the equilibrium of space-curved beams is well-described
in the literature. Reissner [27] used the equations on the vectorial form
dP
+p=0
ds
dM
+ t p+m = 0
ds
(5)
in which P is the sectional beam load, M the sectional moment, p distributed external loads, m
distributed moments and t the beam tangent. It is unclear who first derived the equations, which were
addressed by Clebsch [28], but also contained in Loves book on theory of elasticity [29]. These
equations are in the present project solved in a mixed formulation including both geometry and force
variables for a beam which is partially constrained to a toroid, see section 4.1. This approach leads to a
nonlinear system of six differential equations. The equations of equilibrium for a long and slender
curved beam have been applied to a wide range of problems in numerous publications. Costello [30]
used the equations of equilibrium when addressing large deflections of helical springs. Spillers et. al.
[31] followed a similar approach when calculating stresses in a helical tape on a bent cylinder with the
underlying assumption, that the lay angle was constant. Stump and van der Heijden [32] and Vaz and
Patel [13]-[16] formulated the equations governing the structural behavior of offshore cables and
flexible pipes on basis of equilibrium equations similar to those given in equation (5). Furthermore,
similar equations were used to address the equilibrium of armouring wires in dynamically bent
flexible pipes by Leroy and Estrier [33]. This work constitutes one of very few known examples of
research in which transverse wire slippage is taken into calculation with frictional forces included as
transverse wire loads. However, an experience-based solution form was chosen as basis of the
analysis. Svik [34]-[35] developed a finite element model based on finite-strain continuum
28
mechanics addressing the same problem as Leroy and Estrier, with no restrictions regarding lay
angles and transverse slip. Furthermore, the global radius of curvature could be varied and boundary
effects close to end-fittings taken into calculation. The finite element model was afterwards applied as
basis for computer programs for analysis of armouring wires, see [36], however, assuming slip solely
to occur along curves with constant lay angles. Witz and Tan [6] followed a different approach based
on lineralized strains and potential wire energy, when addressing problems related to slippage along
bent helices. The determined critical curvature causing slippage was also derived by Kraincanic and
Kabadze [8] through a slightly different process. A question often addressed is which curve a wire
within the wall of a bent flexible pipe can be assumed to follow. While the assumption, that no
transverse slip will occur has been applied to a wide extend, Out and von Morgen [37] considered the
geodesic on a toroid as the equilibrium state in bending and tension focusing on constructing the
geodesic curve.
The problem is initially formulated without taking frictional effects into calculation. As a
consequence, the fatigue-like aspect of the problem is neglected by assuming, that the
physical wire equilibrium state, towards which convergence will occur as cyclic loads are
applied, equals the equilibrium state obtained directly without friction when the wire is
loaded.
The radius of curvature R of the modeled pipe will be considered constant. Hence, for the sake
of simplicity curvature variations are neglected.
Contact with adjacent layers will be considered by assuming the modeled wire embedded in a
cylinder bent into a toroid.
The wire dimensions will be assumed small compared to the minor torus radius, r, which is
convenient when formulating the constitutive relations
29
t n
tu
z
x
u (s)
t
b
Figure 23, left: An armouring wire modeled as a curve embedded in a toroid, right: Vector coordinate triads.
With these assumptions, the wire constitutes a curve (s) embedded in a toriod, which is
parameterized by an arclength coordinate u(s) along the torus centerline, and an angular coordinate
(s) along the circumferential direction, see Figure 23. The curve will be assumed parameterized by
arclength, s. A point on the torus surface is in Cartesian coordinates given by
,
(6)
A curve is constructed by establishing a relation between u(s) and (s). Assuming such a relation
established, a curvilinear coordinate triad can be attached to each point on the curve as a wire tangent
t, a normal n and a binormal b by means from basic differential geometry. The wire normal n is
assumed to coincide with the surface normal. As a consequence, the wire rotation around the local
tangent is governed by the underlying toroid. This is deemed to be a fair assumption for flexible pipes
with high-strength tape. In order to describe the curvature of the wire, this is split into a component in
the normal tn-plane, n, a geodesic component in the toriod tangent plane, g, and a torsional
component in the nb-plane, . A result from basic differential geometry known as the Darboux frame is
available for definition of the curvature component in terms of the triad vectors and their derivatives
in s
0
"
!#% &(
)
$
(
0
*
) "
* + !#%
0
$
(7)
Considering the vector triads in Figure 23 and Equation (7), it is noted that for a positive change of
wire arc length, a positive rotation around a local wire axis corresponds to a positive change of
curvature. Inserting proper definitions of the triad vectors in Equation (7), the following curvature
components are derived
( 01,-./
2 1
2
,-./
(8)
30
) 01,-./
2
.3(/
4
.
* 01,-./ 1
2
2
,-./
(9)
(10)
Alternatively, the definitions of curvature given by do Carmo [44] may be applied, which lead to the
same expressions, see Appendix A.
The equilibrium equations for long and slender curved beams have in the past served a wide range of
applications in applied mechanics. The equations were included in Loves famous book on theory of
elasticity, [29], and were given by Reissner [27], on vectorial form. It is desirable to express the
equilibrium equations with the local coordinate triad of the wire as basis
"
$
7
(
8
6 7
(
8
"
#
$ 97 " 9( # 98 $
:
:
7
(
( ( ) 8 97 ; " :
( 7 *8 9( ; #
8
) 7 *( 98 ; $ 0
<
"
$
?7
?(
?8
= 5 > " ?7
?(
?8
"
#
$
@7 " @( # @8 $ 8 # ( $
:
?7
?(
( ?( ) ?8 @7 ; " :
( ?7 *?8 8 @( ; #
(11)
(12)
?8
) ?7 *?( ( @8 ; $ 0
The following system of field equations governing the wire equilibrium can now be derived, if the
toroid is considered frictionless, causing distributed wire loads in the tangent plane to be neglectable
2
1
2
2
2 )
1
(13)
(14)
7
( ( ) 8
(15)
?(
( ?7 *?8 8
(17)
8
) 7 *(
(16)
(18)
31
in which equation (13) and (14) are derived on basis of the wire geometry in the toroid tangent plane,
equation (15) is derived on basis of the expression of the geodesic curvature given in equation
(9),
equation (16) governs the tangential force equilibrium, equation (17) the binormal force equilibrium
and equation (18) the normal moment equilibrium. This system of equations derived in Paper A form
the basis of the theoretical work conducted in the present project. They have been applied for single
wire analysis of perfect and imperfect wire geometries (Paper B), global models of the tensile armour
layers taking the torsional imbalance due to lateral wire buckling into calculation, (Paper C),
preliminar studies of frictional effects (Paper D) and, finally, simplifications of the global tensile
armour models (Paper F). Plasticity has not been simulated in the present work. However, this would
have been possible by applying appropriate constitutive relations.
32
paths, which due to frictional energy dissipation exhibit a loop-like behavior. Obviously, if this loop is
closed after a few bending cycles, the system is stable and further cyclic loading can be assumed not to
cause instability. On the other hand, if the loop is open, wire slippage may occur towards a state, which
leads to lateral wire buckling. Further elaborations of instability, when friction is present, are
contained in Paper D.
Mechanical systems, which can be described assuming rotations and deformations small, may be
analyzed with respect to instability on basis of eigenvalue problems arising from the formulation of
the equilibrium conditions in the deformed state. If rotations and deformations cannot be assumed
small, this must be accounted for when formulating the governing equations. In the present context,
stability considerations are conducted on basis of full non-linear analyses of perfect and imperfect
structures.
P
P
c.
b.
a. P = EI
2
L
d.
P=
EI
L2
Bifurcation point
Figure 24 Examples of equilibrium paths in (compressive load-displacement) diagrams for simply supported
beam-column, reproduced after Brush and Almroth [41], a.) Initially straight beam-column, small deflections
and rotations (Euler solution), b.) Initially straight beam-column with large rotations, c.) Initially slightly
crooked beam, primary unstable path, d.) Initially slightly crooked beam, secondary stable path.
4.3. Paper A
A method for prediction of the equilibrium state of a long and slender wire on a frictionless toroid
applied for analysis of flexible pipe structures
In the first paper, the system of governing equations (13)-(18) is derived on basis of the equilibrium of
space curved beams and the geometry of a wire modeled as a curve on a torus surface. Frictional
effects were neglected assuming, that the wire equilibrium state obtained after a significant number of
bending cycles applied is equal to the equilibrium state, which is obtained for a given load input if
friction is neglected. It is demonstrated, that the derivation of the curvature components can be
performed on basis of concepts from abstract differential geometry, see do Carmo [44]. This approach
corresponds in a mathematical sense to direct application of the Darboux equation (7) to the derived
local vector triad of the wire, but eases the derivation significantly. Furthermore, it is demonstrated
how the system of governing equations can be solved numerically by application of a commercially
available solver. A solution is obtained after having converted the differential equations from being
functions of deformed arclength to functions of undeformed arclength. This is conducted on basis of
the assumption that strains will remain sufficiently small to be described using Cauchys definition of
strain. Obtained results reveal, that the calculated equilibrium state of the wire approaches the
geodesic of a toroid (a curve possessing no geodesic curvature) when the simulated pipe is bent and
tensioned.
33
pure bending
0.93
0.92
0.91
0.9
0.89
0.88
0
3
4
5
Wire arclength, s(m)
Figure 25, left A single armouring wire within the wall of a flexible pipe modeled as a curve embedded in a
torus surface, right: Wire lay angle for different configurations and loads.
4.4. Paper B
Imperfection analysis of flexible pipe armour wires in compression and bending
On basis of the established method for frictionless single wire analysis proposed in Paper A, the
mechanical behavior of a wire in compression and bending is addressed. In order to examine the
sensitivity of the wire to small initial imperfections, a small harmonic perturbation is added to the
initial geodesic curvature. The system of equations governing the static wire equilibrium is solved for
stepwise increased compressive loads with the solution of one load step as initial guess to the
following. By this method, the equilibrium path of the loaded end of the wire was studied. For perfect
wire structures, a linear equilibrium path (Branch A, Figure 26) and a path exhibiting significant
softening behavior (Branch B, Figure 26) were obtained. For small geometrical imperfections, which
were chosen in a manner, so the initial linear response was not influenced, another solution was
obtained. This solution softened at a load level below what was obtained for perfect structures
(Branch C, Figure 26). While equilibrium states along branch A only differs little from a bent helix, both
branch B and C differs significantly, which is interpreted as buckling. However, branch B is
symmetrical with respect to the pipe midpoint, which is not the case for solutions along branch C. The
effect of key parameters is examined and it is determined that decreasing the pitch length of the
modeled wire increases the load carrying ability significantly. On the other hand, it is worth noticing
that the applied pipe curvature seems to have very little effect on the load carrying ability. Finally, it is
demonstrated, that the number of pitches included in the computational model impose significant
influence on the load carrying ability and mode of deformation for low number of pitches. The
analyses may indicate that buckling by bifurcation has been encountered, since the uniqueness of the
solution vanishes after a certain load level even for perfect wire geometries. This, however, is not
demonstrated in an exact mathematical sense. For the imperfect structures analyzed in the present
context, buckling is usually considered likely to occur as limit points rather than bifurcations as
described by Koiter [42]. This corresponds well with the obtained results.
34
1400
A
1200
B
C
1000
D
m=0
m=1, =0.001
m=2, =0.001
800
m=5, =0.001
i
m=20, =0.001
i
600
m=1, =0.001
i
m=2, =0.001
i
m=5,i=0.001
400
m=20, =0.001
i
m=20, =0.01
i
m=20, =0.0001
200
m=20, =0.005
i
Primary path
3
Compressive pipe strain
6
4
x 10
Figure 26 Equilibrium paths for armouring wire in compression and bending (wire geometry shown in Figure
25), m denotes the number of harmonic terms taken into calculation in the applied imperfection, the magnitude
of the imperfection amplitudes (these are set equal to each other).
4.5. Paper C
On modeling of lateral buckling failure in flexible pipe tensile armour layers
On basis of the methods for single wire analysis proposed in paper A and B, a global model of the
armouring layers is proposed. It is shown that the outer layer of armouring wires can be modeled with
the assumption that wire slippage can only occur along curves with constant lay angles. This approach
leads to the same load carrying ability as an analysis in which full lateral slippage is allowed in both
layers. However, neglecting lateral slippage in the outer layer decreases the computational power
necessary to obtain a solution significantly. As softening occurs, the torsional balance of the pipe
structure can no longer be maintained. This leads to a pipe twist causing further straining of the wires
in the inner layer. A model for calculation of the torsional moment in the free end of the pipe for a
specified pipe twist and a given longitudinal pipe strain is established on basis of multiple single wire
analyses. Lateral wire contact is neglected in this context. On basis of this model, the torsional
equilibrium equation for all wires is solved with respect to pipe twist for fixed pipe strain values by
Newton-Raphson iterations. Results are compared to the equilibrium paths obtained for torsionally
fixed-fixed pipes. Significant difference in load carrying ability is detected between the two scenarios.
However, the difference is imposed solely by the forces in the outer layer. These are for a torsionally
fixed-free pipe free to relax, which is not the case for a torsionally fixed-fixed pipe. It is demonstrated,
that the model is capable of simulating modes of deformation, which locally correspond well to the
experimentally triggered modes. Radial elasticity of the pipe wall is accounted for by considering the
minor torus radius in the loaded configuration a function of the applied longitudinal pipe strain. This
approach is equivalent to assigning a Poissons ratio to the global pipe structure. It is demonstrated,
that little effect on the load carrying ability can be detected, when radial deformations are accounted
for in this manner.
35
M
A
1
R=
1
R=
Figure 27 Equilibrium paths of modeled layers of armouring wires, radially stiff structures, left: Torsionally
fixed-fixed pipe, right: Torsionally fixed-free pipe.
4.6. Paper D
Simulation of frictional effects in models for calculation of the equilibrium state of flexible pipe
armouring wires in compression and bending
In the fourth paper, frictional effects on a single wire within the wall of a compressed flexible pipe subjected
to cyclic bending are investigated. Friction was modeled by a regularized Coulomb-law and directed
oppositely to the slip velocity. This is implemented in the system of field equations as distributed transverse
and tangential loads. Frictional effects are addressed in a similar manner by Pfister [38]. The system of
equations is solved stepwise for a prescribed load history. However, for the sake of simplicity, inertia terms
were neglected in the system of equations solved. This was estimated to be fair in the case of cyclic bending
applied at very slow speed. The system is hereby analyzed in a quasistatic manner and the mathematical
formulation of the problem still solved as a boundary value problem.
Papp=2.0 kN
10
0.02
0.04
0.06
Pipe curvature, (1/m)
0.08
0.1
Figure 28, left: Calculated pipe strain as function of load step number from quasistatic frictional analysis, right:
Local moment contribution to global bending hysteresis.
36
The system of equations can only be solved for moderately small transitions from zero to full friction. As a
consequence, the number of bending cycles leading to instability in simulations cannot be expected to
correspond to experimental results. On basis of the present research it cannot be concluded that frictional
effects impose significant influence on neither mode of deformation nor load carrying ability. The model
exhibited a mechanical behavior which in a qualitative sense corresponds well to results published in related
publications with respect to bending hysteresis and transverse vs. binormal slip.
The present paper can due to the unresolved issues only be considered as a preliminary study in frictional
effects on tensile armour wires. Further research is needed in order to formulate the problem by more strict
methodical means.
4.7. Paper E
On lateral buckling failure of armour wires in flexible pipes
The fifth paper is a full length conference paper from proceedings of the OMAE (Offshore, Marine and
Arctic Engineering). It contains a summary of the work presented in Paper A-C, in which a simplified
approach to the single wire model is applied. Only the field equations governing the tangent geometry
are converted from deformed to undeformed arclength. The approach proposed in Paper C is followed
in order to establish a computational model of the armouring layers. This model is used to calculate
the frictionless equilibrium paths of the 6 riser, which is considered torsionally fixed-free. The
obtained load carrying ability is compared with the experimental results from test series I, which are
described in section 3.2. Neglecting the remaining layers of the flexible pipe, it is furthermore
attempted to study how the effect of limited wire slippage close to end-fittings influences the load
carrying ability. This is conducted by setting the model length shorter than the physical length of the
pipe. Furthermore, the maximum inner layer stresses are calculated. It is shown that while the
maximum pipe curvature has very little influence on the calculated load carrying ability of the wires, it
imposes significant influence on the calculated stresses. Finally, it is demonstrated that the modes of
deformation obtained by the model locally correspond well to the buckling modes, which are triggered
during the experiments, see Figure 29.
37
4.8. Paper F
Simplified models for prediction of lateral buckling in flexible pipes armour wires
The sixth paper is like paper E a full length conference paper from proceedings of the OMAE (accepted
for presentation in 2012). In this paper, simplified global models of flexible pipes are proposed in
order to demonstrate means, by which the work contained in Paper A-C and E can be converted to a
form, which is more appropriate for engineering analysis. In order to limit the computational power
necessary to perform lateral buckling analyses, only a single wire in the inner layer of armouring wires
is analyzed, and the torsional moment contribution is scaled in order to model the mechanical
behavior of the entire layer. Two different approaches were followed and denoted A and B.
In the first model simplification A, only the mechanics of the tensile armor layers is addressed. The
mechanical behavior of the inner layer of armourig wires was described as the scaled result from a
single wire analysis. The outer layer of armouring wires is modeled with the linear equations
contained in [4]. It is demonstrated, that while this approach leads to a poor representation of the
force response prior to buckling, the load carrying ability in the buckled state is approximated with
very good accuracy. This is deemed to be due to the fact, that the load carrying ability of the armouring
wires exhibit little dependency on the boundary conditions in the circumferential pipe direction. The
equilibrium paths of the full global pipe model and the present simplification are shown in Figure 30.
By the second model simplification B it is assumed, that the inner layer of armouring wires all behave
linearly in a (force-strain)-diagram until the load carrying ability determined by a single wire analysis
is reached. After this load level, the layer is assumed to soften completely. Results are shown in Figure
30. The main conclusion of the present analysis, despite the simple and rather crude approach chosen,
is that while adjacent pipe layers do not impose significant influence on the load carrying ability prior
to buckling, an influence can be detected as a slope of the equilibrium path in the buckled state.
Obviously, this improves the prediction of the experimental results, since the conservatism of the
calculated load carrying ability is limited. This is obviously related to the fact, that it with this
approach seems possible to let overstressing of the wires define the limit state design. Further results
and elaborations regarding the choice of limit state design are included in section 5.2.
140
120
100
Longitudinal force (kN)
200
150
100
80
60
40
Inner layer of armouring wires
Outer layer of armouring wires
ABC-layer
Outer sheath
Sum of layers
20
50
-20
0
0.2
0.4
0.6
0.8
Longitudinal compressive pipe strain
1
3
x 10
10
15
20
25
30
Pipe twist (deg)
35
40
45
50
Figure 30 Equilibrium paths of 6 riser pipe, left: Model simplification A compared to the full global pipe model
proposed in Paper C, only tensile armour layers are modeled, right: Model simplification B, postbuckling
response taken into calculation. Tensile armour layers, high-strength tape and outer sheath taken into
calculation.
38
39
2.
3.
A full global model of the armouring layers which for a prescribed longitudinal strain by
Newton-Raphson iterations can be used to determine the pipe twist of the free pipe end for
which the torsional equilibrium is fulfilled. The model was based on separate analyses of all
armour wires neglecting lateral wire contact. The method was proposed in Paper C and
numerically simulated results were compared to experimentally determined modes of
deformation and load carrying abilities in Papers E and F.
