A theoretical approach to prediction of service life of unbonded flexible pipes under dynamic loading conditions

Marine Structures 5 (1992) 399-429

A Theoretical Approach to Prediction of Service Life of Unbonded Flexible Pipes under Dynamic Loading Conditions

P. C l a y d o n , G. C o o k , P. A. B r o w n & R. C h a n d w a n i Zentech International Limited. 103 Mytchett Road. Camberley. Surrey. UK. GU 166ES

ABSTRACT One important aspect in the design of unbondedflexible pipe for dynamic service is the prediction of life under cyclic loading conditions. This paper presents the methodology behind flexible pipe life prediction including analytical determination of pipe stiffness, interlayer contact pressures, stress recovery, slip between layers, wear and fatigue. Particular emphasis is placed on the derivation of pipe stiffness and on the mechanism of slip between layers. Key words: damage, fatigue, wear, slip, design, life.

NOTATION A

Cc CI

Cl E

f F G

Individual helical wire cross-sectional area C h a n g e in curvature Extension per unit original length C h a n g e of twist Young's modulus Local helical element force Total layer axial force Shear modulus

I J K l L m

Individual helical wire second m o m e n t of area Individual helical wire polar m o m e n t of area Wear coefficient Length along helix Lead of helix (axial length per revolution) Local helical element moment Multiplicity of the helix

n 399 Marine Structures 0951-8339/92/$05.00 © 1992 Elsevier Science Publishers Ltd. England. Printed in Great Britain.

400

P

P ?.

R t t

w

a Al Ap Ar e

P Claydon, G. Cook, P. A. Brown, R. Chandwani

Radial outward force/unit tangential length on an individual helical element Total pressure differential through pipe thickness Mean radius Radius of curvature Thickness Individual helical wire thickness (=layer thickness) Individual helical wire width Helix angle defined by tan a = 2nr/L Change in length/unit length (axial strain) Pressure differential through layer thickness Change in radius " Strain

p v cr o

Coefficient of friction Effective layer Poisson ratio Stress Poisson's ratio Angle neutral bending axis

Subscripts a Axial component along the pipe int Internal n Normal component to both tangential and radial components o Original unloaded state r Radially outward from the helix axis x Tangentially to the local helical element 0 Circumferential component

1 INTRODUCTION The use of flexible pipe within the dynamic conditions experienced in offshore oil and gas drilling and production systems has steadily increased over the past l0 years. This type of pipe is typically used within floating production systems for high pressure production risers, export risers, chemical/water injection lines and gas lift lines to connect a floating vessel with the seabed or with a fixed platform. The most commonly used type of pipe for high pressure service is the unbonded type characterised by a number of steel and thermoplastic layers. Currently the main manufacturers of this pipe are Coflexip, Wellstream and Furukawa. Flexible riser systems are designed to be compliant in order to withstand both dynamic wave-induced motions of the production vessel and direct dynamic wave loading. The flexible riser hangs from the vessel in a catenary-type configuration, possibly with intermediate buoyant support -- the configuration choice is made based on environmental, economic and operational conditions, t Once the configuration type has been

Prediction of service life of unbonded flexible pipes

401

chosen, the system is usually designed using global dynamic response analyses and requires the use of a specialist software package, for example the three-dimensional pipe dynamic response package FLEXRISER. 2 The design is governed by extreme wave and current loading combined with the vessel at survival draught) Typically, loading is applied from three separate directions. The pipe allowable tension and curvature are specified by the manufacturer. Problems associated with configuration design are addressed in Refs 4-6. Once the system has been designed to withstand extreme survival conditions, an additional, and equally important, requirement is the prediction of the service life of the riser system under operational cyclic loading conditions allowing for the effect of interlayer wear and fatigue damage. The methodology behind this prediction is described within the current paper. This methodology has been incorporated into an in-house software package and will be referred to as 'ZENLIFE'. The objective in developing the ZENLIFE module is to extend and complement the design service capabilities of FLEXRISER. The module is capable of predicting the service lifetime of a flexible pipe by interfacing to the FLEXRISER analysis results and taking the necessary auxiliary input data not required for the FLEXRISER analysis. The module predicts the service life for any particular pipe layer at any pipe axial and circumferential location. Life prediction is based on a wear model dependent on interlayer pressures and slip distances and also mechanical fatigue aggravated by stress increases due to wearinduced cross-sectional area losses. The safe service life can then be predicted based on acceptable wear volume fractions and safety factors to account for uncertainties in loading and scatter in material fatigue data. A number of authors have addressed the problem of predicting stresses and therefore mechanical strength within unbonded flexible pipes. 7-~3 However only a relatively small amount of information is available within the public domain on the behaviour of flexible pipe under cyclic loading and the mechanisms of slip and wear. t4-~6 Reference 14 gives a good general outline on service life prediction. The intention of this paper is to give a theoretical basis to each stage of service life prediction.