This model has been applied in order to analyze the armouring layers of the 6 and 8 risers.
Contributions from adjacent layers were neglected, since these effects were found to slow down
convergence significantly. The 14 jumper has not been analyzed using this model, since the
larger number of wires applied in this pipedesign turned out to slow down convergence to an
extend, by which the method was no longer well-posed to be used on desktop computers.
A simplified iterative model, denoted A, by which the force and moment contributions from the
inner layer was determined by scaling the results from a single wire analysis. The outer layer of
armouring wires was modeled using linear equations. A solution in pipe twist to the torsional
equilibrium of the free pipe end was for fixed values of longitudinal pipe strain obtained in a
manner equivalent to the methods described under 1).
This method was by analysis of the armouring layers of the 6 and 8 risers in paper F shown to
correspond very well to the full global pipe model in the postbuckled state, despite of the fact
that the response of the armouring layers prior to buckling was not estimated accurately. This
method is used to calculate the load carrying ability of the armouring layers of the 14 jumper in
the present chapter.
A simplified linear model, denoted B, was established. In this model it was assumed that the
inner layer of armouring wires exhibited a linear mechanical behavior based on equilibrium on
perfect helices until the load carrying ability of the layer was reached. This was assumed to
cause the load in the layer to soften completely to a constant level, which correlates well with
the results obtained by the models described above. The adjacent layers were modeled linearly.
This model was in Paper F used to estimate the postbuckling response of all three flexible pipes,
which had been tested experimentally. Despite of the fact that the model was coarse and timedependant effects in the polymer and tape layers are not taken into calculation, it was
demonstrated that those layers which prior to buckling generate insignificant contributions to
the load carrying ability and torsional moment, impose significant influence on the postbuckling
response.
In the following, results obtained with the various proposed methods will be presented.
40
Outer sheath *4
6 Riser *1
0.201
1.263
26.2
3 10
52
0.209
1.318
-26.2
3 10
54
0.212
0.075
83.5
1 60
1
0.225
6.0
8 Riser *2
0.276
1.474
30
5 12.5
54
0.289
1.525
-30.3
5 12.5
56
0.292
0.025
88.4
1.8 1.3
8
0.434
10.0
14 Jumper *2
0.442
2.247
31.5
4 15
70
0.452
2.345
-31.0
4 15
72
0.455
0.140
-84.4
1 60
2
0.477
10.0
3.96
3.39
3-34
*1) A basic grade steel used for wires with yield strength of approximately 650 MPa, elastic modulus 210 GPa,
Poisons ratio 0.3
*2) A high strength grade steel used for wires with yield strength of approximately 1350 MPa, elastic modulus
210 GPa, Poisons ratio 0.3
*3) Tape material properties chosen are: elastic modulus 27 GPa. Poisons ratio 0.4
*4) Sheath material properties chosen are: elastic modulus 400 MPa, Poisons ratio 0.4
Table 3 Pipe designs and material properties.
5.2. Discussion of the definition of lateral wire buckling limit state design
As discussed in section 1.2, lateral wire buckling may lead to failure in a flexible pipe structure as the
inner layer of tensile armour deforms to an extend by which the plastic limit of the wire steel is
exceeded. However, this may not always be the case as demonstrated experimentally by testing of the
second 8 riser. This flexible pipe twisted and shortened severely during the experiment, but no sign of
lateral wire buckling could be detected by dissection of the test pipe sample after the experiment had
been concluded, see section 3.4. Considering the computational models proposed in the present thesis,
it is clear that the load carried by the entire pipe structure may still increase in the postbuckled state.
Lateral wire buckling may therefore from a pipe design perspective be defined in two different
manners:
A. The conservative definition of the phenomenon is that softening occurs in the inner layer of
armouring wires due to repeated bending and longitudinal compression. Hence, lateral wire
buckling is governed mainly by the applied compression and occurs when compression causes
the pipe structure to become torsionally unstable. With this definition, lateral wire buckling
can be predicted with reasonable accuracy with computational models only including the
tensile armour wires.
41
B. A less conservative definition of lateral wire buckling taking the postbuckling response into
calculation may also be given. The limit state design can be defined as the state in which the
structure does no longer function in a satisfactory manner when subjected to in-serviceconditions due to plastic deformations in the inner layer of armouring wires. This definition
allows large deformations of the pipe structure and must be considered using models in which
high-strength tape and outer sheath are included.
Obviously, the two definitions are related to the way by which the allowed limit state used for pipe
design is chosen. Considering only the tensile armour layers, definition A will from a computational
point of view give a lower bound for the point at which lateral wire buckling needs not to be
considered in the design process.
Definition B of the limit state design has not been considered in detail by global analysis of the
armouring layers, since the computational effort related to loading the wires until yielding is quite
severe. However, this way of defining the limit state shows large potential, since it enables the
postbuckling response to be taken into account when calculating the load carrying ability.
The ultimate objective of computational modeling of the failure phenomenon would be to accurately
predict the deformation state of a flexible pipe subjected to compression and repeated bending as
function of the number of applied bending cycles. Preliminar studies of cyclic loads are contained in
Paper D, but are with the computational power of the computers usually applied for flexible pipe
designs not a very practical approach. Furthermore, as described in section 6.2, further research is
needed in order to obtain a consistent model capable of predicting the deformation state of a flexible
pipe as function of the number of applied bending cycles.
Linear compressive
pipe response
First yield
P
Postbuckled pipe response,
all layers
Load carrying ability of model
Equilibrium path, only
including only armour layers
armouring layers modeled
Torsional pipe stability can
no longer be maintained
Figure 31 Equilibrium paths calculated by computational models including 1) only the tensile armour layers
(marked in blue, related to definition A), 2) All pipe layers (marked in red, related to definition B).
42
equations are widely applied for prediction of the force or deformation response of straight flexible
pipes, see [4] and [5]. Each layer is initially considered separately and is modeled either as helically
wound or isotropic cylindrical. In the present approach, the change of thickness of each layer will be
neglected, so each layer contributes to the global system of equations with three equations. The
parameters given in Table 4 can pair wise either be considered unknown or specified.
Force parameters
F
Layer of share of longitudinal force
Deformation parameters
Longitudinal pipe strain
L
M
P(i)
P(i+1)
Pipe twist
A number of global equations ensuring equilibrium and compatibility of the pipe structures may be
introduced. Along with equations for each layer, these constitute a linear system of equations which
easily can be solved. For helically wound layers the equations are given by
F
L
P (i ) P (i + 1) sin 2 hel
+
+
r cos 2 hel
r sin hel cos hel
=0
nwires ,i cos hel EA E 2
E
2
r
L
L
r tan hel F M = 0
1
t
t
C E 3 1
E - 1
HI
: ; : ; 1
0
D F 2
F 2
I
rF r ri
r r
+ + P(i ) o P(i + 1) r = 0
EA
E t 2
E t 2
M
=0
GJ
L
in which ri and ro denotes inner and outer radius of the modeled layer and t the layer thickness. It is
noted that the equations are derived using a different coordinate system than in the present work, see
Figure 32. It is necessary to account for this difference in the formulation when combining the two
approaches to simplified global models for lateral buckling prediction. Furthermore, by modeling all
layers except the armour layer prone to buckling with these equations, bending terms are neglected. It
has in Paper F been demonstrated that this is reasonable for the outer layer of tensile armour.
b'
n'
t'
43
Pcr, calc
(kN)
The load carrying ability can now in accordance with definition A be calculated by the full global model
and simplified global model A by considering the determined equilibrium paths. Only the tensile
armour layers are included in the models. The results for the 6 and 8 pipe structures are contained in
paper E. The 14 jumper was due to the very large number of wires only analyzed using the simplified
model. The obtained equilibrium paths are shown in Figure 34. It is interesting to note, that a tensionbending coupling arises in the simplified model. The load carrying abilities are summarized in Table 5.
For the 6 riser, the load carrying ability is within the window determined experimentally in
accordance with definition A. For the 8 riser the simulated load carrying ability is lower than the
experimentally determined values. Hence, the load carrying ability is calculated conservatively.
The postbuckling response has only to full extend been taken into calculation by model simplification
B. This does not imply that this would not have been possible with the full global pipe model, but the
computational effort necessary to perform such analyses is quite severe, and has therefore not been
considered as part of the present work. Values will be presented for respectively 10 and 20 degrees of
pipe twist. In all cases, this increases the load carrying ability significantly. However, for the 6 and 8
risers, the load carrying ability can still be observed to be estimated conservatively. Considering
definition B of lateral wire buckling, stress calculations must be incorporated in the established
methods in order to determine when failure occurs. However, in order for this to be reasonable for the
simplified models, which are based on analyses of only a single wire in the inner layer of tensile
armour, the stress levels must be at least approximately the same in all wires. This is investigated
further in section G6 on basis of the full global pipe model.
44
The experimental results summarized in Table 2 and the load carrying ability calculated with model
simplification B are compared in Figure 33. Modeling the postbuckling response can be observed to
improve the calculated load carrying ability, despite the fact that predictions are still conservative with
respect to the experiments.
700
Slowly increasing, failure
600
500
400
300
200
Slowly increasing
10
15
20
25
30
35
Twist of free end before unloading (deg)
40
45
Figure 33 Experimental results (contained in Table 2) compared with load carrying ability predictions from
model simplification B, comments to measurements in the figure are related to the progression of twist
response as the experiment was concluded.
14" Jumper, simplified pipe model, radially stiff pipe structures
300
180
250
140
200
Longitudinal force (kN)
160
120
100
80
60
150
100
50
40
=0
=1/10 m-1
=0, reduced length
20
0.5
1.5
2
2.5
3
3.5
Longitudinal compressive pipe strain
-50
4.5
5
-4
8
10
Pipe twist (deg)
12
14
16
x 10
Figure 34, left: Equilibrium paths of 14 jumper calculated with simplified global pipe model A, only tensile
armour layers modeled, right: Equilibrium paths of 14 jumper calculated with simplified global pipe model B,
outer sheath and high-strength tape included in model.
45
Test 0
Test I
Test II
Test III
Mode I
Mode II
Mode III
Figure 35 Modes of deformation from experiments and simulations (radially stiff analyses), 6 riser, Mode I:
R=11 m, pipe strain J 2.5 10N , Mode II: R=11 m, pipe strain J 1.0 10NO , 20 imperfection terms with
amplitudes of -0.001, Mode III: Equivalent to mode II, but with positive imperfection amplitudes.
46
Initially, considering the 6 riser the modes of deformation were at a local level shown to correspond
well to simulated results. However, the detected modes of deformation did not localize in the same
manner as detected experimentally, see Figure 35. The result from the initial trial test has deformed to
an extend which seems governed by plasticity. The remaining deformation patterns form localized
large gaps, S-shapes or a combination of the two deformation modes, by which the wires, despite the
formation of gaps, seem to sustain some S-shape. Three different modes of deformation could be
simulated. Mode I is a periodic S-shaped deformation mode, which corresponds well to the wire
configurations detected localized during Test I. The two other simulated modes of deformation are
characterized by large wire gaps localized in each end of the layer. Comparing these modes of
deformation with the modes encountered during dissection of pipes, gaps can be observed to localize
closer to the boundaries in the simulations, while occurring on each side of the pipe midpoint during
experiments. It seems likely, that this discrepancy occurs due to boundary effects in form of limited
wire slippage close to end-fittings.
Considering the modes of deformations of the 8 riser, the simulated modes of deformation were of a
more severe character than encountered during laboratory experiments, see Figure 36. The gaps in the
simulated deformation pattern are localized symmetrically around the pipe midpoint, while gaps only
occurred between the static frame end and the pipe midpoint during the experiment.
The 14 jumper was not analyzed using the full global pipe model. Experimentally triggered modes of
deformation are shown in Figure 37.
Test I
Test II
Figure 36 Modes of deformation obtained experimentally and simulated, 8 riser (radially stiff analysis),
R=12m, J 4.5 10NQ , 20 imperfection terms with amplitudes of -0.001.
47
Test I
Test II
48
n
t
1 =
Pt M n n Eh
+
+
A 2In
2
2 =
Pt M n n Eh
+
A 2I n
2
3 =
Pt M n n Eh
A 2In
2
4 =
Pt M n n Eh
+
A 2I n
2
max = max( 1 , 2 , 3 , 4 )
Figure 38 Maximum stress in wire cross-section, radially inelastic pipe structure, pipe strain R 7 10NQ ,
levels: blue : max < 100 MPa, green : 100 max <200, yellow : 200 max <300, red : 300 max.
Maximum wire stress
450
400
350
300
250
200
150
100
50
ini (rad)
Figure 39 Maximum stresses in wires as function of initial wire angle (related to position in circumferential
direction), 6 and 8 risers, radially inelastic global pipe models.
49
The full global pipe model is incapable of taking lateral wire contact into account. This effect may
cause interlocking effects which prevent shortening or twist, and is therefore estimated to increase the
load carrying ability. However, it is possible to perform computational checks of whether the wires are
in transverse contact or not during post processing of obtained results. An example of such a contact
check is shown in Figure 40. In general, no lateral wire contact occurs prior to softening of the inner
layers and is at larger compressive strains mainly caused by formation of gaps.
Figure 40 Lateral wire contact tjeck, contact violation marked in red, radially inelastic pipe structure, pipe strain
J 7 10NQ .
in which the quantities are given in Figure 9. Gravitational effects have been neglected. The deflection
y of the pipe midpoint was measured during the experiment. However, it is noted that the bending
arrangement in the upgraded test rig is constructed in such a manner, that deflection measurements
by extensometers between the topbar and the test pipe sample will include a horizontal component.
This is caused by a change of the horizontal distance between the pipe midpoint and the extensometer
mounting point on the topbar, as bending is applied. This was not the case in the G1 test rig, in which
this distance remains constant. However, the influence on the calculated moment was estimated to be
below 3% and is therefore neglected in the present context.
The hysteresis loops which by this method are obtained on basis of experimentally measured data are
shown in Figure 41 and Figure 42. The following observations can be made:
The hysteresis loop does due to the presence of gravity on the test pipe sample not intersect
the origin of the (,M)-diagram like the idealized behavior shown in Figure 2. This causes the
hysteresis loop to have a negative offset on the vertical axis. The curvature values also reveal,
that the test pipe sample was not straightened completely due to gravitation.
Differences in compressive load level can be observed to impose very large influence on the
flexural behavior
The hysteresic behavior of the 14 jumper can for the experiment representing extreme load
conditions be observed to change for each bending cycle as the wires slip.
50
8000
Test I, LC 1, 700 kN
Test II, LC 1, 300 kN
Test II, LC 2, 400 kN
=1/12 m
6000
Bending moment (Nm)
x 10
10000
4000
2000
0
-2000
1
0
-1
Test I, LC 1, 265 kN
-4000
Test II, LC 1, 80 kN
-6000
-2
=1/11m
-8000
0.02
0.04
0.06
0.08
Curvature, , of pipe midpoint (1/m)
0.1
-3
0.12
0.01
0.02
0.03
0.04
0.05
0.06
0.07
Curvature, , of pipe midpoint (1/m)
0.08
0.09
Figure 41, Flexural hysteresis of pipe midpoint in curvature-moment diagram, left: 6 riser various load
cycles, right: 8 riser, all load cycles.
4
x 10
4
LC I, 277 kN compression
LC II, 269 kN compression
LC III, 411 kN compression
x 10
2
0
-1
-2
1
0
-1
-2
LC I
LC II
-3
-3
-4
0.04
0.06
0.08
0.1
0.12
Curvature, , of pipe midpoint (1/m)
0.14
0.16
-4
0.12
=1/6m
=1/8m
0.13
0.14
0.15
0.16
Curvature, , of pipe midpoint (1/m)
0.17
0.18
Figure 42, Flexural hysteresis of pipe midpoint in curvature-moment diagram, left: 14 jumper Experiment I,
right: 14 jumper Experiment II.
51
design criteria, definition B given in section 5.2 was chosen. Hence, the single wire model was loaded until
first yield occured. Analyses showed that this was conservative with respect to the results obtained by the
laboratory tests described in chapter 3 in the present thesis. In order to address the lacking one-to-one
correspondence between laboratory experiments and DIP-tests in the best possible manner, the model was
calibrated against available DIP-test results. This is, due to commercial issues, not included in the present
thesis. The developed design tool was named TFlex, see Figure 43.
Figure 43 The logo of the NKT-Flexibles TFlex tool for design of flexible pipes against lateral wire buckling.
The documentation of the TFlex code [71]-[73] and NKT-Flexibles test and analysis reports were
reviewed by Bureau Veritas as third part in order to obtain a certificate confirming the correctness of
the developed method. The review was conducted in autumn 2011. At the time, at which this project
was finalized, a draft revision of a design methodology certification report [74] had been issued, see
Figure 44. The draft section certifies the validity of the method for design of flexible pipes of 2 to 10
bore diameter for water depths not larger than 2000 m. The papers contained in the present thesis
were also reviewed by Bureau Veritas as part of the certification process. These were issued as NKTFlexibles documents [75]-[80].
Figure 44 Frontpage for Bureau Veritas draft revision of type approval certification report.
52
53
6. Concluding remarks
The lateral wire buckling failure mode has been reconstructed under controlled conditions in the
laboratory by use of mechanical test rigs simulating wet annulus conditions. Four 6, two 8 and three
14 pipes were tested by applying cyclic bending and longitudinal compression. Results confirmed the
widely accepted fact that lateral wire buckling occurs at lower load levels in the laboratory than
encountered with field conditions. The limit load carrying ability for pipes subjected to laboratory
testing could roughly be estimated for the 6 and 8 pipes. Furthermore, a global mode, which seemed
elastic, was detected during testing of a 14 pipe. It was also noticeable, that an 8 pipe exhibited a
global deformation behavior, which is usually related to lateral wire buckling, while dissection of the
pipe after unloading revealed no sign of failure at all.
A method for calculation of the equilibrium state of a beam embedded in a frictionless cylinder bent
into a toroid has been developed. The proposed method was used to calculate the equilibrium state of
an armouring wire within the wall of a flexible pipe of finite length bent to a constant radius of
curvature. The analysis was carried out assuming that the equilibrium state, which a wire will reach
after a significant number of bending cycles, equals the state obtained directly if friction is neglected.
With this assumption, it was shown, that the modeled wire approached the geodesic curve on a torus
surface in bending and tension.
This method was used for imperfection analysis of armoring wires within flexible pipes in bending and
compression. It was demonstrated, that significant softening occurred in the equilibrium path of the
modeled wire, and that this behavior corresponded to large change of wire lay angle. Furthermore,
imperfections caused changes of wire lay angle to localize, while perfect wire geometries exhibited a
deformation pattern which was symmetrical around the pipe midpoint. Surprisingly, it was found that
the bending radius of the modeled pipe had little influence on the load carrying ability. The initial lay
angle seemed to be the geometrical parameter which along with the dimensions of the wire cross
section governs the load level at which softening occurs.