2 LIFE PREDICTION M E T H O D O L O G Y An unbonded flexible pipe consists of a number of concentric layers formed and tailored to suit specific service conditions. During their lifetime these risers experience high axial and pressure loading combined with cyclic bending moments. As such. the layers within a riser pipe are

402

P. Claydon. G. Cook. P A. Brown. R. Chandwani

subjected to alternating stresses causing fatigue, which in itself is compounded by wear arising from the layers rubbing against each other. The lifetime prediction scheme, in addition to performing a static strength determination, must be able to predict the service life for any pipe layer at any pipe axial and circumferential position. The life prediction scheme is based on a combined wear and fatigue approach dependent on the interlayer pressures and relative slip between the layers. Failure can occur either statically or through fatigue. Both are compounded by the wear of the layer surfaces which act to increase stresses through reduced layer cross-section and to provide initiation sites for cracks. Failure of a layer and hence effectively the pipe can be assumed to have occurred when one of the following criteria has been reached: 1. The net section equivalent von Mises stress exceeds the notched ultimate static tensile strength. 2. The normalized fatigue damage as defined using the PalmgrenMiner rule reaches a specified value, typically unity. 3. The total wear on both sides of a layer has reached a specified critical level. This is usually quoted as a percentage reduction of between 30 and 50% of the layer thickness. There are two methods for determining the service life of a riser, namely the deterministic and stochastic approaches. The deterministic approach discretises the wave height/period scatter diagram into a manageable number of load cases. This approach thus relies on engineeringjudgement to best rationalize the data to correctly define the most significant sections of the environmental loading. The stochastic approach, on the other hand, takes into account the random wave loading within a single load case. The applied motion then consists of an irregular wave form having amplitudes and frequencies making up the desired wave spectrum, applied in random phase angles. In view of the complexity of predicting the service lifetime of flexible pipe, the simplest method of incorporating the prediction is initially adopted, namely the deterministic approach. The procedure behind this approach will be considered in the following section.

2.1 Life prediction procedure A flexible riser experiences loading from sea currents, waves and forced dynamic motion due to wave-induced motion of the surface production unit. The magnitude, period and direction of the waves vary throughout the lifetime of the system. These wave data are often represented by a

Prediction of service life of unbonded flexible pipes

403

frequency of occurrence scatter diagram of wave height vs period in conjunction with a m e a n wave direct diagram or wave rose diagram. Example scatter and wave rose diagrams are shown in Fig. 1 and Table 1. For a deterministic life prediction analysis, the frequency of occurrence diagram is rationalized into a n u m b e r of discrete wave height/period load cases considered to be most representative of the service loading. Each load case is then further subdivided into a small n u m b e r of discrete wave directions from the wave rose diagram. Such rationalizations can result in 60 or more discrete load cases to be analysed. Each load case is analysed in turn, using the flexible riser response program F L E X R I S E R over a n u m b e r of wave periods until steady state has been reached. The required results from the analyses over the last wave period, namely time histories of the curvature, effective tension and coordinates for every analysis nodal point along the pipe, are written to a file for later use in the life prediction analysis. The procedure is shown in Fig. 2. The service life prediction module, referred to as Z E N L I F E , is a postprocessor to FLEXRISER. It reads the results from the analysed load cases and evaluates the service life of the riser at specified locations along the pipe. The methodology behind the life prediction is presented as two flow charts in Figs 3 and 4. The whole scheme may best be considered as a series of modules, the principles b e h i n d which are briefly explained below. The initial modules must first access the additional information necessary for the life prediction. This information consists essentially of: • • • • •

pipe layer geometrical configuration data; material properties; pipe axial and circumferential positions for life prediction; layer interface data - - friction and wear coefficients; fatigue S - N data, specified fatigue life, allowable fatigue damage, allowable wear damage.

Using this information, the individual layer stiffness relations can be formulated which, using compatibility and equilibrium, yield the global stiffness of the pipe cross-section. The loads module accesses the output file to obtain the necessary time history data at the selected axial locations where the life prediction is to be carried out. For every time step of each load case in turn, the interlayer contact pressures, layer stresses, and the incremental slip and wear during the time step are then calculated at positions a r o u n d the circumference. This information is collated for all time steps to obtain m e a n and alternating stresses, together with the total slip and wear across the wave period for each load case.

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Within the fatigue module, the results from each load case are considered in turn over a time interval of, say, 6 months of the riser's service life. The layer's alternating and m e a n stresses are factored to account for loss of material from wear in the previous time interval. To ensure that the layer has not failed statically, the m a x i m u m stress (factored mean plus alternating) is then checked against the notch factored ultimate tensile strength. The fatigue life, N, is calculated for each load case's alternating and m e a n stresses from the material's S - N d a t a . The n u m b e r ofcycles, n, occurring within that time step for each load case is determined from the frequency of occurrence data. The incremental fatigue d a m a g e given by n/N and the surface wear over a given time step are s u m m e d for all the load cases. If the total fatigue damage and wear on both sides of the layer at the specified point on the pipe exceeds specified limits, the layer and hence the pipe is considered to have failed. The total service life is taken to be the sum of the time steps up to that point. If the layer has not failed, the layer's thickness is updated to account for wear and the next time step is considered. The above incremental process continues until failure occurs statically or critical limits on the wear or fatigue have been reached.

406

P. Claydon, G. Cook. P. A. Brown, R. Chandwani [. ]

Select load cases

1 1

I

Load case 1

[ I

FLEXRISER

Flexible riser configuration, r i s e r properties, wave and current loading, analysis time etc.

flexible riser system response analysis

1 1

I

Write to output file

Next

[ Frequency of occurrence scatter and [ m e a n w a v e direction diagrams

]

FLEXRISERoutput'.DBS'

[

!

!

load case

] I

No

maximum

[STOP]

Fig. 2. Riser response analysis.

Because the effect of the incremental wear on the stresses is only considered after a given time step. the accuracy of this procedure increases with smaller time steps, at the expense of increasing the computing costs.

3 DESCRIPTION

OF UNBONDED

PIPE STRUCTURE

An unbonded flexible pipe consists of a complex composite of a number of layers, each serving a particular function in the integrity of the complete pipe. By varying the geometry and number of these layers, the pipe's overall properties can be tailored to suit the conditions and needs of a great many situations. Figure 5 shows the layers making up a typical unbonded flexible riser. The basic unbonded flexible riser pipe will have

Prediction of service life of unbonded flexible pipes

407

Starmpmodule Pipe configurationfile

[

User inputmodule

Materialsfile

[

1

Fatiguedata file

[

User load data file

I

Stiffnessmodule (calculatelayer stiffnesses)

, o-,..DBs.l

1

Loadsmoduleanalyses(aCcesSresults)FLEXRISER I'

[

1

Load case I

1

i

'1 Calculateinterlayercontactpressures I

1 I

Stress recoverymodule

1 Slip module(relativeslip betweenlayers)

1 Wearmodule

1 Statisticsmodule(timeaveragedstress)

1 I

Next loadcase

No

Fatigue/ weardamagemodule

I

1

C sTo,] Fig. 3. Life prediction methodology flow diagram. some, if not all, of these layers d e p e n d i n g on the operational requirements a n d conditions. A detailed description o f the function a n d nature o f e a c h layer has been m a d e by a n u m b e r of authors, in particular see Ref. 16, a n d so will not be i n c l u d e d here.

408

P Claydon. G. Cook. P. A. Brown, R Chandwani

No

Yes

No

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Time, T = 0

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Cslcuhte no. of occurrences, n, within dme step L

[ Frequency of occummce d~agnma I

| Incremental fatigue damage = n / N

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Yea (no failure within pip* m ~ )

Fig. 4. Fatigue and wear damage model.

Prediction of service life of unbonded flexible pipes Intermediate thenr~oplasticsheaths

409

Steel carcass

External thermoplastic shield Fig. 5. Typical u n b o n d e d flexible pipe structure:

4 PIPE STIFFNESS The total pipe stiffness may be obtained by considering the stiffne6s of each layer in turn. These layers can be divided into two types, the cylindrical elastomer or thermoplastic layers and the helically wound steel wire layers. The loading on the pipe is essentially from internal/ external pressure and axial tension with some limited bending moment. In the absence of any bending moment, relations can be produced for each layer type between the radial and longitudinal displacements and the axial force and pressure differential through the thickness. Using compatibility and equilibrium between the layers and across the end of the pipe. the individual layer relations can be combined to produce overall radial and longitudinal stiffness relations for the complete pipe section.

4.1 Assumptions In deriving the stiffness relations for the cylindrical and/or helical layers the following assumptions have been made: 1. All stiffness relationships have been formulated assuming small displacements and small strains. 2. Only axial and pressure loading have been considered in the stiffness formulation, as under these conditions the loads and hence stresses are constant around the circumference. Stresses from bending moments are evaluated separately and then superimposed on the axial tension and pressure stresses. 3. It is assumed that the cylindrical polymer layers could be idealised using conventional elastic thin-wailed theory.

P Claydon, G. Cook, P. A. Brown, R, Chandwani

410

4. The helix is constrained from rotating about the local helical element axis. This rotation is assumed to be suppressed by the surrounding layers. In combining the individual layer stiffness relations into global relationships for the complete pipe cross-section the following further assumptions have been made: 5. Through-thickness strains are ignored, so that radial compatibility requires all layers to have the same radial displacement. 6. The layer thicknesses are ignored in the development of radial equilibrium, so that the total pressure differential is equated to the sum of the individual layer pressure differentials.

4.2 Polymer layer stiffness The stiffness relationships for the polymer cylindrical layers are formulated using conventional elastic, thin-walled theory. Figure 6 shows the resulting forces acting on a section of a cylindrical layer. The axial and circumferential stresses in a cylinder under axial and pressure loading are given by Fa O- a

--

t~0 -

(1)

2 rt rt Apr

(2)

t

Using the elastic stress-strain relationships, 8a =

A!

-

(3)

°'a-°°'°

E

~0

~

m

Fa

j_ ffa

Fig. 6. P o l y m e r layer loading.

Oa

Prediction of service life of unbonded flexible pipes

e0 -

Ar

o- 0 -

oo- a

(4) r E the following stiffness expressions for the pressure difference through the thickness and the axial force in terms of the radial and longitudinal displacements can be produced: Ap = F.

(

-

411

)

r2(1-E~ -u2 ) A r +

(,>2.e,~

= !t l _ o2) A r

+

r(1-u2)]A1

(2,,e.'~ \ ]---Z-U~,j AI

(5) (6)

4.3 Steel helical layer stiffness

The loading and hence stresses along a helical element under uniform axial and pressure loading are constant along its length. Figure 7 shows

\

\

/>

\

J \li

/

~._t./

Fig. 7. Steel helical layer loading.