On basis of multiple single wire analyses, a model of both armouring layers was obtained. This model
made it possible to calculate the equilibrium paths and load carrying ability taking the effect of the
torsional imbalance of the pipe structure due to buckling of the inner tensile armour layer into
calculation. A comparison of the the calculated load carrying ability with experimentally obtained
results reveals that the method, as expected, is conservative. The reason for this is most likely that
effects due to friction, limited wire slippage close to end-fittings and load carried by the remaining
pipe layers were not considered in this analysis. Modes of deformation obtained experimentally were
shown to correspond well to the simulated buckling modes at a local level.
On basis of the global model of the armouring layers, model simplifications have been proposed. These
have proven capable of predicting the load carrying ability with quite high accuracy, although the
representation of the response prior to buckling may be of poor precision. By this method it was
shown, that while the remaining pipe layers imposed very little influence on the load which leads to
softening, high-strength tape and outer sheath imposed some influence on the force response in the
postbuckled state. Along with end-fitting effects, this may improve the prediction of the result of a
laboratory test. However, further research is needed to improve the models of sheaths and antibirdcaging layer. These may experience large deformations and often exhibit time-dependant
constitutive behavior. These issues are not included in the present implementation of the model.
Comparing the calculated equilibrium paths obtained by the full global model of the armouring layers,
it has been demonstrated that the load carrying ability is estimated conservatively. Furthermore, it has
been demonstrated, that the calculated predictions can be increased if the postbuckling response is
taken into account due to effects from adjacent pipe layers. Limited wire slippage close to end-fittings
may increase the load carrying ability of the computational model further.
Finally, it can be concluded that the objective of the present work formulated in section 1.3 has been
fulfilled in a satisfactory manner.
54
The work conducted in the present project served as basis for a tool for design of flexible pipes against
lateral wire buckling. The methods developed in the project were implemented by NKT Development
Engineer Anders Lyckegaard. The documentation of the method was reviewed by Bureau Veritas. At
the time at which the present project was finalized, a draft section for a type approval certificate
validating the correctness of the method had been issued. The design tool implementation is briefly
described in section 5.8.
55
few DIP-test results, which could be used for calibration of the computational model, are known.
Furthermore, no field failures have been encountered by NKT-Flexibles. The experiments conducted as
part of the present project have shown, that pipes may be installed at water depths corresponding to
compressive loads which could lead to lateral buckling, if only a limited number of bending cycles are
applied. This complicates the problem related to prediction of a DIP-tests result further.
Presently, it does not seem that laboratory experiments conducted inside pressure chambers
correspond better to field conditions than the principle applied in the present work. However,
laboratory experiments with longer test pipe samples than applied during the present experiments
would enable investigation of how boundary effects influences lateral wire buckling. Since it has been
demonstrated that lateral buckling may develop in the elastic regime, so no sign of failure can be
detected during dissection, the experimental results have shown that lateral buckling and failure by
lateral buckling can be considered as distinct terms in the flexible pipe design process. Further
research is needed in order to apply this very interesting observation as basis for design rules and
definition of the limit state design.
Regarding the theoretical work proposed in the present thesis, several issues could be addressed in
future research. First of all, the assumption, that the equilibrium state can be determined neglecting
friction, should be investigated further. Furthermore, the effect of limited wire slippage close to endfittings should be considered. In the present work, the effect has been considered by shortening the
computational model. However, this is strictly speaking not a correct method, and can only be used
crudely to estimate the effect. In order to investigate the interaction between friction and boundary
effects on the buckling load, further research following the approach used in the present work to
model frictional resistance is needed. Furthermore, measurement of the strains and curvature
components with optical monitoring technology would be both interesting and valuable. In bending a
tension, results were obtained which supported the assumptions made by Out and von Morgen [37]
regarding the geodesic curve as wire limit state. These results to some extend contradict the results
obtained by Leroy and Estrier [33], who included friction in the analysis and found cyclic bending to
cause slips which were centered around the geodesic. However, due to the experience-based solution
form, which was not justified, it is difficult to draw conclusions regarding the validity of these results.
Further research is needed in order to examine this issue with experimental and theoretical means.
Finally, the instability analyses in the present project are undertaken on basis of full-nonlinear
analyses of initially perfect and imperfect wire geometries. Future research should include further
investigations of the influence of the chosen imperfections on the obtained results. It would in this
context be both interesting and valuable to apply linearization techniques in order to investigate the
initial postbuckling behavior by analytical means.
While it in the present work has been demonstrated, that the adjacent pipe layers impose little
influence on the load carrying ability of the armouring layers, high-strength tape and polymeric
sheaths may have some influence on the postbuckling response. In the present approach, a method
based on linear responses of the remaining pipe layers has been applied neglecting time-dependent
effects in polymeric layers. Further research should include more detailed models of the adjacent pipe
layers including more sophisticated material models. Effects caused by ovalization of the pipe crosssection in bending have been neglected in the present context. These effects may, however, possibly
generate physical imperfections which should be accounted for in future research. Finally, an
improved method for modeling of radial deformations of the pipe wall taking variations of radial strain
into calculation should be considered.
If frictional effects are included when modeling the single armouring wires, the governing equations
for the wires could be formulated for varied radius of curvature throughout the length of the test pipe
sample. This would make it possible to model the global constitutive behavior in detail on basis of the
internal components. The arising method would obviously be extremely demanding to solve
numerically, but would on the other hand constitute a very valuable multipurpose tool for flexible pipe
analysis.
56
57
References
[1]
API 17J, Specification for Unbonded Flexible Pipes, American Petroleum Institute, 3rd edition,
2008.
[2]
Braga, M.P. and Kaleff, P. : Flexible pipe sensitivity to birdcaging and armor wire lateral buckling,
Proceedings of OMAE 2004, OMAE2004-51090.
[3]
Secher, P., Bectarte, F. and Felix-Henry, A. : Lateral Buckling of Armor Wires in Flexible Pipes:
reaching 3000 m Water Depth, Proceedings of OMAE 2011, OMAE2011-49447.
[4]
[5]
Fret, J.J. and Bournazel, C.L. : Calculation of stresses and slips in structural layers of unbonded
flexible pipes, Journal of Offshore Mechanics and Arctic Engineering, Vol. 109, pp. 263-269,
1987.
[6]
Witz, J.A. and Tan, Z. : On the Flexural Structural Behaviour of Flexible Pipes, Umbillicals and
Marine Cables, Marine Structures, Vol. 5, pp. 229-249, 1992.
[7]
Tan, Z., Quiggin, P. and Sheldrake, T. : Time Domain Simulation of the 3D Bending Hysteresis
Behavior of an Unbonded Flexible Riser, Journal of Offshore Mechanics and Arctic Engineering,
Vol. 131, 031301, 2009.
[8]
Kraincanic, I. and Kabadze, E. : Slip initiation and progression in helical armouring layers of
unbonded flexible pipes and its effect on pipe bending behavior, Journal of Strain Analysis, Vol.
36, No. 3, pp. 265-275, 2001.
[9]
Alfano, G., Bahtui, A. and Bahai, H. : Numerical derivation of constitutive models for unbonded
flexible risers, International Journal of Mechanical Sciences 51, pp. 295-304, 2009.
[10]
Alfano, G., Bahtui, A. and Bahai, H. : Numerical and Analytical Modeling of unbounded flexible
pipes, Journal of Offshore Mechanics and Arctic Engineering, Vol. 31, 2009.
[11]
Dastous, J.B. : Nonlinear Finite-Element Analysis of Stranded Conductors With Variable Bending
Stiffness Using the Tangent Stiffness Method, IEEE Transactions On Power Delivery, Vol. 20,
No.1, pp. 328-338, 2005.
[12]
Jiao, G. : Limit State Design for Flexible Pipes, Marine Structures, Vol. 5, pp. 431-454, 1992.
[13]
Vaz, M.A. and Rizzo, N.A.S. : A finite element model for flexible pipe armor wire instability, Marine
Structures, 2011.
[14]
Vaz, M.A. and Patel, M.H. : Post-buckling behavior of slender structures with a bi-linear bending
moment-curvature relationship, International Journal of Non-linear Mechanics, Vol. 42, p. 470483, 2007.
[15]
Vaz, M.A. and Patel, M.H. : Lateral buckling of bundled pipe systems, Marine Structures, Vol. 12 ,
pp. 21-40, 1999.
[16]
Vaz, M.A. and Patel, M.H. : Initial post-buckling of submerged slender vertical structures subjected
to distributed axial tension, Applied Ocean Research Vol. 20, pp. 325-335, 1998.
[17]
Secher, P., Bectarte, F. and Felix-Henry, A. : Qualification Testing Of Flexible Pipes For 3000m
Water Depth, Proceedings of OTC 2011, OTC-21490.
[18]
Bectarte, F. and Coutarel, A. : Instability of Tensile Armour Layers of Flexible Pipes under
External Pressure, Proceedings of OMAE 2004, OMAE2004-51352.
58
[19]
Tan, Z., Loper, C., Sheldrake, T. and Karabelas, G. : Behavior of Tensile Wires in Unbonded
Flexible Pipe under Compression and Design Optimization for Prevention, Proceedings of OMAE
2006, OMAE2006-92050.
[20]
Custdio, A.B. : Analytical model for instability assessment of flexible pipes armour, Doctoral
thesis, COPPE/UFRJ, 2005 (in Portuguese).
[21]
Brack. A., Troina, L.M.B. and Sousa, J.R.M. : Flexible Riser Resistance Against Combined Axial
Compression, Bending and Torsion in Ultra-Deep Water Depths, Proceedings of OMAE 2005,
OMAE2005-67404.
[22]
Montgomery, D.C. : Design and Analysis of Experiments, John Wiley & Sons 2009, ISBN: 978-0470-39882-1.
Kensche, C. W. : Fatigue of composites for wind turbines, International Journal of Fatigue, pp.
1363-1374, 2006.
[23]
[24]
Imamura, A., Hijikata, K., Tomii, Y., Nakai, S. and Hasegawa, M. : An experimental study on
nonlinear pile-soil interaction based on forces vibration tests of a single pipe and a pile group,
Eleventh World Conference on Earthquake Engineering, 1996, Paper No. 563, ISBN: 0 08
042822 3.
[25]
[26]
Terndrup Pedersen, P. : Equilibrium of offshore cables and pipelines during laying, International
Shipbuilding Progress, Vol. 22, pp. 399-408, 1975.
[27]
[28]
Clebsch, A. : Theorie der Elasticitt Fester Krper, Druck und Verlag von B.G. Teubner, Leipzig,
1862.
Love, A.E.H. : A treatise on the Mathematical Theory of Elasticity, Dover Publications Inc., N.Y.,
1944.
[29]
[30]
Costello, G.A. : Large deflections of helical spring due to bending, Journal of the Engineering
Mechanics Division, Vol. 103, No. 3, pp. 481-487, 1977.
[31]
Spillers, W. R., Eich, E.D., Greenwood, A.N. and Easton, R., J. : A helical tape on cylinder subjected
to bending, Eng. Mech. Div. Proc. ASCE, Vol. 109, 1983.
[32]
Stump, D.M and van der Heijden, G.H.M. : Matched asymptotic expansions for bent and twisted
rods: applications for cable and pipeline laying, Journal of Engineering Mechanics, Vol. 38, pp.
13-31, 2000.
[33]
Leroy, J.M. and Estrier, P. : Calculation of stresses and slips in helical layers of dynamically bent
flexible pipes, Oil and Gas Science and Technology, REV. IFP, Vol. 56, No. 6, pp. 545-554, 2001.
[34]
Svik, S. : A finite element model for predicting stresses and slip in flexible pipe armouring
tendons, Computers and Structures. Vol. 46, No.2, pp. 219-230, 1993.
[35]
Svik, S. : On stresses and fatigue in flexible pipes, Doctoral Thesis, NTNU, Trondheim, 1992.
[36]
[37]
Out, J. M. M. and von Morgen, B. J. : Slippage of helical reinforcing on a bent cylinder, Engineering
Structures, Vol. 19, No. 6, pp. 507-515, 1997.
[38]
Pfister, J. : Elastic Multibody Systems with Frictional Contacts, Doctoral Thesis, University of
Stuttgart, 2006.
59
[39]
[40]
Weppenaar, N., Kosterev, A., Dong, L., Tomazy, D. and Tittel, F. : Fiberoptic Gas Monitoring of
Flexible Risers, Proceedings of OTC 2009, OTC-19901.
[41]
Brush, D.O. and Almroth, B.O. : Buckling of bars, plates and shells, Mcgraw-Hill, 1975.
[42]
Koiter, W.T. : Current Trends in the Theory of Buckling, Buckling of Structures, Symposium
Cambridge/USA, June 17-21, 1974, IUTAM, Springer-Verlag, pp. 1-16.
[43]
Timoshenko, S. and Geere, J.M. : Theory of Elastic Stability, McGraw-Hill, 2nd Ed. 1961.
[44]
do Carmo, M.P. : Differential Geometry of Curves and Surfaces, Prentice-Hall Inc., 1976.
Lateral buckling laboratory Testing Dissection Procedure, NKT doc. No. 2895-DOC-TST-501.
[46]
Test procedure for lateral buckling: Gomez 6 export riser, NKT doc. No. 8062-DEV-3018.
[47]
Test report for lateral buckling: Gomez 6 export riser, initial trial test, NKT. doc. No 8062-DEV3019.
[48]
Dissection report Lateral buckling test Gomez 6, initial trial, NKT doc. No. 8062-DEV-3161.
[49]
Radius of curvature estimation - Gomez 6" - initial trial, NKT doc. No. 8062-DEV-3019.
[50]
Lateral Buckling Laboratory Testing Procedure, 6" Gomez, Test I, NKT doc. No. 2895-DOC-TST502.
[51]
Lateral buckling laboratory testing report, Gomez Test I, NKT doc. No. 2895-DOC-TST-504.
[52]
Lateral buckling laboratory testing dissection report, Gomez Test I, NKT doc. No. 2895-DOC-TST503.
[53]
Lateral Buckling Laboratory Testing Procedure, 6" Gomez, Test II, NKT doc. No. 2895-DOC-TST515.
[54]
Lateral buckling laboratory testing report, Gomez Test II, NKT doc. No. 2895-DOC-TST-516.
[55]
Lateral buckling laboratory testing dissection report, Gomez Test II, NKT doc. No. 2895-DOC-TST514.
[56]
Lateral Buckling Laboratory Testing Procedure, 6" Gomez, Test III, NKT doc. No. 2895-DOC-TST518.
[57]
Lateral buckling laboratory testing report, Gomez Test III, NKT doc. No. 2895-DOC-TST-519.
[58]
Lateral buckling laboratory testing dissection report, Gomez Test III, NKT doc. No. 2895-DOCTST-517.
[59]
Lateral buckling laboratory testing procedure, NKT doc. No. 2876-DOC-TST-120 (14" jumper).
[60]
[61]
[62]
[63]
Lateral buckling test program for Block-15 - 906, NKT doc. No. 8098-DEV-5423.
60
[64]
Lateral buckling laboratory testing report, Block-15 906, NKT doc. No. 8098-DEV-5807.
[65]
Lateral buckling laboratory testing dissection report, Block-15 906, NKT doc. No. 8098-DEV5808.
[66]
Lateral buckling laboratory testing procedure, 8" DWD, NKT doc. No. 8098-DEV-5595.
[67]
Lateral buckling laboratory testing report, 8" DWD, TEST I, NKT doc. No. 8098-DEV-5791.
[68]
Lateral buckling laboratory testing dissection report, 8 DWD, TEST I, NKT doc. No. 8098-DEV6349.
[69]
Lateral buckling laboratory testing procedure, 8" DWD, TEST II, NKT doc. No. 8098-DEV-5716.
[70]
Lateral buckling laboratory testing report, 8" DWD, TEST II, NKT doc. No. 8098-DEV-5792.
[71]
Lateral buckling laboratory testing dissection report, 8 DWD, TEST II, NKT doc. No. 8098-DEV6350.
[72]
A simplified model for the axial compression properties of a flexible pipe, NKT doc. No.
8098-DEV-5943.
[73]
A model for a beam defined on a frictionless torus, NKT doc. No. 8098-DEV-5820.
[74]
[75]
Lateral buckling design methodology certification, Bureau Veritas doc. No. E&P11190Z-R2011-01 NKT Lateral buckling, Rev. Draft 2011 12 23.
[76]
[77]
[78]
[79]
[80]
[81]
]
Engineering Structures
journal homepage: www.elsevier.com/locate/engstruct
A method for prediction of the equilibrium state of a long and slender wire
on a frictionless toroid applied for analysis of flexible pipe structures
Niels Hjen stergaard a,, Anders Lyckegaard b, Jens H. Andreasen c
a
a r t i c l e
i n f o
Article history:
Received 11 April 2011
Revised 5 September 2011
Accepted 3 October 2011
Available online 9 November 2011
Keywords:
Mechanics of flexible pipes
Curved beams
Armour wire equilibrium
a b s t r a c t
This paper concerns the behavior of a helical wire on a frictionless cylindrical surface subjected to bending, such that the wire in the deformed state constitutes a curve on a toroid. In order to determine the
equilibrium in the loaded state, a sixth order system of differential equations based on curved beam equilibrium and concepts from differential geometry will be presented. On this basis, a method for analysis of
curved beams on frictionless toroids is established. Helically wound steel wires are widely used in flexible pipes, which have various applications in the offshore industry, in order to ensure the structural
integrity against axial loads. The research presented in this paper constitutes a contribution to the field
of wire mechanics, since it enables calculation of the frictionless wire equilibrium state within the wall of
a flexible pipe.
2011 Elsevier Ltd. All rights reserved.
1. Theory
In the present paper, the mechanics of a helically wound wire
modeled as a curved beam is investigated. The wire will be considered embedded in a frictionless cylindrical surface, which is bent
into a toroid. Helically wound wires are widely used in steelpolymercomposite structures such as flexible pipes and umbilicals.
Such structures are usually unbonded and often used in the offshore
industry for transport of liquid or gas. The helical windings constituting the tensile armour of these structures have as primary function to ensure the structural integrity against longitudinal loads,
see Fig. 1. In most known designs two layers of armour wires are applied and designed such that axial strain and twist do not, or to a very
low extend, couple. When a pipe structure with helical windings as
structural element within the pipe wall is subjected to bending and
longitudinal loads, the wires will slip towards an equilibrium state in
which the lay angle may not be constant, which is the case in the initial unloaded state. However, since friction on the wires limits sliding, multiple load cycles must be applied in order for the wires to slip
towards the limit equilibrium curve. Established models of this
mechanical behavior have mainly been based on prescribed geometrical wire configurations. It is often assumed either, that the lay
angle remains constant in bending, which for constant radius of curvature yields a loxodromic curve on a torus surface. Another widely
Corresponding author. Tel.: +45 41908457.
E-mail addresses: Niels.HojenOstergaard@nktflexibles.com (N.H. stergaard),
Lyckegaard@nktflexibles.com, Anders.Lyckegaard@nktflexibles.com (A. Lyckegaard),
jha@m-tech.aau.dk (J.H. Andreasen).
0141-0296/$ - see front matter 2011 Elsevier Ltd. All rights reserved.
doi:10.1016/j.engstruct.2011.10.012
392
Nomenclature
r
R
Lpitch
/
/hel
j
t
n
b
h
u
a
ds
ds0
L
SL
jn
jg
s
Pt ; Pn ; Pb
Mt ; Mn ; Mb
pt ; pn ; pb
mt ; mn ; mb
DL
L
Dw
L
E
G
In ; I b ; J
A
2. Methods
A consistent system of differential equations for prediction of
the wire equilibrium state will now be derived by considering a
single armour wire. For the sake of simplicity, the radius of curvature for the centroid of the underlying surface will in the loaded
state be considered constant. The initial stress-free wire state will
be assumed helical.