P. Claydon, G. Cook. P A. Brown. R. Chandwani

412

the resulting forces acting on a section of a helical element arising from such loading. The three normal directions have been chosen to be radially outward from the helix axis and tangential to the helical element, with the final direction normal to the helical element taken from the cross product o f the other two. Force a n d m o m e n t equilibrium of the section produces the following: f , = faCOS a + rpsin a

(7)

f , = fasin a - rpcos a

(8)

f~ = 0

(9)

mx=

m,cosa - r2pcosa +farsina

(10)

COS 2 a

m ° = m a s i n a + r2p s i n a

farcosa

mr = 0

(11) (12)

The corresponding change in curvature, change in twist and extension of the helical element can be equated to the local b e n d i n g moment, twisting m o m e n t and axial force respectively: C h a n g e in curvature: Cc - m , _ sin2a E1 r

sin2ao ro

(13)

C h a n g e in twist: Ct -

GJ

cos a sin a r

cos a o s i n ao ro

(14)

Axial extension per unit length: G-

A. _ //cos a - / o / c o s ao EA /o/COS a o

(15)

The stiffness relations are to be restricted to small displacements a n d strains, i.e. Ar r = ro + A r and << 1 (16) r

a = ao + A a

and

Aa

-a

<< 1

(17)

Prediction of service life of unbonded flexible pipes

413

A further kinematic constraint requires that the number of revolutions of wrap of the helix remains unchanged, i.e. /tan a

2zrr

-

/otan ao 2trro

(18)

which reduces to Aa = (A-~-or-Alo)sinaocOSao

(19)

Substituting for m,, mx,f~ and Aa into the expressions for changes in curvature, twist and extension produces the following stiffness expressions for the pressure difference through the thickness and axial force in the helix:

2rtrAp

(20)

P - ntana

= I ( sin3 a c°s2 ctc°s 2a E1) - ( 2cOs4ctr4sin3a GJ + (si-~------~aEA)] Ar

+ [-{ -2sins 1_k

acos4aE1)

r3

-- (sin3ctc°s2ct(sin2ct --c°s2a) GJ) + (sinctc°s2ct r EA fa

-

-

Fa ?/

AI (21)

(22)

= [(-sin4ocosocos ocos3o o ,)+ ,sin, a

[ / 2 s i n ' a cos 3a

o,) El)

+(sin4ac°sa(sin2a-c°s'-a) ) r2 GJ +(cos 3aEA)

]

AL

(23)

For each helical element the following additional expression is produced for the axial moment in terms of the radial and longitudinal displacements:

P Claydon, G. Cook. P A. Brown, R. Chandwani

414

ma

~--

[(sin3ctc°s2a E1) + ~

~

+ [(-2sin3ac°s2a E1) + (

GJ)] Ar a(sin2a-c°s2a)r GJ)] A! (24)

This twisting moment when summed for all helical elements in all helical layers should be close to zero, so that under axial and pressure loading little or no twisting of the pipe occurs.

4.4 Total pipe stiffness The individual layer stiffnesses are formulated between the axial and radial displacements and the axial force and pressure difference through the thickness. In each case the layer relationships reduce to

Ap = p,Ar + ptAl Fa = Fa Ar + Fa,AI

(25) (26)

Combining the individual layer stiffnesses can be achieved by considering equilibrium and compatibility of the pipe. In deriving the stiffness relationships any through thickness effects were ignored. Radial compatibility requires therefore that the radial displacement for each layer will be the same. Similarly, radial equilibrium requires that the total pressure differential is equal to the sum of all the individual layer pressure differentials, i.e. P = Z A(p),

(27)

Compatibility also requires each layer to have the same axial displacement. Similarly axial equilibrium requires that the total axial force is equal to the sum of all the individual layer axial forces, i.e. F = Z (Fa)i

(28)

These requirements simply result in summing the individual layer stiffness expressions into two equations. These relate the total pressure differential through the thickness and the total axial force to the radial and longitudinal displacements:

Prediction of service life of unbonded flexible pipes

415

For a given internal/external pressure difference and axial load, it is just a simple process of solving these two simultaneous equations for the corresponding radial and longitudinal displacements. Special attention must be applied to the innermost layer, normally called the carcass layer. It consists of a stainless steel strip of S-shaped profile spiralled and interlocked into a helix with a large lay angle (80 degrees). The purpose of this layer is to maintain a constant internal profile and prevent internal collapse under high external pressures. Because of a certain amount of play between each spiral the internal fluid totally surrounds this layer and the internal pressure can be assumed to act directly on the next layer out (a polymer sheath). If, under axial and pressure loading, the radial displacement is positive (i.e. outwards) then due to its large lay angle the carcass can be effectively ignored. If, however, the radial displacement is negative the carcass must be included in the total stiffness derivation to prevent internal collapse. 5 INTERLAYER CONTACT PRESSURES Solving the resulting set of global simultaneous equations produces radial and longitudinal displacements for a given internal/external pressure difference and axial load. Back substitution into the individual layer relations gives the individual layer pressure differentials. The interface pressure between any two layers is just simply the sum of the internal pressure and all the layer pressure differentials up to the layer concerned. The interface pressure between layers i and i + 1 is given by i

A(p)i

Pi = Pi,. + ~ j=

(31)

I

The calculated interface pressures are not in general equal to the contact stresses on each layer. This is due to a reduced contact area arising from gaps between the wires making up the helical layers, as shown in Fig. 8. In such cases the contact area is given by till + I Contact areaAc - sin (ai + ai + 1) _

(32)

Likewise the gap area is given by Gap areaAg

=

gigi+ i d- tigi+ i -t- li+ igi

sin (ai + ai+ i)

(33)

416

P. Claydon, G. Cook, P. A. Brown. R. Chandwani

ri

Fig. 8. C o n t a c t a r e a b e t w e e n h e l i c a l e l e m e n t s .