2.1. Surface parameterization and tangent geometry
Having assumed the radius of curvature R j1 constant, the wire
will in the loaded state constitute a curve on a toroid. Initially, a
parameterization of the torus surface must be established. An arclength coordinate, u, along the torus centerline, and an angular
coordinate, h, along the minor torus radius, will be chosen as
parameterization, see Fig. 2. The torus equation is then
2 1
6
xu; h 4
3
cos h cosju j1
7
1
j r cos h sinju 5
r sin h
j r
393
da
ds
xu xh
kxu xh k
btn
in which the vectors xu and xh are the surface derivatives with respect to the torus coordinates, which also span the tangent space
of the torus surface, see Fig. 3
xu
@x
@u
xh
@x
@h
tu
xu
kxu k
th
xh
kxh k
The definition given in Eq. (4) enables the following alternative definition of the tangent of the curve, which the wire constitutes
da
du
dh
xu
xh
ds
ds
ds
cos /tu sin /th
This corresponds to stating
dh sin /
ds kxh k
These equations govern the kinematics of the wire. The norms kxu k
and kxh k are determined by calculating xu and xh as defined in Eq.
(3) on basis of the torus geometry given in Eq. (1)
6
xu 4
Since kinematic equations for the wire have now been derived,
curvature components must be determined. Having followed an
approach by which a curvilinear coordinate triad is attached to
each point on the wire, it is desirable to express curvature components in terms of a geodesic and a normal component, jg and jn,
respectively, along with a component representing the geodesic
torsion, s. A result from basic differential geometry known as the
Darboux frame, based on the FrenetSerret differential formulas,
is available for relating the triad vectors to their first order derivatives with respect to arclength in terms of curvature components.
The Darboux frame is defined as
2 3 2
0
t
d6 7 6
n
4 5 4 jn
ds
jg
b
jn
jg
s 0
32 3
t
76 7
54 n 5
b
10
Assuming the unit triad and corresponding arclength derivatives given, this corresponds to the following definitions of the curvature
components
dt
dn
t
ds
ds
db
dt
jg t b
ds
ds
dn
db
s b n
ds
ds
jn n
du cos /
ds
kxu k
j j1 r cos h sinju
1
j j r cos h cosju 7
5
r cosju sin h
6
7
xh 4 r sinju sin h 5
2
11
The curvature components can on basis of this definition be calculate in terms of u, h and / along with appropriate derivatives of
those. This direct approach was followed both by Svik [6], and
Leroy and Estrier [8]. However, direct application of Eq. (11) to
the coordinate triad given in Eq. (2) leads to very long expressions
which are difficult to handle. Therefore, concepts from abstract differential geometry will be applied. The approach has been shown in
a mathematical sense to be similar to calculating the dot products
and derivatives given in Eq. (11), but eases the task significantly.
Initially, it is noted that the normal curvature component can be
determined on basis of Eulers formula
12
In this equation, j1 and j2 denotes the principle curvature components of the torus surface. These are by definition given by
Sxu j1 xu
Sxh j2 xh
13
r cos h
kxu k 1 r j cos h
kxh k r 2
Sxu nu
9
@n
@u
Sxh nh
@n
@h
14
cosju cos h
6
7
n 4 sinju cos h 5
15
sin h
3
j sinju cos h
6
7
Sxu 4 j cosju cos h 5
0
2
cosju sin h
6
7
Sxh 4 sinju sin h 5
cos h
16
17
394
Sxu
j cos h
xu
1 rj cos h
1
Sxh xh
r
18
Reviewing Eq. (14), it is now clear that the principle curvature components of the torus surface are given by
j1
j cos h
1 rj cos h
j2
1
r
19
jn
j cos h
1
2
cos2 / sin /
1 rj cos h
r
20
s j1 j2 cos / sin /
21
j cos h
1
cos / sin /
1 r j cos h r
22
jg
d/
jg 1 cos / jg 2 sin /
ds
23
jg
3
2
3 2
cos u
sin h cos u
1
7
6
7 6
in which the scalar factors and the first vectorial term represents
the change rate of the curve tangent in arclength and the second
term represents a vector in the tangent plane normal to this tangent. Performing the scalar product the following expression is
obtained
2
jg 1
R r cos h
R r cos h
jg 2 0
24
j sin h
d/
jg
cos /
1 r j cos h
ds
25
P Pt t Pn n Pb b
M Mt t Mn n Mb b
p pt t pn n pb b
m mt t mn n mb b
With the current definition of an orthonormal basis, the equations
of equilibrium can be written as
dP
dt
dn
db
dPt
dPn
dPb
p Pt Pn
Pb
t
n
b
ds
ds
ds
ds
ds
ds
ds
pt t pn n pb b 0
27
dM
dt
dn
db
dM t
dMn
m t P Mt Mn
Mb
t
n
ds
ds
ds
ds
ds
ds
dM b
b
mt t mn n mb b Pb n Pn b 0
ds
28
Applying Eq. (10), the derivatives in Eq. (28) can be calculated, so
the componentwise equations of equilibrium are
dPt
jn Pn jg Pb pt 0
ds
dPn
jn Pt sPb pn 0
ds
dPb
jg Pt sPn pb 0
ds
dM t
jn Mn jg M b mt 0
ds
dM n
jn Mt sM b Pb mn 0
ds
dM b
jg Mt sM n Pn mb 0
ds
29
30
31
32
33
34
The equilibrium equations for a curved beam segment is derived on vectorial form by Reissner [10]
dP
p0
ds
dM
tPm0
ds
26
Pt EA
35
M t GJ Ds
36
M b EIb Djn
37
M n EIn Djg
38
in which the changes of curvature are given with respect to the initial helical state in which the structure is unloaded
ini
Ds s sini Djn jn jini
n Djg jg jg
sin /hel
r
sin /hel cos /hel
ini
s
r
jini
g 0
jini
n
The angle /hel is the initial wire lay angle defined in terms of the
minor torus radius and the pitch length Lpitch
39
du
cos /
ds 1 r j cos h
dh sin /
ds
r
d/
j sin h
cos / jg
ds
1 r j cos h
dPt
jn Pn jg Pb
ds
dPb
jg Pt sPn
ds
dMn
jn Mt sM b Pb
ds
40
dMb
djn
EIb
ds
ds
42
43
44
47
djn
ds
can be obtained by analytical means. Having obtained a solution to the differential equations, unknown quantities, pn and mt
are given in terms of known functions. The external loads are geometrically specified by the two remaining equations of equilibrium governing the normal force and torsional equilibrium of
the wire
48
49
t
Unknown terms dPdsn and dM
can be derived analytically. The physical
ds
explanation for the presence of the moment reaction mt, which is
assumed induced by adjacent pipe layers is, that it constrains the
modeled wire to the toroid such that the rotation around the local
tangent is governed solely by the underlying surface. This is equivalent to assuming that the material coordinatesystem and the
Darboux frame do not rotate relative to each other.
41
s0
rh
sin/hel
50
ds 1 ds0
51
45
dMb
Pn
jg M t sMn
ds
can be determined by
dP n
jn Pt sP b
ds
dM t
jn Mn jg M b
mt
ds
2pr
Lpitch
dM b
ds
pn
tan/hel
in which
395
46
Considering the derived curvature components, these can be rewritten on the form
jn n
jg t
sb
dt
dt ds0
ds0
n
j0n
ds
ds0 ds
ds
db
db ds0
ds0
t
j0g
ds
ds0 ds
ds
dn
dn ds0
ds0
b
s0
ds
ds0 ds
ds
396
du ds0
cos /
ds0 ds
1 rj cos h
52
dh ds0 sin /
ds0 ds
r
53
d/ ds0
j sin h
cos / jg
ds0 ds
1 rj cos h
54
dP t ds0
jn Pn jg Pb
ds0 ds
55
dP b ds0
jg Pt sP n
ds0 ds
56
dM n ds0
jn M t sMb Pb
ds0 ds
57
du
cos /
1
ds0
1 r j cos h
58
3. Results
dh
sin /
1
ds0
r
59
d/
j sin h
cos / j0g
1
ds0
1 r j cos h
60
dP t
j0n Pn j0g Pb
ds0
61
dP b
j0g Pt s0 P n
ds0
62
dM n
j0n Mt s0 Mb Pb 1
ds0
63
P t EA
ds0
sini
ds
ds0
EIb j0n
jini
ds
M t GJDs GJ
M b EIb Djn
s0
Pn
1
dM b
j0g Mt s0 Mn
1
ds0
64
dM b
d
EIb
ds0
ds0
j0n
ds0
ds
EIb
0
n
dj ds0
d ds0
EIb jn
ds0 ds
ds0 ds
65
d ds0
d
1
d
1
ds0 1 2
ds0 ds
ds0 1
dP t
1
1
j0n Pn j0g P b
2
ds0 EA1
EA1 2
u0 0 h0 hAini / 0 /hel
DL
Dw
uSL L 1
hSL hBini
L /SL /hel
L
L
68
dh sin /
ds
r
d/
j sin h
cos /
ds
1 rj cos h
69
70
Alternatively, the geodesic can be constructed on basis of the following derivative, which is given in [13]
p
dh m m2 B2
du
rB
66
67
71
tan /
dh
r
du 1 r j cos h
72
397
3
dg dh
jg sin h2r2 g R2 g2 g 3 rRg
dh
ds
2
dh
jn r Rg cos hg 2
ds
2
dh
s Rg
ds
73
74
Table 1
Straight pipe in tension and torsion.
DL
L
0.001
0.001
0.001
0
p
180
p
180
2
ds
r 2 R2 g2 g 2
dh
dx
g
dh
r
g 1 cos h
R
in which u = xR. Considering a loxodromic curve with pitch length,
Lpitch = 1 m, on a toroid with minor radius r = 0.2 m and major radius
R = 10 m, changes of curvature with respect to the initial helical
configuration as presented on Fig. 6 are obtained. The two sets of
curvature components can be observed to correspond to each other.
3.2. A wire on a tensioned cylindrical surface
Considering a 5 10 mm wire of 1 m pitch length on a cylindrical surface (j = 0) of radius r = 0.2 m subjected to tension, the load
in the axial direction of the cylinder is given by
76
2.535
8.585
13.655
77
Results are presented in Table 1. Results can be observed to correspond well, since the method derived in this paper predicts no
transverse slip in tension and with zero pipe curvature, and the
equations used for references are derived for perfect helices.
pure bending
bending and tension,=0.001
loxodromic curve
geodesic curve, FDalgorithm
bending and tension,=0.002
geodesic curve ref. [13]
0.93
ref.[8]
g
n ref.[8]
ref.[8]
0.02
0.04
0.92
2.537
8.587
13.642
F
DL
Dw
cos2 /hel
r sin /hel cos /hel
cos /hel EA
L
L
0.02
F(kN)
0.04
P(kN)
75
in which a minus is added to jg and s in order to obtain correspondence between the sign conventions, which can be derived
comparing the applied Darboux frames. The unknown terms are
given by
0.06
Dw
L
0.91
0.9
0.89
0.06
0.88
0.5
1.5
398
0.1
0.2
n
0.08
0.06
0.04
0.02
0
0.1
0
0.1
0.2
0
1
0.02
x 10
0.04
0.06
0
Pipe strain 4
30
Pt 103
20
Pn
Pb
0.1
Fig. 11. Change of normal curvature for a wire in bending and varying tensile pipe
strain.
0.15
0.05
10
0
10
20
30
0.05
40
0.1
0
50
20
0.2
15
0.1
25
0
0.1
0.2
0
10
5
0
5
10
15
x 10
Pipe strain
10
mt (Nm/m)
20
pn103 (N/m)
25
0
399
]
Applied Ocean
Research
Applied Ocean Research 1 (2012) 111
Anders Lyckegaard
NKT-Flexibles, Priorparken 480, Brndby, Denmark
E-mail: Anders.Lyckegaard@nktflexibles.com
Jens H. Andreasen
Department of Mechanical and Production Engineering, Aalborg University, Denmark
E-mail: jha@m-tech.aau.dk
Abstract
The work presented in this paper is motivated by a specific failure mode known as lateral wire buckling occurring
in the tensile armour layers of flexible pipes. The tensile armour is usually constituted by two layers of initially
helically wound steel wires with opposite lay directions. During pipe laying in ultra deep waters, a flexible pipe
experiences repeated bending cycles and longitudinal compression. These loading conditions are known to impose
a danger to the structural integrity of the armouring layers, if the compressive load on the pipe exceeds the total
maximum compressive load carrying ability of the wires. This may cause the wires to buckle in the circumferential
pipe direction, when these are restrained against radial deformations by adjacent layers.
In the present paper, a single armouring wire modeled as a long and slender curved beam embedded in a frictionless
cylinder bent into a toroid will be studied in order to gain further understanding of this failure mode. In order to study
the compressive behavior, both perfect beams as well as beams with small geometrical imperfections are studied. The
mathematical formulation of the problem is based on curved beam equilibrium and allows large deflections to be
taken into calculation.
1. Introduction
Marine structures such as flexible pipes and umbilicals are usually unbounded steel-polymer composites.
Such structures are often used in the offshore industry
during field development at large water depths. In most
known pipe designs, two layers of helically wound steel
wires are applied in order to ensure the structural integrity with respect to longitudinal loads, see Figure 1.
The mechanical behavior of these armouring tendons
subjected to longitudinal loads and bending have been
widely investigated in academic as well as industrial research in the past few decades. Established methods for
2. Theory
2.1. Wire equilibrium state with transverse slip
In this section, the method for determination of the
wire equilibrium state presented in [18] is summarized.
In order to obtain the limit equilibrium state of an armour wire, friction on the wires will be neglected. It
is assumed that the wire being modeled will reach the
frictionless configuration after a significant number of
bending cycles. Furthermore, the analysis of a single
tensile armour wire will be based on the assumption,
cos
sin (u)
r sin
(1)
t=
d
ds
n=
xu x
kxu x k
b=tn
(2)
in which xu and x constitute a basis for the toroid tangent plane. These vectors are given by
xu =
x
u
x =
(3)
d cos
=
ds
kx k
(4)
t 0
n g t
d
n = n 0
n
(5)
ds b
b
0
g
dM
+tP+m=0
ds
(6)
cos
1 + r cos
sin
=
r
sin
=
cos + g
1 + r cos
(7)
(8)
(9)
= n Pn g Pb
(10)
= g Pt Pn
(11)
= n Mt + Mb + Pb
(12)
Unknown functions are given in terms of known quantities after a solution has been obtained.
In order to obtain a solution by numerical means, it
is desirable to convert the system of field equations
from deformed arclength s to a system in undeformed
arclength s0 . Assuming that axial strains are small,
Cauchys definition of strain applies. Substituting this
into the field equations ensures that a solution is obtained on a regular mesh. A solution to the wire equilibrium is obtained using a commercially available BVPsolver. The model is, since generalized loads are applied
as longitudinal strain L
L and pipe twist L , deformation
controlled. Applying boundary conditions corresponding to the mechanics of an armour wire within the wall
of a flexible pipe, the following can be stated
A
u(0) = 0 (0) = ini
(0) = hel
u(S L ) = L 1 +
L
L
B
(S L ) = ini
(13)
L
L
(14)
(S L ) = hel
in which S L denotes the total wire arclength and L the
total pipe length.
2.2. Nonlinear imperfection analysis
Nonlinear systems may be sensitive to small initial
imperfection, which may trigger buckling. In the following it will be examined if small initial imperfections
cause a mechanical behavior which may be related to
instability of the wire being modeled. The imperfection
will be added directly to the geodesic curvature and is
chosen in such a manner that it may represent a wide
range of physically possible deformation modes
g = g (g,ini g,imper )
(15)
g,ini = 0
(16)
g,imper =
m
X
i=1
i sin
is
L
(17)
=
=
=
=
Pt t tu + Pb b tu
Pt cos + Pb sin
Pt t t Pb b t
Pt sin Pb cos
(18)
(19)
3. Results
3.1. Wire imperfection analysis
In order to examine if buckling of a simple structure can be triggered by the present approach, a plane
straight clamped-clamped steel beam of 3 10mm rectangular cross section and length 1.5m will be analyzed
using the suggested method. Youngs modulus for steel
will be set to E = 210GPa. The analysis is performed
by reformulating the angular torus coordinate as an arclength coordinate w = r along the minor torus radius
and setting r = R = . The Euler buckling load, PE for
a clamped-clamped beam is given by the well-known
formula
PE =
42 EIn
L2
(20)
150
100
B
50
0.5
1.5
2
2.5
3
Compressive strain
3.5
4.5
5
4
x 10
Figure 7: ( L
L , Pu )-equilibrium paths for clamped-clamped beam in compression, see Figure 6, A: perfect initial geometry, B: imperfect initial
geometry
1400
A
1200
B
C
1000
800
m=0
m=1,i=0.001
m=2,i=0.001
m=5,i=0.001
m=20,i=0.001
600
m=1,i=0.001
m=2,i=0.001
m=5,i=0.001
400
m=20,i=0.001
m=20,i=0.01
m=20,i=0.0001
200
Figure 8: Model of armour wire within the wall of a flexible pipe subjected to bending and longitudinal loads
m=20,i=0.005
Primary path
0
3
Compressive pipe strain
Figure 9: ( L
L , Pu )-equilibrium paths of modeled wire for R = 15m, see
Figure 8. A: perfect wire geometry, primary path, B: perfect wire geometry, secondary path, C: equilibrium path, small initial imperfections, D-E:
equilibrium paths, large initial imperfections, all imperfections measured
in m1
x 10
4
Figure 10: Wire equilibrium states (prior to nonlinear softening), R = 15m, L
L = 1.05 10 , blue curve: bent helix ( = hel in bending),
red curve: Wire equilibrium state, m = 20, 1...20 = 0.001m1 , green curve: Wire equilibrium state, m = 20, 1...20 = 0.001m1
3
Figure 11: Wire equilibrium states (nonlinear force-deformation regime), R = 15m, L
L = 10 , blue curve: bent helix ( = hel in bending),
1
red curve: Wire equilibrium state, m = 20, 1...20 = 0.001m , green curve: Wire equilibrium state, m = 20, 1...20 = 0.001m1
librium paths will be considered. The detected equilibrium paths are of a shape which correspond well to
tendon behavior detected by finite element analysis in
[5]. The force in pure bending can be observed to be
negative, hence, if the arc length of the torus centerline
is considered constant, a bending-compression coupling
occurs. To give an impression of the magnitude of the
chosen imperfections, the change of normal curvature,
n , with respect to the initial helical state for a wire in
pure bending is shown in Figure 14. The chosen amplitudes are observed to be relatively small.