The interface pressure p; thus acts to produce the contact stress Pc, through the relation: Pc~ = Pi

A~+A Ac

(34)

6 STRESS RECOVERY For a selected axial position along the pipe, axial load, internal over external pressure difference and curvature time histories are available from each riser dynamic analysis load case performed using FLEXRISER. Solution of the pipe stiffness equations using these loads yields time histories of the radial and longitudinal displacements. Back substitution of these displacements into the selected layer's individual stiffness relationships produces the pressure differential through the thickness and the layer's component of the axial load. The stresses acting within the layer can then be evaluated using relations produced in the derivation of the layer stiffnesses. Because the derived stiffness relations assumed no bending, these stresses are constant around the circumference. The bending moments introduce important additional stresses that vary around the circumference.

Prediction of service life of unbonded flexible pipes

417

Within the cylindrical polymer layers the axial and circumferential stresses from just the pressure and axial loading are given from simple elastic thin-walled theory, i.e. Axial stress era

-

-

Circumferential stress ero -

Fa 2rtrt

Apr

(35) (36)

For the helix layers, back substitution of the radial and longitudinal displacements into the individual layer stiffness relations gives the helical element's component of the pressure loading, axial force and twisting moment, p,fa and rn~ respectively. From these the axial, shear, twisting and bending moments local to the helical element can be determined (f.~,f,,, mx and m, respectively). The maximum axial tensile stress in the helical element is then mnt ~ + era"'~ - fA 21

(37)

and the maximum shear stress from the twisting moment is m,.(3w + l-St) Vmax

w2t 2

=

(38)

The stresses arising from bending vary around the circumference and within the polymer layers: these stresses are evaluated using Engineer's Theory of Bending (ETB). They manifest themselves as an additional axial stress proportional to the radial displacement from the neutral bending axis: Ersin 0 oa

-

R

(39)

Within the helical layers, the tendency of the wires to slip under bending must be considered in evaluating the bending stresses. If the alternating curvature is less than a critical value, friction prevents the wires from slipping and the additional axial stress arising from bending, resolved to the local helical element axis. is also evaluated using ETB: era - Ersin R 0 cos 2 a

(40)

If the alternating curvature from the bending is greater than critical, slippage of helical layers takes place. The frictional forces (assumed to be

P Claydon, G. Cook, P. A. Brown, R. Chandwani

418

limiting) act against the local axial slip and are considered to constrain local normal slip. Axially the frictional forces induce an additional axial force into the helical elements given by rq~ of = ~ i - 1P¢,_, + PiP¢,)Asin a

0<~
~

(41)

Constraint of the normal slip induces a local bending moment into the helical elements from which the maximum local bending stress is given by (42)

bmiix

where S(v/)

-

sin a

qt =

~ +

k 2 _ f,EIr Aw 2 I,. -

12

The maximum axial stress within a helical element is then the sum of the stresses above induced from the pipe bending together with that arising from the pressure and longitudinal pipe loading. A secondary factor not included in any of these stress evaluations is that as the pipe bends the contact stresses between the layers no longer remains constant around the circumference. With varying circumferential contact pressures, the pressure and axial force stresses, slip distribution and hence the bending stresses will all be altered. The magnitude of this effect depends very much on the bending distribution along the pipe and is difficult to evaluate for a general riser configuration.

7 SLIP BETWEEN LAYERS The pipe may be considered as a composite of polymer cylindrical and steel helically wound layers. As the pipe bends, the polymer cylindrical layers will follow Engineer's Theory of Bending (ETB) where plane sections remain plane. The helical elements, on the other hand, will try to follow their geodesic paths. A geodesic defines the shortest distance along a curved three-dimensional surface between any two points. Along such a path the path's principal normal vector is normal to the three-

Prediction of service life of unbonded flexible pipes

419

dimensional surface, i.e. there is no component of the path's curvature tangential to the surface. A tensioned wire following any other path between two points will thus have some component of curvature along the surface and hence a component of force trying to move it back towards its geodesic. As such, when the pipe bends the helical layers will try to slip relative to the cylindrical layers. Slippage between any two layers produces surface damage and wear, which in turn provide initiation sites for fatigue cracks to develop. Acting against slip are frictional forces arising from this tendency to slip and the layer interface pressures. Up to a limiting point, where the limiting frictional forces have not been exceeded, the steel helical layers will be constrained to closely follow ETB. Past this critical point, slip between the layers will start and the helical layers will try to follow a more geodesic path. During slippage, part of the pipe's internal strain energy is dissipated through friction. This decreases the bending stiffness of the pipe, resulting in a hysteretic bending moment-curvature relationship, an example of which is shown in Fig. 9. The following sections consider the relative pipe longitudinal and circumferential slip between the helical and cylindrical layers. Following that the effect of friction is considered to determine the slip's initiation point and magnitude.

7.1 Assumptions 1. The slip is evaluated in the helix layer relative to an adjacent cylindrical polymer layer, i.e. relative to ETB of a cylinder. Slip between Moment, M

re K

Fig. 9. Hysteretic moment-curvature relationship.

420

P Claydon, G. Cook, P A. Brown, R. Chandwani

two adjacent cylindrical layers, if such a case occurs, is thus taken to be zero. 2. Expressions are produced for the slip between two layers at some circumferential position relative to the neutral axis. In turn, the angle of this neutral axis to some fixed reference datum on the pipe will be a function of both the time and position along the pipe. Evaluating and keeping track of this reference datum throughout a dynamic analysis is very difficult. It has been assumed instead that the angle between the neutral axis and this reference datum is constant, i.e. the pipe system lies in a plane. This assumption will lead to conservative designs due to the maximum slip occurring continually at the same position on the circumference. 3. Restrained slip is assumed to take place between the layers; i.e. under cyclic bending, slip takes place when the change in curvature exceeds twice the critical curvature. In making this assumption a further assumption has been made in that on the surfaces the limiting friction forces have been reached and remain at these values under increasing curvature. Reversing the direction of curvature increment would mean that the limiting friction in the opposite direction would have to be reached before slip would again take place -- hence twice the critical curvature to cause slip. 4. Slip starts to take place at all points round the circumference at the same time. Inherent in this assumption is that limiting friction is reached at all points along the helix at the same time.

7.2 Slip model formulation The slip of any point on a helix relative to a surrounding cylinder may be considered as comprising two parts, the longitudinal and circumferential slip. These in turn may be locally resolved axially and normally to the helical element, as shown in Fig. 10. A number of publications have produced expressions for the slip between helical and cylindrical layers. In particular Feret and Bournazel 8 considered a helix and the path it traced on a cylindrical core. As the pipe bent, they took the slip to be the distance between the geodesic formed by the helix and the core traced path. The expressions they produced for longitudinal and circumferential slip, however, did not take into account the effects of radial strain from the axial load (Poisson's ratio effect). Some account of this effect has been taken in the expressions below and can be considered as representative for the slip occurring in a pipe. The axial component of slip can be considered as arising from a strain mismatch existing between the helical element and the nominal bending

Prediction of service life of unbonded flexible pipes

421

Total slip

Sc j/

/. -/ J ' / ~ i \ S,/

Helix ] \ , ~

St

f Total slip

Sa Sn

= =

St

= ~ - - -

S/.cosct + Sc.sinct Sc'coset - S/.sinct

~a+S 2

Fig. 10. Slip components.

strain of a surrounding cylinder. The total axial slip along the helix is simply the summation of this strain mismatch. W h e n summed, the effect of the radial strain manifests itself as an additional term to that produced in Ref. 8 as shown below: . /.2 COS t~

S~ = (cos 2 a - v;sin 2 a ) ~

a

(43)

r2c°s ~ from Ref. 8) cf. cos-' a R sin-----a Due to some difficulty in formulating the normal c o m p o n e n t of slip to include Poisson's ratio effects, the normal component has been determined by first considering the longitudinal c o m p o n e n t produced in Ref. 8: $1 -

2r2cos tp Rtana

(44)

The circumferential slip has been taken as that magnitude which when combined with the longitudinal slip produces the axial slip given above arising from the strain mismatch: Sc = Sa - S i c o s a sin a

(45)

7.3 Total slip In the previous section, expressions for the axial and normal slip components were produced for a helix slipping relative to a cylindrical

422

P Claydon, G. Cook, P. A. Brown, R. Chandwani

layer. When a helical layer is adjacent to another helical layer the total slippage is simply the vector difference between the individual layer slippages as shown in Fig. 11. This is of particular importance when the two armour layers are adjacent to each other as their windings are always contra-wound to minimise torsional effects. The slip of each layer is thus in opposing directions and because of the relatively high friction factor of steel acting against steel, wear can be a significant problem with these layers. This together with high cyclic stresses means that wear and fatigue within these layers could dominate the life prediction of the total riser. As stated above, restrained slip is assumed between the layers; i.e. frictional forces between the layers will prevent slippage until at a critical curvature the limiting frictional forces have been exceeded. As plane sections of a cylinder try to remain plane under bending, both surfaces of any non-cylindrical intermediate layer such as a helix will try to slip at about the same time. A reasonable approximation to make, then, is that the frictional forces on both sides of the layer at critical curvature are their limiting values and that under increasing curvature the friction forces remain at these values. Reversing the direction of curvature increment would mean that the friction forces would have to unload first and then increase in the opposite direction to their limiting value before slip in the opposite direction initiates. Therefore, for any load case where the curvature is cyclic, only the increment in curvature needs to be considered and that slip will only take place when the increment in curvature exceeds twice the critical curvature. This can be best

1/t //

Fig. I i. Total relative slip between two helical layers.

Prediction of service life of unbonded flexible pipes

423

demonstrated by considering the two possible situations that might arise, that of no slip and that of limited slip. The first case is that where the change in curvature about its m e a n is less than critical. A demonstrative curvature history and m o m e n t curvature relationship relating to this case are shown in Fig. 12. Increasing the curvature of an initially straight pipe will cause slip between the layers after the critical curvature Kc has been reached. Slip will continue with increasing curvature until point B at m a x i m u m curvature. Decreasing the curvature from point B to D will reverse the direction of the friction forces, but not sufficiently to cause slip in the opposite direction. Increasing the curvature back up to point F is not sufficient to cause slip in the original direction. The net effect is no slip for this load

Curvature,

KI

Ki/l-

~x~

Km- /z~¢ -/ / / /

o/ time. t Moment, M B F

A

CE

Curvature, K

Fig.