In order to study the effect of the chosen imperfections
m=0
hel
0.5
m=1,i=0.001
m=0,i=0.001
0.52
0.49
m=20,i=0.001
m=1,i=0.001
m=2,i=0.001
0.47
m=5,i=0.001
m=20,i=0.001
0.46
m=20,i=0.001
m=1,i=0.001
0.5
Wire lay angle, (rad)
0.48
m=5,i=0.001
m=2,i=0.001
m=5,i=0.001
m=2,i=0.001
m=1,i=0.001
m=2,i=0.001
m=5,i=0.001
m=20,i=0.001
0.48
Primary path
0.46
A
0.45
0.44
0.44
3
4
Wire arclenght, s(m)
0.42
3
4
Wire arclength, s(m)
on the wire geometry, the wire lay angle will be considered, see figure 12 and 13. It is interesting to note,
that switching the sign of the imperfection amplitudes
causes the changes in wire lay angle to localize opposite, see the curves denoted C + and C corresponding
respectively to positive and negative amplitudes.
is different for different values of . The compressive wire force in pure bending ( L
L = 0) can
be observed to increase, when the curvature is increased. Hence, the pipe curvature does not effect the load in the nonlinear regime after softening
has occurred, but is again detected as a bendingcompression coupling.
3. A and B in equation 13 and 14 representing where
in the pipe crosssection the wire starts and ends has
little detectable influence on the calculated wire
equilibrium paths.
R=15 m
1400
0.2
Longitudinal compressive wire force, Pu (N)
A
0.15
0.1
0.05
0.05
1200
1000
A
B
800
600
400
Lpitch=1.0 m
200
0.1
Lpitch=1.25 m
Lpitch=1.5 m
0.15
0
3
4
Wire arclength, s(m)
Figure 14: Change of normal wire curvature n with respect to the initial
1
, the magnitude of the initial
helical state, state of pure bending to = 15m
helical normal curvature is 2.02m1
0.5
1.5
2
2.5
3
3.5
Compressive pipe strain
4.5
5
4
x 10
Figure 15: ( L
L , Pu )-wire equilibrium paths for varied pitch lengths, A:
perfect wire geometry, B: imperfect wire geometry, state of bending to
1
= 15m
and compression
R=15
1100
Longitudinal compressive wire force, Pu (N)
1200
1000
800
600
400
200
=0
=(20 m)
800
700
600
A=B=0
500
A=B=/2
A=B=
A=B=3/2
300
0
=(10 m)
Figure 16:
vature
=(15 m)1
1
200
0
A
900
400
1000
0.1
0.2
( L
L , Pu )-wire
0.3
0.4
0.5
0.6
0.7
Compressive pipe strain
0.8
0.9
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Compressive pipe strain
0.8
0.9
1
3
x 10
x 10
4. Conclusions
Based on a mathematical formulation of the mechanical behavior of a long and slender beam embedded in
a frictionless cylindrical surface bent into a toroid, the
equilibrium state of a single armouring wire within the
wall of a flexible pipe can be determined. The model
allows curvature components and deformations to be
large, but is in the present approach restricted to the
small strain regime. On this basis, the behavior of a
tensile armour wire in compression and bending has
Figure 17: ( L
L , Pu )-wire equilibrium paths for varied pipe bending radii
(for the sake of simplicity only imperfect wire structures are considered),
A: perfect wire geometry, B: imperfect wire geometry
been examined, including both initially perfect and imperfect wire geometries. Significant nonlinear softening
has been observed in the calculated force-displacement
responses. It has been demonstrated, that when a certain
load level is exceeded, the uniqueness of the solution to
the equilibrium state of an initially perfect wire geometry is lost. The effect of various imperfections was found
to have noticeable effects on the obtained solution.
The effect of key model parameters was examined. The
input parameter which had the largest influence on the
obtained solution, was the pitch length of the wire, since
10
0.55
2500
0.5
2000
1500
3000
1000
0.45
0.4
0.35
500
0.2
0.4
0.6
0.8
1
1.2
1.4
Compressive pipe strain
1.6
1.8
x 10
Figure 18: ( L
L , Pu )-wire equilibrium paths detected when the number of
modeled pitches is varied, imperfect wire geometry
6
8
Wire arclength, s (m)
10
12
14
Figure 19: Wire lay angles, , detected when the number of modeled
pitches is varied, imperfect wire geometry
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
11
Marine Structures
Marine Structures 1 (2012) 112
Anders Lyckegaard
NKT-Flexibles, Priorparken 480, Brndby, Denmark
E-mail: Anders.Lyckegaard@nktflexibles.com
Jens H. Andreasen
Department of Mechanical and Production Engineering, Aalborg University, Denmark
E-mail: jha@m-tech.aau.dk
Abstract
In the present paper, a mathematical model which is capable of representing the physics of lateral buckling failure
in the tensile armour layers of flexible pipes is introduced. Flexible pipes are unbounded composite steel-polymer
structures, which are known to be prone to lateral wire buckling when exposed to repeated bending cycles and longitudinal compression, which mainly occurs during pipe laying in ultra-deep waters. On basis of multiple single wire
analyses, the mechanical behavior of both layers of tensile armour wires can be determined. Since failure in one layer
destabilizes the torsional equilibrium which is usually maintained between the layers, lateral wire buckling is often
associated with a severe pipe twist. This behavior is discussed and modeled. Results are compared to a pipe model,
in which failure is assumed not to cause twist. The buckling modes of the tensile armour wires can be obtained by the
presented method.
1. Introduction
Unbounded Flexible pipe structures are widely used
as risers or flowlines for development of subsea oil
or gas reservoirs in deep waters. In order to obtain a
structural design capable of resisting large curvature,
internal and external pressure and longitudinal loads,
flexible pipes are constructed as composite structures
comprised by a number of layers with different mechanical properties, see Figure 1. The pipe bore denoted the
carcass is constructed in such a manner that it retains
flexibility while capable of resisting pressure. On the
outside of this layer, a helically wound pressure armour
comprised by steel profiles is applied. Two polymer
liners constitute fluid barriers which ensure that the
pipe is tight. Furthermore, two layers of helically
The lateral buckling failure mode is from a design perspective of a more complex nature than buckling by
birdcaging. Both failure modes can be observed as localized wire buckling, but while birdcaging can be detected by visual inspection without dissection of the
failed pipe, lateral buckling is significantly harder to detect without precise measurements.
Lateral buckling of tensile armour wires of flexible
pipes are known mainly to occur during pipe laying in
deep-waters, see Figure 4.T1. In this scenario, the flexible pipe is in a free-hanging position from an installation vessel to the seabed. Furthermore, the pipe is often
empty during installation. The conditions, which are
known to cause lateral buckling in a flexible pipe are
1. Axial compression due to hydrostatic pressure on
the end cap of an empty pipe
2. Repeated bending cycles due to waves, current and
vessel movements
3. Often breached outer sheath, so the pipe annulus is
flooded, so hydrostatic pressure does not introduce
contact stresses which limits wire slip
The wire buckling failure modes were described by
Braga and Kaleff, [2]. A mechanical test principle for
reconstruction of the failure mode in the laboratory was
described. However, it seems to be a widely accepted
fact that results obtained experimentally in the laboratory by use of mechanical test rigs do not correspond
to observations made in the field, since failure occurs at
lower load levels in mechanical test rigs. The test principle was developed further by use of hyperbaric chambers by Bectarte and Coutarel, [1], Bectarte et. al.,[3],
Figure 2: Localized lateral deformations in the inner layer of tensile armour wires, generated by NKT-Flexibles by laboratory testing
Figure 4: Flexible pipe configurations, T.1: Pipe laying scenario, T.2: Idealized scenario, used during laboratory testing, see [2],[1] and [19]
2. Theory
In order to construct a global model of a flexible pipe,
the means which are needed for modelling of a flexible
pipe will initially be introducted. Afterwards, it will be
desribed how multible single wire analyses form a basis
for a global pipe model.
2.1. Wire equilibrium state with transverse slip
Initially, a brief summary of the method derived for
single wire analysis in [17] is given. A single tensile
armour wire within the wall of a flexible pipe is modeled. Assuming the pipe curvature = R1 constant, the
wire can be assumed to constitute a curve on a torus surface with minor radius r, see Figure 5, parameterized
by the coordinates u and . A point on the torus surface with specified (u, )-coordinates has the cartesian
coordinates
1
+ r cos cos (u) 1
1
(1)
x(u, ) =
+ r cos sin (u)
r sin
d
ds
n=
xu x
kxu x k
b=tn
(2)
d cos
=
ds
kx k
(3)
(4)
(5)
(6)
cos sin
(7)
=
1 + r cos r
dM
+tP+m=0
ds
(8)
=0
(9)
=0
(10)
=0
(11)
=0
(12)
dMn
+ n Mt Mb Pb + mn = 0
ds
dMb
g Mt + Mn + Pn + mb = 0
ds
(13)
(14)
Assuming the wire dimensions small with respect to minor and major torus radii, it is fair to neglect curved
beam terms and assume the constitutive relations linear
Pt
Mt
Mb
Mn
= EA
= GJ = GJ( 0 )
= EIb n = EIb (n n,0 )
= EIn g = EIn (g g,0 )
(15)
in which denotes tangential wire strain. Now assembling the unknowns, the following system of equations
is derived
du
ds
d
ds
d
ds
dPt
ds
dPb
ds
dMn
ds
cos
1 + r cos
sin
=
r
sin
=
cos + g
1 + r cos
=
(16)
(17)
(18)
= n Pn g Pb
(19)
= g Pt Pn
(20)
= n Mt + Mb + Pb
(21)
u(S L ) = L 1 +
L
L
L
L
(S L ) = hel
(S L ) = ini
(22)
(23)
=
=
=
=
Pt t tu + Pb b tu
Pt cos + Pb sin
Pt t t Pb b t
Pt sin Pb cos
(24)
(25)
with the conventional assumption, that the wire lay angle remains constant. Slip in this model may only occur
tangentially along a bent helix. For a wire with constant
, the term d
ds vanishes. On this basis the system of field
equations can be reduced to
du
cos
=
(26)
ds
1 + r cos
d
sin
=
(27)
ds
r
d
= 0
(28)
ds
dPt
= n Pn g Pb
(29)
ds
The corresponding boundary conditions are
A
u(0) = 0 (0) = ini
!
L
B
(S L ) = ini
u(S L ) = L 1 +
L
L
(30)
(31)
All observations made during test execution have supported the assumption, that transverse wire slip is very
limited in the outer layer of tensile armour, see [2] and
[19]. This simplified system of equations may therefore
be used in order to model the mechanical behavior of
the outer layer of armour wires.
A layer of armouring wires has this far been assumed
radially inelastic. In order to take effects of a radially
elastic pipe wall into calculation, the minor torus radius
will be set as function of the applied compressive strain.
The following relation specifying the deformed radius
rd was derived in [18] and [19].
!
!
ka L
r
r = 1
r
(32)
rd = 1 +
r
kr L
in which ka and kr denotes respectively an axial and an
radial linear pipe stiffness.
(33)
i=1
nX
wires
i=1
nX
sheets
Gi J i
i=1
=
=
nX
wires
i=1
nX
wires
i=1
Piu +
nX
sheets
Piu,sheets
(34)
i=1
nX
sheets
i=1
E i Ai
L
L
longitudinal load by Equation 34, see Figure 9. It is initially noted that the layer of wires exhibits a behavior
equivalent to what was observed for a single armouring wire, see Figure 9. Hence, the buckling analysis
performed for a single armouring wire can now be performed for an entire layer.
By equivalent methods the Equilibrium paths of a torsionally fixed-fixed pipe can be determined, see Figure
12. The inner layer of armouring wires, which for a
pipe subjected to longitudinal compression is prone to
lateral buckling, will be modeled by means described
in section 2.1, so transverse slip is allowed. The outer
layer of armouring wires will be described as described
in section 2.2. Hence, the wire lay angle remains constant when the pipe is loaded. The force in the inner
layer can be observed to exhibit significant softening
behavior, while the force in the outer layer can be observed to increase linearly, since this layer can not slip
transversely and thereby exhibit buckling behavior.
Now considering the pipe torsionally fixed-free with
the inner and outer layer of armour wires modeled in a
manner equivalent to what was applied during analysis
of a torsionally fixed-fixed pipe, the forces in the pipe
layers are shown on Figure 13. The computational time
necessary to perform this analysis is significantly larger
than for a fixed-fixed pipe, since the torsional equilibrium must be fullfilled by solving equation 33 iteratively
for all wires. The consequence of the softening behavior is, that applying compressive loads at levels larger
than the maximum value determined by the analysis,
will lead to transverse wire buckling in the inner layer
of armouring wires, which causes a severe pipe twist,
see Figure 16.
The effect of a radially elastic pipe wall can now be
estimated on basis of equation 32. By methods established on basis of equations derived in [8], the ratio between linear longitudinal and radial stiffness kkar is calculated to 2.52 which corresponds well to what has been
measured during a compression test of a pipe with the
given design. The test setup was described in [19]. Considering the equilibrium paths shown on Figure 15, the
radial elasticity taken into calculation can be observed
to have major influence on the longitudinal pipe stiffness, which prior to buckling corresponds well to the expected linear behavior. However, it can be observed that
the compressive load, which can be carried by the pipe
structure, only to a very low extend is effected by radial
deformations. The corresponding pipe twist is shown
on Figure 16. The behavior of the armouring layers in a
twist-longitudinal force diagram can be observed not to
be effected by radial elasticity.
In Figure 14 the equilibrium path for a radially stiff
1500
1000
500
0.5
1.5
2
2.5
3
3.5
Longitudinal compressive pipe strain
4.5
5
4
x 10
Figure 9: ( L
L , Pu )-equilibrium paths of analyzed armour wires, see Figure 10 (pipe modeled torsionally fixed-fixed), all imperfections measured
in m1
Figure 10: Model of armour wire within the wall of a flexible pipe subjected to bending and longitudinal loads
Outer diameter(m)
L pitch (m)
Size (mm)
Number of windings
Inner layer
0.2012
1.263
3 10
52
Outer layer
0.209
1.318
3 10
54
High-strength tape
0.2117
0.075
1 60
1
Wire steel
High-strength tape
Youngs Modulus
210 GPa
27 GPa
torsionally fixed-free pipe in which transverse slip is allowed in both layers is presented. It can be concluded,
that the maximum load, which can be carried by the
structure, only to a very low extend is influenced by that
transverse slip is allowed in both layers of armouring
wires. However, the longitudinal stiffness can be concluded to be effected, since the strains are larger than in
results obtained with a transversely fixed outer layer, see
Figure 13. Since this is the case, it is desirable to model
Poissons ratio
0.3
0.3
180
50
Longitudinal compressive force (kN)
60
40
30
20
imperfect wire geometry, 1...20=0.001
10
160
140
120
100
80
60
40
20
0.5
1.5
2
2.5
3
3.5
Longitudinal compressive pipe strain
4.5
Figure 11: ( L
L , Pa )-equilibrium paths of inner layer of armouring wires,
analysis comprized of mulitiple single wire analyses, torsionally fixedfixed boundary conditions on layer, all imperfections measured in m1
110
100
100
90
90
80
80
70
60
50
40
30
20
inner layer
outer layer
sum of layers
10
0
0.5
1.5
2
2.5
3
3.5
Longitudinal compressive pipe strain
0.5
1.5
2
2.5
3
3.5
Longitudinal compressive pipe strain
50
40
30
20
inner layer
outer layer
sum of layers
x 10
5
4
x 10
60
Figure 13: ( L
L , Pa )-equilibrium paths of analyzed flexible pipe (torsionally fixed-free, see Figure 8),transverse slip taken into calculation in inner
layer, neglected in outer layer
4.5
70
10
4.5
Figure 12: ( L
L , Pa )-equilibrium paths of analyzed flexible pipe (torsionally fixed-fixed, see Figure 7), transverse slip taken into calculation in
inner layer, neglected in outer layer
5
4
x 10
0.5
1.5
2
2.5
3
3.5
Longitudinal compressive pipe strain
4.5
5
4
x 10
Figure 14: ( L
L , Pa )-equilibrium paths of flexible pipe (torsionally fixedfree, see Figure 8), transverse slip is taken into calculation in both layers
of armouring wires
10
110
100
100
50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Longitudinal compressive pipe strain
0.8
0.9
80
70
60
50
40
1
x 10
Figure 15: ( L
L , Pa )-equilibrium paths of flexible pipe, radially stiff and
radially elastic structures
A
90
2
Pipe twist, /L
5
3
x 10
Figure 16: Pipe twist of torsionally free pipe end, all imperfections measured in m1 , radially stiff and radially elastic structures
C
11
Figure 18: Closer examination of buckling modes, R = 11m, E: Localized buckling detected as large wire gaps, F: Periodic deformation mode
L
found at large pipe strain ( L
L = 0.01) for torsionally fixed-fixed pipe, G: Periodic deformation mode found at large pipe strain ( L = 0.025) for
torsionally fixed-free pipe
5. Conclusions
Based on a mathematical model for determination of
the wire limit equilibrium state within the wall of a flexible pipe, a global model of a flexible pipe subjected
to bending to a constant radius of curvature and compressive longitudinal loads has been proposed on basis of multiple single wire analyses. On this basis, the
limit compressive load of a pipe structure, in which the
pipe interior is assumed flooded, can be calculated along
with deformation modes, which correspond well to preliminar experimental results. The wire models have
proven sensitive to initial imperfections, which cause
the gaps in the inner layer of armouring wires to localize
when lateral buckling occurs. The presented approach,
in which a flexible pipe is modeled torsionally fixedfree, may however on the shown form serve as basis
of conservative estimates of the maximum compressive
load, which can be carried by the pipe structure. Experimental validation and assessment of how friction effects
the triggered buckling modes have not been presented in
this paper. Further research is needed in order to draw
final conclusions regarding how design rules and methods for flexible pipe design can be formulated on basis
of the proposed model, due to the conservatism included
in the present approach.
[16]
[17]
[18]
[19]
[20]
1211
[1] Bectarte,F. and Coutarel,A. (2004): Instability of Tensile Armour Layers of Flexible Pipes under External Pressure. Proceedings of OMAE, 2004
[2] Braga, M.P. and Kaleff, P. (2004): Flexible Pipe Sensitivity to
Birdcaging and Armor Wire Lateral Buckling. Proceedings of
OMAE, 2004
[3] Secher, P., Bectarte, F. and Felix-Henry, A. (2011): Lateral
Buckling of Armor Wires in Flexible Pipes: reaching 3000 m
Water Depth. Proceedings of OMAE, 2011
[4] Tan, Z., Loper, C., Sheldrake, T. and Karabelas, G. (2004): Behavior of Tensile Wires in Unbonded Flexible Pipe under Compression and Design Optimization for Prevention Proceedings
of OMAE, 2006
[5] Brack. A., Troina, L.M.B. and Sousa, J.R.M. (2005): Flexible
Riser Resistance Against Combined Axial Compression, Bending and Torsion in Ultra-Deep Water Depths. Proceedings of
OMAE, 2005
[6] Vaz, M.A. and Rizzo, N.A.S. (2011): A finite element model for
flexible pipe armor wire instability Proceedings of OMAE, 2005
[7] Feret, J.J. and Bournazel, C.L. (1987): Calculation of stresses
and slips in structural layers of unbonded flexible pipes. Journal
of Offshore Mechanics and Arctic Engineering, Vol. 109
[8] CAFLEX Theory manual. IFP/SINTEF, 1991
[9] McIver, D.B. (1995): A method of modeling the detailed component and overall structural behaviour of flexible pipe sections.
Engineering Structures, Vol. 17, No. 4, pp. 254-266
[10] Custodio, A.B. and Vaz, M.A. (2002): A nonlinear formulation
for the axisymmetric response of umbilical cables and flexible
pipes. Applied Ocean Research 24, 21-29
[11] Alfano, G., Bahtui, A. and Bahai, H. (2009): Numerical derivation of constitutive models for unbonded flexible risers. International Journal of Mechanical Sciences 51,295-304
[12] Vaz. M.A. and Patel, M.H. (2007): Post-buckling behaviour of
slender structures with a bi-linear bending moment-curvature relationship. International Journal of Non-linear Mechanics, Vol.
42, p.470-483
[13] Witz, J.A. and Tan, Z. (1992): On the Flexural Structural Behaviour of Flexible Pipes, Umbillicals and Marine Cables. Marine Structures, Vol. 5
[14] Kraincanic I. and Kabadze E.. (2001): Slip initiation and progression in helical armouring layers of unbonded flexible pipes
and its effect on pipe bending behavior. Journal of Strain Analysis, Vol. 36, No. 3
[15] Svik, S. (1993): A finite element model for predicting stresses
12
!""
#$%%
&
%$#
Introduction
Unbounded flexible pipes are steel-polymer composite structures with a wide range of
applications in the offshore industry. A flexible pipe structure is usually constituted
by numerous layers with different properties, see Figure 1. The pipe bore, denoted the
carcass, is constituted by helically wound profiles surrounded by a pressure armour. These
layers ensure the structural integrity against external and internal pressure. The pressure
armour is surrounded by a polymeric liner, which like the external pipe sheath, is a fluid
barrier layer. The space between liner and outer sheath is usually denoted the pipe
annulus. In the pipe annulus, the tensile armour layers are located, usually constituted
by two layers of oppositely wound steel wires. Usually, the total number of wires is
80 150. These layers ensure the structural integrity against longitudinal and torsional
loads. The tensile armour layers are in flexible pipes for deep-water applications usually
surrounded by a high strength tape in order to prevent radial deflections. Flexible pipes
are usually designed in accordance with the specifications given in the API17J-standard,
[1].
In the present paper, only the mechanics of the tensile armour wires are addressed.
During pipe laying, the flexible pipe is in a free-hanging configuration from an installation
vessel to the seabed, see Figure 3. Furthermore, the pipe is empty, in order to ease
the installation process, and hydrostatic pressure on the end cap causes longitudinal
243
compression. Due to vessel movements, wave loads and current the flexible pipe is also
exposed to repeated bending cycles. This is known possibly to lead to lateral wire buckling
failure, especially, if the outer sheath of the pipe is breached such that the pipe annulus
is flooded. This leads to, that external pressure no longer induce sufficient frictional
resistance to prohibit wire slippage. The failure mode was first described by Braga and
Kaleff, [2], who reproduced it experimentally in the laboratory. Further experimental
investigations were conducted in [3]-[6].
In repeated bending the wires within the pipe wall may slip towards a configuration
in which the wire lay angle is not constant, like in the initial helical configuration. For a
pipe subjected to longitudinal compression, the geometrical configurations of the wires obtained after a significant number of bending cycles, may be associated with wire buckling
within the pipe wall leading to a reduced load carrying ability of the pipe structure.
The mechanics of armouring wires in flexible pipes have been subject of both academic
and industrial research in the past few decades. Feret and Bournazel, [7], derived expressions for prediction of the global response of straight flexible pipes on basis of analysis of
internal components. The methods were implemented in a computerprogram, see [8], in
which the armouring tendons were described as perfect helices. The global behavior of
flexible pipes has been investigated further in numerous publications, see [9]-[12].
Witz and Tan, [13], and Kraincanic and Kabadze, [14], considered progression of wire
slippage along curves with constant lay angles for flexible pipes in bending. Svik, [16],
addressed the same problem, but based his analysis on a finite element formulation based
on finite-strain continuum mechanics. Out and von Morgen, [15], considered wire slip
towards the geodesic of a toriod in bending. Leroy and Estrier, [17], simulated wire
slippage due to cyclic bending based on curved beam equilibrium with frictional effects
taken into calculation. However, a prescribed experience-based solution form was applied.
The approaches to wire mechanics are obviously all incapable of predicting transverse
wire slippage for a flexible subjected to bending and compression. A method for calculation of the equilibrium state of a wire which was free of geometrical constraints was
proposed in [6] and elaborated further in [18]. The problem of a curved beam embedded in
a frictionless toroid was addressed, assuming that the wire equilibrium state reached after
244
The pipe will be assumed bent to a constant radius of curvature. Hence, a wire
constitutes a curve on a cylindrical surface, which is bent into a toroid with major
radius R = 1/.
A wire will be modeled as a long and slender curved beam of rectangular cross
section. The dimensions of the cross sections are assumed small compared to both
minor and major torus radii.
Wire friction will be modeled using Coulombs law. Hence, the frictional load is
assumed speed independent.
Wire inertia terms are neglected, since these are estimated small compared to stiffness related terms. A similar approach was followed in [17].
The wires in the inner layer of tensile armour have responses which in terms of stable/unstable behavior are sufficiently equivalent to only consider a single armouring
wire.
Frictional effects will be accounted for by applying transverse loads. However, since inertia
terms are small, second order terms related to the wire slip acceleration in the equilibrium
equations will be neglected. Furthermore, applying Coulomb friction, the transverse wire
loads constituting frictional effects are governed only by the normal wire load and the
frictional coefficient. The direction of the frictional loads will be determined on basis of
the previous load step.
Six differential equations in torus coordinates u and , wire lay angle , tangential
wire force Pt , Shear force in the binormal wire direction Pb and normal moment Mn as
functions of wire arclength s is derived.
cos
du
=
ds
1 + r cos
d
sin
=
ds
r
246
(1)
(2)
d
ds
dPt
ds
dPb
ds
dMn
ds
sin
cos + g
1 + r cos
(3)
= n Pn g Pb p t
(4)
= g Pt Pn pb
(5)
= n Mt + Mb + Pb
(6)
The system is derived on basis of Kirchhoffs equations for curved beam equilibrium
given on vectorial form by Reissner, [19], and concepts from differential geometry for
mathematical description of curves on surfaces.
In order to discretize the system on a known regular mesh, the unknown arclength s
in the deformed state is converted to initial helical arclength s0 . Assuming strains small,
this can be done by applying Cauchys definition of strain, , which is given by
ds
= (1 + )
ds0
(7)
The initial arclength is given by the well-known relation valid for a helix
s0 =
r
sin(hel )
(8)
A
(0) = ini
(0) = hel
B
(SL ) = ini
(SL ) = hel
(9)
(10)
in which Papp is the external load on the wire in the longitudinal pipe direction and SL
B
A
is the total arclength of the wire. ini
and ini
denotes the circumferential wire angles in
both end of the pipe.
Wire stability in dynamic bending
Stability problems have to a wide extend been investigated and are well-described in the
literature. In general, compressive loads are known possibly to cause the equilibrium
equations of a given structure to be fulfilled in buckled geometrical configurations associated with large deflections and rotations. The corresponding equilibrium paths in
force-displacement diagrams may exhibit softening, bifurcation or limit point behavior.
Neglecting friction on the wires, a classical stability approach to the lateral wire buckling
problem was followed in [6]. In the present approach, simulation of frictional loads encaptures an additional physical effect, namely, that cyclic loads must be applied in order
for the wire to slip. A different definition of stability must therefore be considered. Considering the equilibrium paths of a point on the modeled wire, these will, except for the
points s = 0 and s = SL , exhibit a loop-like behavior (examples of such loops are given
247
in Figure 11 and 12). If the wire when subjected to cyclic loads converges towards an
equilibrium state in which this loop is closed, the wire will in the following be considered
stable. If this is the case, the pipe strain obtained by the analysis after each bending cycle
has been completed, will be constant after a number of bending cycles have been applied.
On the other hand, if the (load-strain) loops are not closed, the slip with respect to the
initial configuration will increase for each bending cycle. This may lead to, that the yield
strength of the wire steel is exceeded and failure occurs due to formation of plastic hinges.
In [6] it was demonstrated that buckling could be triggered by adding a small harmonic
response to the initial helical geodesic curvature, such that g can be determined by
m
is
Mn X
(11)
i sin
+
g =
EIn i=1
L
In the following, the imperfection will be calculated by setting 1...20 = 0.001 in accordance with [6].
Frictional forces
The wire loads in the toroid tangent plane, pt and pb , will be defined such that they
constitute frictional resistance. In order to do so, the problem will be defined and solved
stepwise for a prescribed load history. First, the wire will be loaded longitudinally. Afterwards cyclic bending will be simulated. An example of such a definition of loads is
presented in Figure 7 and 8. Since the mass of the wire is small and assuming bending to
be applied slowly, inertia terms can be neglected. Coulomb friction is for a given speed v
defined as
v
(12)
pfric pn
kvk
(13)
The slip speed can be observed only to provide the direction of the frictional force. However, the formulation given in equation (12) in inconvenient for implementation in numerical solvers. A regularization will therefore be applied by assuming a transition, z,
between zero frictional force for v = 0 to full frictional force at v = z
v
(14)
v < z : pfric = p(v)
kvk
v
v z : pfric = pn
(15)
kvk
in which z is the length of the transition zone and p(v) is a polynomial of second order
determined on basis of the conditions
dp(z)
=0
(16)
dv
The slip D is calculated with respect to the previous load step, see Figure 6. For the load
step i and the curvature fixed to = i the slip is given by
p(0) = 0
p(z) = pn
(17)
Di
t
248
(18)
Figure 6. Definition of wire slip, which is calculated with respect to the deformed underlying layer.
0.1
0.09
0.08
Pipe curvature, (m1)
500
1000
1500
2000
0.07
0.06
0.05
0.04
0.03
0.02
2500
0.01
3000
0
0
500
1000
1500
load step number
2000
2500
500
1000
1500
load step number
2000
2500
(19)
(20)
The normal load is obtained from equation 42 governing the normal force equilibrium
pn =
dPn
n Pt + Pb
ds
(21)
In the present approach, radial elasticity of the pipe wall being modeled, is not taken into
account in an exact manner. Since the effect is crusial to the magnitude of the frictional
forces, the minor torus radius will be assumed a function of the applied load. Neglecting
ovalization due to bending, the minor torus radius is given by
r
ka L
r = 1
r
(22)
rd = 1 +
r
kr L
in which ka and kr constitute respectively an axial and a radial spring coefficient of the
pipe being modeled. This approach is equivalent to the methods described in [7] and are
based on the equilibrium of perfect helices. The consequence of calculating the normal
249
load in this manner is, that pn is estimated in a fair manner prior to buckling, while the
value of pn may be inaccurate after occurrence of instability, since buckling leads to large
changes of wire lay angle.
Since the problem is solved numerically by a commercially available BVP-solver, the
length of the slip transition zone, z , must be chosen in such a manner, that convergence
can still be obtained. All time steps will be set to t = 1 s. With this assumption,
the slip speed is related to the slip by Di = vi [1 s]. By numerical experiments it was
determined, that a solution could not be obtained for very short values of z. A transition
length of z = 0.005 was the smallest value, for which the analysis could be performed with
reasonable precision. An obvious consequence of this choice, is that the wires when loaded
may not experience full friction, since the slip speed does not cause, that the length of the
transition zone is exceeded. In the present analysis, stick effects are therefore simulated
in a manner, so the wire has a small speed.
Results
An armouring wire with rectangular cross section within the wall of a flexible pipe will
be modeled on basis of the following geometrical input
r = 0.2762m
Lpitch = 1.474 m
width = 12.5 mm
hel = 30 deg
ka
= 1.9
height = 5 mm
kr
(23)
(24)
The wire steel will be considered isotropic with elastic modulus E = 210 GPa and Poissons ratio = 0.3. Five analyses will be conducted for compressive wire loads, 2.0 kN,2.5
kN, 2.75 kN, 3.0 kN and 3.5 kN. 20 bending cycles from = 1/1000 m1 configuration
(almost straight) to = 1/11 m1 will be simulated. The frictional coefficient will be set
to = 0.1, which corresponds well to values chosen in [9] and [17]. The length of the
frictional transition zone will be set to z = 0.005 m.
Initially, the geometry obtained at the last load step of the simulation will be considered for the compressive load levels 3.0 kN and 3.5 kN, see Figure 9. While the
configuration of the wire obtained for the first load level can be observed not to differ
significantly from the initial helical shape, the wire configuration obtained for the second
load level can be observed to have changed. Obviously, the conclusion can be drawn, that
the second load level has caused the wire to exhibit buckling behavior. However, it is
not possible solely on this basis to determine, if the first load level considered is stable or
not. The wire geometry with maximum curvature for the last simulated bending cycle is
shown in Figure 10.
The average pipe strain will now be considered. This is given by
u(SL ) u(0)
L
=
L
L
(25)
In Figure 11 and 12 examples of the loops formed by the equilibrium paths due to cyclic
bending are presented. Since it is difficult to draw conclusions regarding stability of these
loops, this will be studied on basis of the pipe strain.
In Figure 13 the pipe strain for all analyzed load levels are plotted as functions of
the load steps number. Yet, it is still difficult to draw conclusions regarding stability of
a specific load level. Furthermore, it is on this basis not possible to draw conclusions
regarding if buckling will occur if further bending cycles are applied. Therefore, the
250
=2.0 kN
app
=3.0 kN
app
2065
2070
2470
2075
Longitudinal load (N)
2480
2080
2085
2090
2095
2100
2500
2510
2520
2105
2110
2490
2530
1.7
1.65
1.6
Pipe strain
1.55
1.5
1.45
3
x 10
change of strain after each bending cycle has been concluded with respect to the strain
obtained after the first bending cycle will be considered, see Figure 14. The slope of these
curves can now be taken as basis for consideration of if the wire will remain in a stable
configuration, or if instability may occur after a larger number of bending cycles. The
magnitude of slope for the analyses with Papp set to 2.0 kN and 2.5 kN is decreasing
while this value for the remaining analyses is increasing. The conclusion can therefore be
drawn, that the wire for the two first load levels seem to converge against a closed loop in
(force-strain)-diagrams, while the geometry of the equilibria obtained with the remaining
load levels do not converge towards closed loop behavior. The limit compressive load for
the wire can on basis of this method be estimated to lie between 2.5 and 2.75 kN.
It is interesting to compare this measure for the maximum load carrying ability of
a single wire with the limit load obtained from a frictionless analysis of both layers of
armouring wires by methods proposed in [6]. Calculating the compressive load per wire,
an equilibrium path as shown on Figure 15 is obtained. A limit load of 2.33 kN is
251
0.001
0.5
0.002
0.003
1.5
Change of strain, i1
Pipe strain
0.004
0.005
0.006
Papp=2.0 kN
0.007
0.008
Papp=2.75 kN
=2.5 kN
2
2.5
3
P
Papp=2.5 kN
0.009
Papp=2.75 kN
=3.0 kN
app
500
1000
1500
Load step number
=3.0 kN
=3.25 kN
app
5
0
app
4.5
Papp=3.25 kN
=2.0 kN
app
3.5
app
0.01
x 10
2000
6
8
10
12
14
Number of applied bending cycles
16
18
20
obtained as maximum load carrying ability by the analysis. This is slightly less than the
value determined on basis of the present analysis.
In order to compare the wire mode of deformation associated with instability obtained
by the present method with the buckling mode determined with no friction, these are
shown in Figure 17. It is noted that the frictionless buckling mode is calculated with a
deformation controlled model and that direct comparison of the magnitude of the two
responses is not possible. Furthermore, the two responses do not represent the same load
level, since this cannot be ensured due to significant differences in the chosen means for
controlling the model. However, it can be concluded that the two deformation modes
have approximately the same shape. Hence, inclusion of friction in the model can not be
concluded to have changed buckling modes significantly. In order to investigate the effect
of the frictional coefficient, three analyses with Papp = 2.75 kN and frictional coefficients
0.05, 0.1 and 0.15 were carried out. The results are available in Figure 16. As expected,
the strain rate increases after a lower number of applied bending cycles if the frictional
coefficient is decreased.
P
0.5
2500
Pipe strain
=0.15
=0.10
=0.05
3000
2000
1
1500
=2.75 kN
app
x 10
=1/12 m
along path
1.5
=0 along path
2.5
1000
500
Frictionless model
Model including friction
Frictionless limit load
bending
extension
coupling
3.5
0
0.5
1.5
Pipe strain
2.5
3
3
x 10
500
1000
1500
Load step number
2000
2500
252
Mz = M k
(26)
The behavior detected by the present approach corresponds well to the expected hysteresic
flexural behavior despite only the contribution from a single wire is considered. However,
it is noted that a force term should be added in equation 26 if the total contribution from
the analyzed wire to the global pipe moment is desired, see [6].
Papp=2.0 kN
10
0.65
0.6
Moment contribution from wire (Nm)
0.55
0.5
0.45
0.4
0.35
After 1 bending cycle
After 10 bending cycles
After 15 bending cycles
After 18 bending cycles
Frictionless equilibrium, L/L=0.001
Frictionless equilibrium, L/L=0.002
0.3
0.25
0.2
0.15
0.02
0.04
0.06
Pipe curvature, (1/m)
0.08
0.1
Figure 18. Wire contribution from local moments to global pipe hysteresic flexural behavior for Papp = 2.04 kN, s = SL /2.
For small wire slips, it is reasonable to calculate slippage in terms of a tangential and a
transverse components by projecting the wire slip for load step i onto the initial tangent,
t0 , and initial binormal, b0 for fixed curvature
Dt = kD t0 k
Db = kD b0 k
The two slip components are plotted versus each other, see Figure 19 and 20. Similar
results are presented in [17] and [21].
Conclusions
On basis of an established model for determination of the equilibrium state of an armoring
wire within the wall of a flexible pipe, means for inclusion of frictional effects have been
presented. Solutions are obtained as the solution to a boundary value problem solved
for each step in a predefined load history. Friction has been modeled as tangential and
transverse distributed wire loads with magnitudes based on a regularized Coulomb law
and the normal distributed wire load. The directions of the frictional loads have been
calculated on basis of the wire slippage with respect to the the previous load step. Despite
the choice of slip speed transition is arguable, the proposed method has proven capable
253
Papp=3.0 kN
Papp=3.25 kN
25
20
10
15
0.5
1.5
2
tangential slip (mm)
2.5
0.5
1.5
2
tangential slip (mm)
2.5
3.5
of limiting the wire slippage in dynamic loading and representing key-effects which are
known to be caused by friction.
The proposed method was applied to a specified pipe design and the stability of a
single wire subjected to prescribed cyclic loads was examined. It was found, that when
simulating only a limited number of bending cycles, an estimation of if the wire would
remain in a stable configuration could be found by considering the change of strain obtained after each bending cycle with respect to the strain found after the first bending
cycle. The sloop of the obtained curves may serve as basis for stability considerations,
since they reveal if wire slippage converges towards a stable configuration or not. The
buckling modes determined were approximately of the same shape as buckling modes
found if friction was neglected. The load carrying ability was slightly larger than the
limit load determined when friction was neglected.
With larger computational power, than used for conducting the present analyses, the
proposed method may be used to model all wires within the wall of a flexible pipe. However, coupling the stick-slip effects to the global flexural pipe constitutive relations, which
due to friction is known to exhibit hysteresic behavior leading to variations of the radius
of curvature, is a task which calls for further research. Due to the assumption, that the
global curvature is constant, this cannot be conducted, without extending the present
formulation. Furthermore, it is desirable to investigate means for implementation of a
shorter transition zone and possibly a frictional law based on measured parameters. Inclusion of such means in the analysis are likely to limit wire slippage further and represent
the modeled physics in a more accurate manner. In order to do so, further research and
severe computational power is needed. However, very little research in slip mechanics
allowing transverse slips limited by friction has been conducted, and the present research
may therefore serve as a valuable basis for further research.