12. Curvature amplitude less than critical.

P. Claydon, G. Cook, P. A. Brown, R. Chandwani

424

case (the slip from O to B is covered by another load case within a deterministic analysis). The second case is that where the change in curvature about its mean is greater than critical. A demonstrative curvature history and m o m e n t curvature relationship relating to this case are shown in Fig. 13. Again like the previous case increasing the curvature of an initially straight pipe will cause slip once the critical curvature has been exceeded up to the m a x i m u m curvature at point B. Decreasing the curvature from point B

Curvature, K Km

B

+ AKc

gm

m

/ / Km

-

AKc

-

-

/ I / I I Ol

E

time, t

Moment, M A

G

rB

Curvature, K

Fig. 13. Curvature amplitude greater than critical.

Prediction of service life of unbonded flexible pipes

425

will cause no slip until point D, where friction forces have reversed and reached their limiting values in the opposite direction. Slip will then proceed in the opposite direction until point E, at m i n i m u m curvature. This repeats itself in the original direction, not slipping until point G. This pattern of slip produces the hysteretic bending moment-curvature relationship shown in Fig. 13.

8 INTERFACE WEAR In any design, static and dynamic failure must be considered together with possible failure through fatigue and wear. The latter two modes of failure are usually considered together, since surface damage caused by wear can act as initiation points for fatigue cracks to propagate (fretting fatigue). Consequently, although the loss of material through wear may not be a problem structurally, the initiation of fatigue crack growth may be greatly accelerated. Calculations for wear itself must be considered approximate as the process of wear is not fully understood. It is known, however, to be a strong function of friction, and hence contact stress, and the materials on either side of the interface. There is little data available on wear, and due to its strong friction dependency any data used in any calculations must be from tests under the environmental conditions at which slip is taking place. Within the pipe itself, the difficulties in calculating the wear are further compounded by the fact that most of the pipe layers form a sealed system. Except for the small gaps between the helix wires, the worn material (especially the polymer) has nowhere to escape to and may start lubricating the sliding surfaces. 8.1 Wear model

Despite possible inaccuracies and general lack of data, wear is presented as the amount of material removed for a given number of cycles. The formulation for this is based on 'adhesive' wear from Rabinowitz's work Is and is accepted as a valid description of wear in many cases. This model assumes that the ultimate tensile strength characterises the hardness of the material and is quoted as Thickness of = K X contact normal stress X slip distance material removed ultimate strength

(46)

426

P. Claydon. G. Cook. P. A. Brown, R. Chandwani

9 FATIGUE Fatigue is concerned with failure that occurs under alternating stresses where the same peak load would be safe if applied statically. The number of cycles to failure for a given mean and alternating stress is called the fatigue life. The fatigue life is very much dependent on the initiation of cracks at a microscopic level and as such is dependent on the surface treatment and the environmental conditions. Any rubbing between two components in contact will act to provide initiation sites for cracks to propagate which could greatly reduce its life (fretting fatigue). This has been found to be of particular importance within flexible pipes where the two steel armour layers are in contact. The two helical layers under bending slide relative to each other and are known to be subject to wear and fretting fatigue. In most engineering cases, components are not subjected to just a single alternating-mean stress pair, but a range of varying stress amplitudes. A means for predicting the life of components under such loading is required.

9.1 Fatigue model A linear cumulative damage rule as suggested by Palmgren and Miner is used in the majority of engineering fatigue life calculations. This rule assumes that the damage in the specimen from n load cycles is proportional to n/N, where N is the number of cycles to failure for that stress ratio. For varying stress ratios failure is assumed to have occurred when ~r = C,a constant

(47)

Simple damage considerations by Miner ~9 suggested this constant should be unity. However, a large number of tests have shown that this constant is dependent on the order in which the stresses are applied and values between 0.3 and 3 have been obtained. This would imply that the linear cumulative damage assumption is not valid and in fact there is no physical basis behind why such an assumption can be made. Despite this, Miner's rule has widespread acceptance and is used in nearly all cases of early design to predict the fatigue life. Although Miner's rule is straightforward to use, the scatter in fatigue test data and the variability ofsum ofn/Nto failure mean that great care has to be taken when using it. Other techniques, such as proposed by Kaechele and Swanson and Corten and Dolan, ~6assume various relationships between the damage

Prediction of service life of unbonded flexible pipes

427

and number of cycles and try to take into consideration the order in which the stresses are applied. Although they can be markedly better in predicting the fatigue life they are generally too cumbersome and complex to use and are often restricted to specific conditions. Because of its simplicity and ease of use, fatigue life calculations will be performed using the Palmgren-Miner damage rule above.

9.2 Fatigue life calculation A number of load cases exist arising from different environmental load conditions such as different wave height, wave direction and wave period. One steady state cycle of each load case is analysed from which an alternating and mean stress are determined at a point on the pipe. These stresses are factored to account for a reduction in thickness from wear and the number of cycles Nto failure for each load case is determined. This can be performed by either interpolating from S - N data curves or preferably by using empirical expressions obtained from material fatigue tests and supplied by Heywood) 7These expressions consider the notched fatigue strength, and hence the fatigue life, through stress concentration and strength reduction factors. More importantly, by using some high characteristic strength reduction factor, the reduced fatigue life time under fretting conditions can be estimated. From the frequency of occurrence diagram, over a given time interval a load case will be repeated n times. The normalised damage arising from the load case is given by n / N from- the Palmgren-Miner rule. If this is repeated for all the load cases the total normalised damage can be determined over the time step as the sum of the individual n/N. The fatigue damage calculations continue over subsequent time steps until the fatigue damage reaches a specified value at which failure is assumed to have occurred. Because of the wide range of critical fatigue damage constants experienced in tests, the value used in the calculations has to be obtained from experimental tests.

10 SUMMARY AND RECOMMENDATIONS This paper has presented the theoretical background to assessment of service life o f u n b o n d e d flexible pipes under deterministic cyclic loading. These formulations have been coded into a computer tool known as ZENLIFE. The recommendations for further work in this area are as follows:

428

P. Claydon. G. Cook. P A. Brown, R. Chandwani

• verification of the computer model against mechanical test work and manufacturers' data; • further understanding of the wear m e c h a n i s m and the effect ofspalled material on interlayer friction; • further understanding of the lubrication effect of the polymer layers in reducing fretting fatigue; • establishing the effects of stress concentration factors on fatigue life due to geometric effects of notches at m a x i m u m wear sites; • stochastic modelling incorporating r a n d o m statistical models (Rayleigh, Gaussian) and 'rainfall' analysis to determine fatigue loading cycles from r a n d o m time history data; • effects of torsional loading on stresses and interface pressures taking into account the lazy and stiff directionality; • effect of circumferential variation of interface pi'essure due to bending on the slip model; • accounting for finite layer thicknesses in determining radial equilibrium; • investigation of usefulness of fracture m e c h a n i c s techniques to demonstrate residual life for postulated crack initiation u n d e r fretting conditions; • finite element analysis of pipe using non-linear three-dimensional contact algorithms to verify slip distances, contact pressures and stress recovery modules.

REFERENCES 1. Brown, P.A., Chandwani, R. & Larsen, 1., Flexible riser dynamics modelled in 3 dimensions. Proceedings of International Conference on Offshore Structures, City University, London, 1987. 2. Brown, P.A., Soltanahmadi, A. & Chandwani, R., Application of the finite difference technique to the analysis of flexible riser systems. CIVIL-COMP 89, Civil and Structural Engineering Conference, City University, London, 1989. 3. Brown, P.A., Soltanahmadi, A., Chandwani, R. & Larsen, I. Investigation into optimised design of flexible riser systems. Institute of Marine Engineering, London, 1989. 4. Brown, P.A., Soltanahmadi, A. & Chandwani, R., Problems encountered in detailed design of flexible risers. International Seminar on Flexible Risers, University College, London, 1988. 5. Brown, P.A., Chandwani, R. & Larsen, I. Interaction between flexible risers and mooring lines within a floating production system. Latin American Petroleum Engineering Conference (LAPEC), Rio de Janeiro, 1990. 6. Hoffman, D., lsmail, N. M., Nielsen, R. & Chandwani, R., The design of flexible marine risers in deep and shallow water. OTC 6724, 1991. 7. Lotveit, S.A. & Often, O., Increased reliability through a unified analysis tool

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for bonded and non-bonded pipes. In Advances in Subsea Pipeline Engineering and Technology, Vol. 24, edl C.P. Ellinas. Society for Underwater Technology, 1990, pp. 79-110. 8. Feret, JA. & Bournazel, C.L., Calculation of stresses and slip in structural layers of unbonded flexible pipe. Jnl of OMAE, 109, (1987) 263-9. 9. McNamara, J.F. & Harte, A.M., Three dimensional analytical simulation of flexible pipe wall structure. OMAE 89-744, 1989. 10. de Oliveira, J.G., Goto, Y. & Okamoto, T., Theoretical and methodological approaches to flexible pipe design and application. OTC 5021, 1985. 11. Out, J.M.M., On the prediction of the endurance strength of flexible pipe. OTC 6151, 1989. 1~ Mallen, J., Estrier, P. & Amilhau, S., The Quality and Reliability of Flexible Steel Pipes. Institute of Marine Engineering, London, 1989. 13. Serta, O.B. & Brack, M., Stress and strain assessment of multi-layer flexible pipes. Proceedings of International Conference EUROMS-90, Norway, 1990. 14. Nielsen, R., Coiquhoun, R.S., McCone, A., Witz, J.A. & Chandwani, R., Tools for predicting service life of flexible risers. Proceedings of International Conference EUROMS-90, Norway, 1990. 15. Feret, J.J., Bournazel, C.L. & Rigaud, J., Evaluation of flexible pipes" life expectancy under dynamic loading conditions. OTC 5230, 1986. 16. Coflexip, Coflexip flexible pipe design description -- fatigue. 1987. 17. Heywood, R.B., Designing against fatigue. Chapman & Hall, 1962. 18. Rabinowitz, E., The wear coefficient magnitude scatter, uses. Journal of Lubrication Technology, (1980). 19. Miner, M.A., Cumulative damage in fatigue. Journal of Applied Mechanics. 12, (1945).