References
[1] API Spec 17J, Specification for Unbonded Flexible Pipe American Petroleum Institute,
2nd edition, 1999
[2] Braga, M.P. and Kaleff, P. Flexible Pipe Sensitivity to Birdcaging and Armor Wire
Lateral Buckling. Proceedings of OMAE, 2004
254
[3] Bectarte,F. and Coutarel,A. Instability of Tensile Armour Layers of Flexible Pipes
under External Pressure. Proceedings of OMAE, 2004
[4] Secher, P., Bectarte, F. and Felix-Henry, A. Lateral Buckling of Armor Wires in Flexible Pipes: reaching 3000 m Water Depth. Proceedings of OMAE, 2011
[5] Tan, Z., Loper, C., Sheldrake, T. and Karabelas, G. Behavior of Tensile Wires in
Unbonded Flexible Pipe under Compression and Design Optimization for Prevention
Proceedings of OMAE, 2006
[6] stergaard, N.H., Lyckegaard, A. and Andreasen, J. On lateral buckling failure of
armour wires in flexible pipes. Proceedings of OMAE, 2011
[7] Feret, J.J. and Bournazel, C.L. Calculation of stresses and slips in structural layers of
unbonded flexible pipes. Journal of Offshore Mechanics and Arctic Engineering, Vol.
109, 1987
[8] CAFLEX Theory manual. IFP/SINTEF, 1991
[9] McIver, D.B. A method of modeling the detailed component and overall structural
behaviour of flexible pipe sections. Engineering Structures, Vol. 17, No. 4, pp. 254-266,
1995
[10] Custodio, A.B. and Vaz, M.A. A nonlinear formulation for the axisymmetric response
of umbilical cables and flexible pipes. Applied Ocean Research 24, 21-29, 2002
[11] Alfano, G., Bahtui, A. and Bahai, H. Numerical derivation of constitutive models
for unbonded flexible risers. International Journal of Mechanical Sciences 51,295-304,
2009
[12] Vaz. M.A. and Patel, M.H. Post-buckling behaviour of slender structures with a bilinear bending moment-curvature relationship. International Journal of Non-linear Mechanics, Vol. 42, p.470-483, 2007
[13] Witz, J.A. and Tan, Z. On the Flexural Structural Behaviour of Flexible Pipes, Umbillicals and Marine Cables. Marine Structures, Vol. 5, 1992
[14] Kraincanic I. and Kabadze E. Slip initiation and progression in helical armouring
layers of unbonded flexible pipes and its effect on pipe bending behavior. Journal of
Strain Analysis, Vol. 36, No. 3, 2001
[15] Out, J. M. M. and von Morgen, B. J. Slippage of helical reinforcing on a bent cylinder.
Engineering Structures, Vol. 19, No. 6, pp. 507-515, 1997
[16] Svik, S. A finite element model for predicting stresses and slip in flexible pipe armouring tendons. Computers and Structures. Vol. 46, No.2, 1993
[17] Leroy, J.M. and Estrier, P. Calculation of stresses and slips in helical layers of dynamically bent flexible pipes. Oil and Gas Science and Technology, REV. IFP, Vol. 56,
No. 6, pp. 545-554, 2001
255
1
+ r cos cos
(u) 1
1
+ r cos sin (u)
(27)
x(u, ) =
r sin
A curve is defined by specifying a relation in (u, )-coordinates. Assuming that such a
relation is given, the following norms are defined
xu =
x
u
x =
(28)
d
du
d
= xu
+ x
ds
ds
ds
n=
xu x
kxu x k
b=tn
(29)
In equation (29), the wire normal has been defined equal to the surface normal. Hereby, it
is assumed that adjacent pipe layers are sufficiently stiff to prohibit the wire from rotating
256
freely around the local tangent. Hence, the rotation around t is geometrically governed
by the underlying toroid.
Adressing the definition of the wire tangent geometry, an alternative definition can be
based on the following vectors spanning the tangent space of the toroid
t = cos tu + sin t
(30)
in which tu and t , which span the toroid tangent space, are given by
tu =
xu
kxu k
t =
x
kx k
(31)
In order for this definition to be consistent with equation (29), the following two differential
equations must hold
du
cos
cos
=
=
ds
kxu k
1 + r cos
d
sin
sin
=
=
ds
kx k
r
(32)
These equations govern the wire geometry in the surface tangent plane.
Transformation formulaes
Having defined two orthonormal frames, (t, n, b) and (tu , t , n), see Figure 5, it is desirable
to relate those by a transformation formula
t
cos sin 0
tu
n = 0
0
1 t
(33)
b
sin cos 0
n
Furthermore, considering the (t, n, b)-frame, it is desirable to relate the triad vectors to
their derivatives in arclength. Defining a normal curvature component, n (curvature
in the (t, n)-plane), a geodesic curvature component, g (curvature in the (t, b)-plane)
and a wire torsion component, (in the (n, b)-plane), this transformation, known as the
Darboux frame, is given by
t
t
0
n g
d
n
n = n 0
(34)
ds
g
0
b
b
It is noted, that the transformation contained in equation (34) implies that a positive
rotation about a given triad axis corresponds to a positive change of curvature for a
positive change of arclength. This is sufficient to specify the signs in the constitutive
relations for the wire.
Equilibrium equations
The equations of equilibrium for a curved Bernoulli-Euler beam segment were formulated
by Kirchhoff and included in Loves book on theory of elasticity, [20]. On vectorial form,
the equilibrium equations were given by Reissner, [19]
dP
+p=0
ds
dM
+tP+m=0
ds
257
(35)
=
=
=
=
P t t + Pn n + Pb b
Mt t + Mn n + Mb b
pt t + pn n + pb b
mt t + mn n + mb b
(36)
(37)
(38)
(39)
(40)
=0
(41)
=0
(42)
=0
(43)
=0
(44)
=0
(45)
=0
(46)
Applying the transformation given in equation 34, the following expressions are derived
dt
dn
= t
ds
ds
dt
db
= b
g = t
ds
ds
dn
db
=b
= n
ds
ds
n = n
258
(47)
The wire curvature components can now be calculated on basis of the chosen geometry
cos
1
n =
cos2 sin2
r
1 + r cos
sin
d
cos +
g =
1 + r cos
ds
1
cos
cos sin
=
1 + r cos r
(48)
(49)
(50)
Constitutive relations
In order to relate the changes of curvature with respect to the initial helical wire state
( = 0) to sectional wire moments, the constitutive relations will be assumed linear. This
is a reasonable assumption if the wire cross sectional dimensions are small compared to
the minor torus radius, which is the case when modeling a flexible pipe. Furthermore, it
will be assumed that the wire strains, , are small, so Cauchys definition of strain applies.
The following constitutive relations can then be assumed valid
Pt = EA
Mb = EIb n
Mt = GJ
Mn = EIn g
Similar constitutive relations have to a wide extend been applied when investigating the
mechanics of armouring wires, see [8, 13, 17].
Field equations
In order to determine the geometry of the wire which on basis of the chosen constitutive
relations satisfy the equations of equilibrium, a sixth order system of first order differential
equations can be derived by considering the following:
Equation (32) governing the wire geometry in the toroid tangent plane provides two
differential equations in u and .
The definition of the geodesic curvature, equation (49), provides one differential
equation in .
The equilibrium equations in tangential force, equation (41), in binormal force, equation (43) and normal moment, equation (45), provides three differential equations
in Pt , Pb and Mn .
This yields the system of six first order differential equations (1-6).
259
! "#
"$%&%' #"()"$ "
$$%" #
# #
!"#$$%&'()*"+*,-$*./01*2344*53,-*6',$!'7,&"'78*9"'+$!$'#$*"'*:#$7';*:++)-"!$*7'%*.!#,&#*1'(&'$$!&'(*
:0.12344*
<='$*4>?2@;*2344;*A",,$!%7B;*C-$*D$,-$!87'%)
:0.12344?@>358
Niels H. stergaard
NKT-Flexibles /
Aalborg University, Department of Mechanical
and Production Engineering
Denmark
Email: Niels.HojenOstergaard@nktflexibles.com
Anders Lyckeggaard
NKT-Flexibles
Jens H. Andreasen
Aalborg University,
Department of Mechanical and Production Engineering
ABSTRACT
This paper introduces the concept of lateral buckling of tensile armour wires in flexible pipes as a failure mode. This phenomenon is governed by large deflections and is therefore highly
non-linear. A model for prediction of the wire equilibrium state
within the pipe wall based on force equilibrium in curved beams
and curvature expressions derived from differential geometry is
presented.
On this basis, a model of the global equilibrium state of the armour layers in flexible pipes is proposed. Furthermore, it is
demonstrated how this model can be used for lateral buckling
prediction. Obtained results are compared with experiments.
INTRODUCTION
Flexible riser pipes are widely used in the offshore industry
for oil and gas extraction from subsea reservoirs at water depths
so large, that it is not possible or feasible to place a traditional
jacket supported oil rig on top of the reservoir. In this case, flexible risers may connect a floating platform to a subsea reservoir.
In order to obtain a structural design which provides sufficient
structural integrity against external and internal pressures, axial
loads and large deflections, flexible pipes are usually designed
as unbonded steel-polymer composite structures comprised by a
number of layers with different mechanical properties and structural functions. Due to the extreme loading conditions a flexible
pipe may experience both during installation and in operation,
multiple failure modes have been identified. Most failure modes
can today be reconstructed experimentally under controlled con1
c 2011 by ASME
Copyright
FIGURE 2. Touch-down zone of flexible pipe during DIP-testing simulating the installation scenario
1 METHODS
1.1 Single wire mechanics
The geometry, equilibrium and constitutive relations for a
single tensile armour wire subjected to axial loads and bending will be considered in this section. The radius of curvature
will, for the sake of simplicity, be assumed constant. A single
armour wire can therefore be considered as constituting a curve
on a torus surface.
Friction will, in order to determine the limit equilibrium state
of a wire subjected to given loads, be neglected. Slip towards
this limit state will occur, since the pipe annulus is considered
flooded, so friction does not restrain the wires to a loxodromic
2
c 2011 by ASME
Copyright
surface
tu =
xu
xv
tv =
kxu k
kxv k
(3)
xu =
x
x
xv =
u
v
(4)
d
du
dv
= xu
+ xv
ds0
ds0
ds0
= cos tu + sin tv
configuration.
While the curvature components of a single tensile armour wire
in the analysis are allowed to be large, the axial strain of the wire
is assumed sufficiently small to determine using Cauchys definition of strain. It will furthermore be assumed that the curvature
components can be determined on basis of the geometry of an
inextensible curve due to small axial strains.
A parameterization of the torus by an arc length coordinate u
along the torus centerline and an angular coordinate v is chosen,
see figure 4. The torus surface is then, for pipe curvature = R1
and radius r given by
x(u, v) =
1
+ r cosv cos
(
u)
1
+ r cosv sin ( u)
r sin v
d
ds0
n=
xu xv
kxu xv k
b = tn
(6)
dv
sin
=
ds0
kxv k
(7)
(1)
ds = (1 + )ds0
t=
(5)
(8)
(2)
t t = (1 + )2
(9)
t = (1 + ) cos tu + (1 + ) sin tv
3
(10)
c 2011 by ASME
Copyright
sin
dv
= (1 + )
ds
kxv k
(11)
kxv k=r
g derived
0.04
Change of curvature, (1/m)
du
cos
= (1 + )
ds
kxuk
(12)
n derived
Leroy and Estrier
derived
0.02
0.02
0.04
0.06
dM
+tP+m= 0
ds
1
1.5
wire arclength, s(m)
2.5
(13)
P = Pt t + Pnn + Pbb
M = Mt t + Mnn + Mb b
p = pt t + pnn + pb b
m = mt t + mnn + mbb
(18)
Applying equation 13, the equilibrium equations can be
rewritten on component form
dPt
n Pn + g Pb + pt = 0
ds
dPn
+ n Pt Pb + pn = 0
ds
dPb
g Pt + Pn + pb = 0
ds
dMt
nMn + gMb + mt = 0
ds
(14)
(15)
(16)
The obtained curvature components can be compared with components derived by Leroy and Estrier, [5], chosing a loxodromic
curve on a torus surface as reference curve, see Fig. 5. The
curvature components can be observed to be of the same magnitude and differences in signs can be observed to correspond
to different sign conventions. Since the wire geometry has now
been considered, a set of equilibrium equations must be derived.
The equilibrium equations of a curved beam are given by Reissner, [8], on vectorial form
dP
+p = 0
ds
0.5
cos sin
1 + r cos v r
dMn
+ nMt Mb Pb + mn = 0
ds
dMb
gMt + Mn + Pn + mb = 0
ds
(19)
(20)
(21)
(22)
(23)
(24)
(17)
4
c 2011 by ASME
Copyright
(25)
Mt = GJ
Mb = EIb n
g =
Mn = EIn g
cos
1 + r cos v
sin
= (1 + )
r
sin v
=
cos + g
1 r cos v
= (1 + )
du
ds
dv
ds
d
ds
dPt
ds
(26)
(27)
(28)
= n Pn g Pb
(29)
= g Pt Pn
(30)
= n Mt + Mb + Pb
(31)
L
v(S) = vBini
L
u(S) = L 1 +
L
L
(S) = hel
cos
1 + r cos v
sin
= (1 + )
r
= (1 + )
(35)
(36)
=0
(37)
= n Pn g Pb
(38)
(34)
Now considering the equations governing the tangent wire geometry in equation 11, rearranging the obtained definition of the
geodesic curvature and considering equilibrium in the tangent
plane, the following consistent sixth order system is obtained
du
ds
dv
ds
d
ds
dPt
ds
dPb
ds
dMn
ds
sin v
cos
1 + r cosv
L
u(S) = L 1 +
L
v(S) = vBini
(39)
(40)
(32)
(33)
in which S denotes the total wire arc length, hel the initial helical
wire lay angle and vAini and vBini the specified v-coordinate of the
wire, respectively, for s = 0 and s = S. Furthermore, L
L denotes
the pipe strain and L the pipe twist, which correspond to the
applied generalized loads.
A system of equations for prediction of the wire equilibrium state
has now been derived allowing for large transverse slips.
Modeling an armour wire with the conventional assumption, that
the wire angle remains constant such that the wire constitutes a
c 2011 by ASME
Copyright
nwires
i=1
nsheets
Gi J i
i=1
=0
L
Pa =
i=1
nwires
Pui +
nsheets
i=1
i=1
i
Mu,sheets
=
nsheets
E i Ai
i=1
L
L
(Mui Pvi ri ) +
(42)
i=1
i=1
nwires
i
Pu,sheets
Pa = ka
r
L
= kr
L
r
(43)
in which ka and kr denotes the axial and radial stiffness of a flexible pipe modeled with linear global properties. Defining the
radial strain and rearranging equation 43 yields
ka L
r
rd = 1 +
r = 1
r
r
kr L
(44)
The radius in the torus model can therefore be considered a function of the applied axial loading, while initial curvature components are calculated on basis of the radius in the unloaded state.
Since the system of governing equations for the wires in the inner layer is non-linear, an imperfection must be added to the geometry, in order to trigger stability phenomena if present. The
(41)
6
c 2011 by ASME
Copyright
Inner layer
Outer layer
OD(m)
0.2012
0.209
L pitch (m)
1.263
1.318
3 10
3 10
Number of wires
52
54
150
100
50
R=11 m
R=16 m
R=11 m, radially elastic
g,pert = g + i sin
i=1
i s
L
(45)
0.2
0.4
0.6
Axial compressive pipe strain
0.8
1
3
x 10
2 RESULTS
A 6 flexible pipe with tensile armour properties given in
table 1 will be modeled. Effects from other pipe layers are neglected. It is noted, that polymer sheaths, insulation and highstrength tape layers of flexible pipes may in some cases contribute significantly to the torsional- and compressive pipe stiffness. However, for the present pipe design, this is not the case.
The wires are made of steel with elastic modulus 2.1 105 MPa,
yield stress 765MPa and are considered isotropic. In order to
study the structural behavior in compression, the (load-strain)curve of the loaded end of the pipe will be presented for various
model parameters, see Fig. 8. Two different bending radii will be
studied for radially stiff pipe structures. Furthermore, a radially
elastic pipe structure will be analyzed by applying equation 44.
All responses exhibit significant softening behavior, which is interpreted as limit point buckling. Considering the obtained equilibrium state of the wires, the added imperfection can be observed to cause wire gaps to localize in one end of the analyzed
pipe, see Fig. 9. Both the pipe curvature and the effect of radial expansion can be observed to have very limited influence on
the buckling load in the analyzed flexible pipe. However, the approach by which radial expansion is taken into calculation, can
for obvious reasons not be considered exact and can therefore
only be considered as an estimate.
Considering the twist angle of the free end of the flexible
pipe, this can be observed also exhibit limit point behavior, see
Fig. 10. The physical interpretation of this result is, that when
the pipe is subjected to loads equal to or larger than the limit
point buckling load, it will cause a severe twist. Calculating the
stresses in all wires in the equilibrium state, the maximum stress
in the armour layers can be determined, see Fig. 11.
3 MODEL-EXPERIMENT COMPARISON
In order to reconstruct the lateral buckling failure mode in
the laboratory, experiments were carried out on three 5 meter
long 6 pipe samples with armour layer design given in Tab. 1 in
mechanical test rigs, see Fig. 12. The test setup was quite similar to the one applied by Braga and Kaleff, [1]. The pipe samples were mounted with geometrical boundary conditions corresponding to the ones shown in Fig. 7. Compression was applied
by mounting a smaller flexible pipe inside the test pipe. Subjecting this to tension caused a compressive reaction in the test
sample. Cyclic bending from neutral position to a specific maximum pipe curvature was applied by rotating the pinned frames
on which the pipe endfittings were mounted. In one case, a large
number of bending cycles were applied without sign of failure
in the test sample. After the initial test cycle had been concluded, the test pipe was tensioned in cyclic bending in order
7
c 2011 by ASME
Copyright
110
100
90
80
70
60
R=11 m
R=16 m
R=11 m, radially elastic
50
0.5
0.5
1
Twist of free pipe end (deg)
1.5
350
300
250
200
150
100
50
0
Test Series I
Test Series II
Test Series III
60
400
0.2
0.4
0.6
Axial compressive pipe strain
40
30
20
10
R=11 m
R=16 m
0
50
0.8
1
3
x 10
50
100
150
200
250
number of bending cycles
300
350
400
to straighten the wires within the pipe wall, and the test pipe was
finally subjected to compressive loads larger than during the initial test cycle. This caused failure by lateral buckling in the test
pipe.
The key difference between results obtained from the model
and experiments is, that while modeled wires fail immediately
when critical loads are applied, cyclic bending must be applied
in the experiments to overcome frictional effects before failure
occurs. The measured twist of the free end of the test sample is
presented in Fig. 13. It is during laboratory testing observed, that
the pipe curvature has large influence on the number of bending
cycles which must be applied in order to trigger failure by lateral
buckling.
In Fig. 14 buckling mode shapes determined experimentally and
by modeling are compared. Buckling mode A. corresponds to
large wire gaps, while buckling mode B. corresponds to a geometrical state with large deviations from the initial helical angle
but with small gaps. It is noted, that while gaps in mode A. localize similar in test and model results, mode B. is detected as
localized buckling during experiments, but as a periodic solution
throughout the pipe length in the model. However, this mode has
in the current case occurred in the model after the yield strength
has been exceeded in the wires. This may explain the difference
8
c 2011 by ASME
Copyright
R=11 m
L=2.526 m
L=5 m
L=1.263 m
Test Result:No Failure
Test Result:No Failure
possibly not infinite lifetime
Test Result:Failure
250
200
150
100
50
0.5
1.5
2
2.5
3
3.5
Axial compressive pipe strain
4.5
5
4
x 10
c 2011 by ASME
Copyright
cluded in the model, only the limit load for which lateral buckling
will never occur can be determined.
The friction may along with end-fitting effects cause, that the
wires in each end of the analyzed pipe will never experience slip,
since they are locked in their positions. The impact of this effect
on the results can be estimated by setting the length of the mathematical model lower than the physical length of the test pipe.
Results show that this effect may impose severe impact on the
buckling load.
Future research will therefore include a method for determination of the length of the slip-free zones in each end of the test
sample. Furthermore, additional experiments will be conducted
in order to validate the obtained model.
REFERENCES
[1] Braga, M.P. and Kaleff, P. (2004): Flexible Pipe Sensitivity
to Birdcaging and Armor Wire Lateral Buckling. Proceedings of OMAE, 2004
[2] Bectarte,F. and Coutarel,A. (2004): Instability of Tensile
Armour Layers of Flexible Pipes under External Pressure.
Proceedings of OMAE, 2004
[3] Feret, J.J. and Bournazel, C.L. (1987): Calculation of
stresses and slips in structural layers of unbonded flexible
pipes. Journal of Offshore Mechanics and Arctic Engineering, Vol. 109
[4] Witz, J.A. and Tan, Z. (1992): On the Flexural Structural
Behaviour of Flexible Pipes, Umbillicals and Marine Cables. Marine Structures, Vol. 5
[5] Leroy, J.M. and Estrier, P. (2001): Calculation of stresses
and slips in helical layers of dynamically bent flexible
pipes. Oil and Gas Science and Technology, REV. IFP, Vol.
56, No. 6, pp. 545-554, 2001
[6] Svik, S. (1993): A finite element model for predicting
stresses and slip in flexible pipe armouring tendons. Computers and Structures. Vol. 46, No.2
[7] Brack. A., Troina, L.M.B. and Sousa, J.R.M. (2005): Flexible Riser Resistance Against Combined Axial Compression, Bending and Torsion in Ultra-Deep Water Depths.
Proceedings of OMAE, 2005
[8] Reissner, E. (1981): On finite deformations of spacecurved beams. Journal of Applied Mathematics and Physics
(ZAMP), Vol. 32
4 CONCLUSIONS
In order to develop a method, which can predict lateral buckling of the tensile armour wires in flexible pipes, theoretical and
experimental studies have been conducted.
A mathematical single wire model based on equilibrium of
curved beams and curvature expressions derived on basis of differential geometry has been presented. Since the wire is assumed to rest on a frictionless surface, the equilibrium state of
the wire is reached immediately when loads are applied, while
cyclic loadings must be applied to a physical pipe structure in
order to overcome frictional effects so the equilibrium state is
reached.
On basis of the single wire model, a mathematical model of an
entire flexible pipe can be obtained by multiple single wire analyses, if transverse contact between the wires is neglected. This
model can be used to determine the torsional equilibrium state
for a flexible pipe subjected to given compressive and bending
loads.
This model exhibits behavior quite similar to the observations
made during experiments and can be applied in order to obtain
a conservative estimate of the limit buckling load, which can be
carried by the pipe structure. However, since friction is not in10
c 2011 by ASME
Copyright
!"#"$ %&!
!!"
'! '
(
Proceedings of the 31th International Conference on Ocean, Offshore and Arctic Engineering
OMAE2012
June 10-15, 2012, Rio de Janeiro, Brazil
OMAE2012-83080
Niels H. stergaard
NKT-Flexibles /
Aalborg University, Department of Mechanical
and Production Engineering
Denmark
Email: Niels.HojenOstergaard@nktflexibles.com
Anders Lyckeggaard
NKT-Flexibles
Jens H. Andreasen
Aalborg University,
Department of Mechanical and Production Engineering
ABSTRACT
In the present paper, simplifications of methods developed
for modeling of lateral wire buckling in the tensile armour layers
of flexible pipes are proposed. Lateral wire buckling may occur
during pipe laying in ultra-deep waters. In this scenario a flexible pipe is subjected to repeated bending and axial compression
due to hydrostatic pressure on the end cap of an empty pipe. If
the outer sheath is breached, these loads may cause wire slippage
towards states in which the load carrying ability is reduced and
wire buckling in the circumferential pipe direction occurs. This
leads to characteristic deformation patterns, which may compromise the structural integrity of the entire pipe structure. On the
other hand, these loads may cause overstressing of the wires, if
the outer sheath is intact.
Simplifications of established models for calculation of the load
carrying ability are in the present context proposed in a manner,
by which the effect of adjacent pipe layers on the postbuckled
response can be estimated. The simplifications enables significant reduction of the computational time, which is necessary to
calculate the load carrying ability of a given pipe structure.
structures are application as riser pipes for oil and gas extraction
from subsea reservoirs at large water depths. The complex mechanical behavior of flexible pipes has for the past few decades
been subject of industrial and academic research in order to develop methods capable of predicting the loads, which may lead
to the numerous failure modes, that have been identified. Most
failure modes can be prevented using well established methods
for engineering analysis.
Among the many layers constituting a flexible pipe are the tensile armour layers, which ensure the structural integrity against
longitudinal loads. The tensile armour layers are in most known
pipedesigns constituted by two layers of helically wound steel
wires. During pipe laying in deep-waters, a flexible pipe is in
a free-hanging configuration from an installation vessel to the
seabed, see Figure 1 and 2. In this scenario, the flexible pipe
is empty, such that hydrostatic pressure on the end cap causes
longitudinal compression. Furthermore, the flexible pipe is subjected to repeated bending cycles due to waves, current and vessel movements. These loads are known possibly to cause a failure mode referred to as lateral wire buckling, by which the armouring wires become unstable in the circumferential direction,
which leads to large deformations confined within the pipe wall
with respect to the initial helical state. Instability is known to occur in the inner layer of armouring wires, since the compressive
loads in this layer is larger than in the outer layer of armouring
wires. Furthermore, instability is known to cause, that the torsional equilibrium between the armouring layers can no longer
be maintained in the twist free pipe configuration, which may
1 Introduction
Unbonded flexible pipes are composite steel-polymer structures capable of withstanding large tensile loads, bending and
external as well as internal pressure. Flexible pipe structures are
usually designed in accordance with the specifications given in
the API 17J-code, [1]. Among the numerous applications of such
1
c 2012 by ASME
Copyright
FIGURE 1. Flexible pipe during Deep Immersion Performance tests simulating pipe laying
FIGURE 2. Schematic drawing of the principle which is usually used during pipe laying
2 Theory
2.1 Local model for determination of the frictionless
equilibrium of armouring wires
Initially, a single wire within the wall of a flexible pipe assumed bent to a constant curvature will be modeled as constituting a curve on a frictionless toroid, see Figure 4. The initial
helical configuration will be assumed stress-free. The toroid with
major radius R and minor radius r will be parameterized by an
2
c 2012 by ASME
Copyright
g =
du
ds
d
ds
d
ds
dPt
ds
dPb
ds
dMn
ds
cos
1 + r cos
sin
=
r
sin
=
cos + g
1 + r cos
=
m
Mn
i s
+ i sin
EIn i=1
L
(7)
(1)
in which m = 20 and i = 0.001 were used as parameters governing the imperfection. The system was solved by a commercially available BVP-solver with respect to boundary conditions
chosen in accordance with flexible pipe end-fittings and general
ized pipe loads in form of pipe strain L
L and pipe twist rate L .
Means for estimation of effects caused by radial elasticity were
presented and found not to influence the load carrying ability significantly. In order to model the outer layer of armouring wires,
in which failure does not occur, the pitch angle of the wires in this
layer was assumed constant, which reduced the governing equations to a system of fourth order. This enables determination of
the equilibrium state of the wires within the pipe wall.
(2)
(3)
= n Pn g Pb
(4)
= g Pt Pn
(5)
= n Mt + Mb + Pb
(6)
c 2012 by ASME
Copyright
FIGURE 4. Model of armour wire within the wall of a flexible pipe subjected to bending and longitudinal loads
j=1
i=1
2.2
nlayers nwires ( j)
Pu =
j=1
L
,
L L
nlayers nwires ( j)
j=1
(8)
=0
nlayers nwires ( j)
(Mui, j Pi, j r j )
i=1
j=1
Pui, j
i=1
(9)
i=1
c 2012 by ASME
Copyright
3.1 Simulations
It will now be assumed, that the mechanical behavior of the inner layer of armouring wires can be modeled as the scaled result
from a single wire analysis performed on basis of equation 16. The outer layer of tensile armour will be modeled using the
CAFLEX equations. Neglecting the effect of the outer sheath and
anti-birdcaging tape and denoting contributions from the outer
layer of tensile armour with index 2, equation 8 simplifies to
(Mt1 cos 1 + Mb1 sin 1 )n1wires
((Pt1 sin 1 Pb1 cos 1 ) r1 )n1wires + M2 = 0
(10)
This equation can also be solved in an equivalent manner to equation 8. This approach constitutes the first simplification, which
will be presented in the present work. The second simplification
is described in section 3.3 on basis of the results presented in
section 3. It is noted, that despite force and torsional moment
contributions from outer sheath and anti-birdcaging layers are
neglected in the present approach in order to ensure that obtained
results can be compared with results obtained by the full global
pipe model presented in [5], these can in principle be taken into
calculation by adding appropriate contributions to Equation 10.
3 Results
In the following, a 6 and an 8 riser are studied, see table
1. Both samples were 5 meter long. The polymeric sheaths will
be modeled with elastic modulus 400 MPa. The 6 riser had
5
c 2012 by ASME
Copyright
1...20
300
250
Longitudinal compressive force (kN)
200
150
100
50
200
150
100
Inner layer, radially inelastic
Outer layer, radially inelastic
Sum of layers, Radially inelastic
Inner layer, radially elastic
Outer layer, radially elastic
Sum of layers, radially elastic
Offset linear response
Model simplification A, sum of layers,
radially elastic
Analysis terminated
50
0
50
100
150
200
0.2
0.4
0.6
0.8
Longitudinal compressive pipe strain
0.2
0.4
0.6
0.8
1
1.2
1.4
Longitudinal compressive pipe strain
1.6
1.8
3
x 10
x 10
FIGURE 8. Equilibrium paths obtained for 8 riser with various methods, contributions from separate layers shown (only tensile armour layers
modeled)
FIGURE 7. Equilibrium paths obtained for 6 riser with full global pipe
model and model simplification A (only tensile armour layers modeled)
TABLE 1. Flexible pipe designs, abbreviations) TA1, inner layer of tensile armour, TA2, outer layer of tensile armour, ABC, anti-birdcaging tape
3.2
6,TA1
6,TA2
6,ABC
6,sheath
8,TA1
8,TA2
8,ABC
8,sheath
Outer diameter(m)
0.2012
0.209
0.2117
0.2253
0.2762
0.289
0.292
0.4336
L pitch (m)
1.263
1.318
0.075
1.474
1.525
0.025
26.2
26.2
83.5
30
30.3
88.4
Thickness (mm)
1.8
10
Number of windings
52
54
54
56
Laboratory experiments
Mechanical test rigs constructed specifically for experimental reconstruction of the lateral wire buckling failure mode was
used to test three 6 and two 8 pipe samples , see Figure 10. A
more detailed description of the test principle is included in [5].
It is widely accepted that lateral wire buckling is associated with
shortening, twist and change of circumference of a flexible pipe,
see Figure 11, 12, 13 and 14. The test programs are described in
table 1, while a summery of the results obtained is contained in
table 2.
Two very interesting observations were made during testing of
the 8 risers. Firstly, despite the experiment denoted Test II, LC
2 exhibit behavior indicating that failure had occurred in the test
sample, no signs of lateral wire buckling could be detected during dissection. Since the 8 riser has wires of high-strength steel,
this suggests that lateral buckling may developed in the elastic
6
c 2012 by ASME
Copyright
300
8" Riser, full length
8" Riser, reduced length
6" Riser, full length
6" Riser, reduced length
250
200
150
100
50
0.5
1.5
2
2.5
3
3.5
Longitudinal compressive pipe strain
4.5
4
x 10
FIGURE 10. Mechanical test rig applied for lateral wire buckling laboratory experiments
200
50
6" Riser, Test I, LC 1, 265 kN, 11 m bending
6" Riser, Test II, LC 1, 80 kN, 11 m bending
6" Riser, Test II, LC 2, 210 kN, 11 m bending
6" Riser, Test III, LC 1, 160 kN, 11 m bending
6" Riser, Test III, LC 2, 265 kN, 8 m bending
8" Riser, Test II, LC 2, 400 kN, 12 m bending
8" Riser, Test II, LC 1, 300 kN, 12 m bending
8" Riser, Test I, LC 1, 700 kN, 12 m bending
40
35
30
150
Compressive stroke (mm)
45
25
20
15
100
50
10
5
0
50
0
500
1000
1500
Number of bending cycles
2000
2500
500
1000
1500
Number of bending cycles
2000
2500
buckling could be detected by dissection after the DIP-test, failure occured during the laboratory experiment. This supports the
observation made by Braga and Kaleff, [2], that failure occurs at
lower compressive load levels during laboratory experiments in
mechanical rigs than during pipe installation in the field.
ering the pipe equilibrium paths in Figure 7 and 8, it can be concluded, that after buckling occurs, the load carried by the structures softens to a level, which with very good accuracy can be
assumed constant. This load is determined on basis of methods
described in section 2.2 and 2.3. It will now be assumed that the
mechanical behavior of the layer prone to buckling in a forcedisplacement diagram exhibits a bilinear trend. The mechanical
behavior prior to buckling is then based on the expected linear
compressive pipe response. The postbuckling behavior is based
on a constant force in the inner layer. The remaining layers will
be modeled using the CAFLEX equations, [11], however, neglecting the change of thickness in the layers, this yields three
3.3
c 2012 by ASME
Copyright
4.5
4
Change of circumference (mm)
5
Change of circumference (mm)
Gauge 1
Gauge 2
Gauge 3
Gauge 4
Gauge 5
3.5
3
2.5
2
1.5
1
0.5
0
50
100
150
200
250
Number of bending cycles
300
350
500
1000
1500
number of bending cycles
2000
TABLE 2. Test results, abbreviations) LC: load cycle, P: applied compressive load, n: number of bending cycles applied during experiment
Pipe ID
Test ID
LC ID
P(kN)
R(m)
Results
6 riser
Test 1
265
11
204
Failure ( 45deg)
6 riser
Test 2
80
11
800
6 riser
Test 2
II
200
11
392
No failure ( 45deg)
6 riser
Test 3
160
11
1200
No failure ( 3deg)
6 riser
Test 3
II
265
151
Failure ( 45deg)
8 riser
Test 1
700
12
1200
Failure ( 27deg)
8 riser
Test 2
300
12
1200
8 riser
Test 2
II
400
12
2400
No Failure ( 15deg)
pared to iteratively solved simplified models for established moment equilibrium and were found to correspond well. It is noted
that while friction is neglected in the local wire model enabling
computation of the wire limit states, friction is in the present simplification assumed to cause all pipe layers in the cross section to
experience the same twist and strain.
4 Discussion
Considering the equilibrium paths calculated with the various presented methods, the following can be observed:
1. If only the armouring layers are modeled, the load carrying
ability of a given pipe structure is estimated conservatively.
8
c 2012 by ASME
Copyright
140
500
Inner layer of armouring wires
Outer layer of armouring wires
ABClayer
Outer sheath
Sum of layers
450
120
80
400
Inner layer of armouring wires
Outer layer of armouring wires
ABClayer
Outer sheath
Sum of layers
100
60
40
350
300
250
200
150
100
20
50
0
10
20
30
Pipe twist (deg)
40
50
10
20
30
Pipe twist (deg)
40
50
5 Conclusions
In the present paper, two simplifications of global pipe models for modeling of lateral buckling of flexible pipe armour wires
have been proposed. While the first simplification leads to a poor
representation of prebuckled response the postbuckled response
is estimated with good accuracy. The second method has been
applied in order to represent the global behavior of the pipe structure both before and after softening of the inner layer of tensile
armour due to lateral wire buckling. However, the load carrying ability of the layer prone to buckling must be established by
other means prior to the analysis. Compared to experimentally
obtained results, the present approach limits the conservatism of
the original model for lateral buckling prediction, which served
as basis for the present simplifications. The accuracy of the prediction may be increased further by inclusion of a better measure
of boundary effects.
REFERENCES
[1] API Spec 17J, Specification for Unbonded Flexible Pipes.
American Petroleum Institute, 3rd edition, 2008
[2] Braga, M.P. and Kaleff, P. Flexible Pipe Sensitivity to Birdcaging and Armor Wire Lateral Buckling. Proceedings of
OMAE, 2004
[3] Secher, P., Bectarte, F. and Felix-Henry, A. Lateral Buckling of Armor Wires in Flexible Pipes: reaching 3000 m
Water Depth. Proceedings of OMAE, 2011
9
c 2012 by ASME
Copyright
[4] Tan, Z., Loper, C., Sheldrake, T. and Karabelas, G. Behavior of Tensile Wires in Unbonded Flexible Pipe under Compression and Design Optimization for Prevention. Proceedings of OMAE, 2006
[5] stergaard, N.H., Lyckegaard, A. and Andreasen, J. On lateral buckling failure of armour wires in flexible pipes. Proceedings of OMAE, 2011
[6] stergaard, N.H., Lyckegaard, A. and Andreasen, J. A
method for prediction of the equilibrium state of a long and
slender wire on a frictionless toroid applied for analysis of
flexible pipe structures. Engineering Structures, Vol. 34, pp.
391-399,2012
[7] stergaard, N.H., Lyckegaard, A. and Andreasen, J. Simulation of frictional effects in models for calculation of the
equilibrium state of flexible pipe armouring wires in compression and bending. Rakenteiden Mekaniikka (Journal of
Structural Mechanics), Vol. 44, 2011, No. 3, 2011
[8] Vaz, M.A. and Rizzo, N.A.S. A finite element model for
flexible pipe armor wire instability Marine Structures, 2011
[9] Brack, A., Troina, L.M.B. and Sousa, J.R.M. Flexible Riser
Resistance Against Combined Axial Compression, Bending
and Torsion in Ultra-Deep Water Depths. Proceedings of
OMAE, 2005
[10] Reissner, E. On finite deformations of space-curved beams.
Journal of Applied Mathematics and Physics (ZAMP), Vol.
32, 1981
[11] CAFLEX Theory manual. IFP/SINTEF, 1991
[12] Feret, J.J. and Bournazel, C.L. Calculation of stresses and
slips in structural layers of unbonded flexible pipes. Journal
of Offshore Mechanics and Arctic Engineering, Vol. 109,
1987
10
c 2012 by ASME
Copyright