OTe 6333

A Dedicated Finite-Element Model for Analyzing UpheavalBuckling Response of Submarine PipelinesF.J. Klever, L.C. van Helvoirt, and A.C. Sluyterman, Shell Research B.V.

Copyright 1990, Offshore Technology Conference

This paper was presented at the 22nd Annual OTC in Houston, Texas, May 7-10, 1990.

This paper was selected for presentation by the OTC Program Committee following review of information contained in an abstract submitted by the author(s). Contents of the paper,as presented. have not been reviewed by the Offshore Technology Conference and are subject to correction by the author(s). The material, as presented, does not necessarily reflectany position of the Offshore Technology Conference or its officers. Permission to copy is restricted to an abstract of not more than 300 words. Illustrations may not be copied. Theabstract should contain conspicuous acknowledgment of where and by whom the paper is presented.

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ABSTRACT

Offshore developments, for example, in the NorthSea, are showing a clear trend towards the use ofsatellite technology, associated with an increasingnumber of small-diameter pipelines and flowlinestransporting multiphase fluids at high temperaturesand pressures.

Upheaval buckling of a pipeline at a criticalfoundation undulation may occur when thermalexpansion is axially restrained and lateral movementis restricted (burial).

Classical design of protective covers is oftenover-conservative. More recently, models have beendeveloped that take realistic imperfections and theactual, non-linear response of the cover intoaccount. However, only idealised, symmetricfoundation imperfections were considered, thusrendering these models unsuitable for fitness-for­purpose assessments of actual cases.

Therefore, a computer model has been developedthat takes all the relevant non-linear parametersinto account, such as elastic/plastic materialbehaviour, axial friction and sand/clay/rock coveruplift resistance. Furthermore, both idealisedimperfections and arbitrary foundation profiles arepossible.

In the paper the theoretical formulation will bedescribed and results will be shown of the analysisof a characteristic North Sea pipeline using both theconventional design method and the newly developednumerical model. The effects of the most importantparameters will be demonstrated and the potentialbenefits of using such an advanced tool illustrated.

This new PC-based computer program enables theengineer to analyse the buckling response of apipeline accurately and fairly quickly. Variousdesign options can be assessed at different levels ofcomplexity, as can be the analysis of fitness-for­purpose and integrity during service using conditionassessment techniques.

References and illustrations at end of paper.

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INTRODUCTION

Offshore developments, for example, in the NorthSea, are showing a clear trend towards the use ofsubsea completion technology and minimum facilitiesconcepts. This trend is associated with an increasingnumber of small-diameter pipelines and flowlinestransporting untreated hydrocarbons from deep wellsinto adjacent facilities. These lines may operate athigh temperatures (even beyond lOO·C) and pressures.

Recently, several upheaval buckling failures ofsmall-diameter flowlines that were designed for hightemperatures have occurred in the North Sea, thisamplifying the need for research into this phenomenonand the development of effective protective measuresagainst it.

When a pipeline, after its installation, isoperated at higher than ambient temperatures andpressures, it will try to expand. If the line is notfree to expand, but axially restrained by friction,the pipe will be subjected to an axial compressiveload. When in such a case, at a critical foundationundulation, the force exerted by the pipe on the soilcover exceeds the vertical uplift restraint createdby the pipe's submerged weight, its bending stiffnessand the soil cover, the pipe will tend to move in thevertical plane (or along the trench side slope whenthe pipe is not covered), and considerable verticaldisplacements may result. This phenomenon is calledupheaval buckling (offshore) or overbend instability(onshore) and shown in Fig. 1. The pipeline response(see, for example, Fig. 8) might then be unacceptablein terms of vertical displacements (the pipeprotruding through the cover or moving out of thetrench), excessive yielding of the pipe material, orlocal buckling. Upheaval buckling is hence a failuremode that has to be taken into account for the designand in-service assessment of trenched and buriedpipelines.

TO facilitate safe and economic design,installation and protection of high-temperaturepipelines, and to enable an accurate assessment of

529

(9 )

(7)

(8 )

(10)

OTC 6333

dvds

Ij\ - /3so

€ - /(R sine

€du

+ 1: (dv)2ds 2 ds

/(d

2v

ds2

~

/(

X s + U R sine sinlj\

y V + R sine coslj\

Z R cose (3)

y(l+2€) coslj\ 1 + dUds

y(l+2€) sinlj\ dVds

1: [(12

~ds

Introducing (4)-(7) into (9) yields:

s + U sl + UI

+ (S-sl+u) cos/3 - v sin/3( 4)

V VI + (S-sl+U) sin/3 + v cos/3

where

/3V

2-V

Iarctan( U U ) (5 )

s2+ 2-s l- 1

where the axial strain € and the bending strain /( arefunctions of s only:

since, obviously, the above properties (7) can alwaysbe ensured by taking P

IP

2short enough.

CompatibilityFor the non-linear analysis of slender beams the

Lagrangian strain ~ is used, defined by

Here (x). denotes the value of (x) at s = s .• Toensure c6mpatibility of the u and v fields it pointsPI and P2 ' we put

Although the above transformation may lookartificial, the crux is that we may put

I ~~ I « 1

I Ij\ - /3 1«1

This two-dimensional deformation of the pipe isgoverned by the displacements U and V of the centreline in the x- and y-directions, respectively, andthese can also be defined in terms of the axialstrain € and the tangent angle Ij\ with respect to thex-axis.

Without loss of generality we may assume that forpoints P in between arbitrary points PI and P thepipe axis deforms only a little further away trom thechord P

IP

2• Hence, the displacements U and V can be

expressed in terms of u and v with respect to a localcoordinate system x' and y' based upon the chord P

IP

2as shown in Fig. 2:

530

(2 )

(1)~

+ R cose iz

~

+ Z iz

~

+ Y iy

~

+ R sine iy

FINITE ELEMENT MODEL FOR ANALYSING UPHEAVAL BUCKLING RESPONSE OF SUBMARINE PIPELINES

~

X ix

~

s ix

~

R

~

r

where

2

on the unit base vectors along the x, y and z axes.The pipe, when deforming, is assumed to behave as aslender beam which may undergo finite displacementsand rotations, but the elastic/plastic strains remainsmall. The deformed position of point P is then

PROBLEM FORMULATION

Kinematics ~

The reference position vector r of a materialpoint P of the pipe wall middle surface (at meanradius R) can be expressed in terms of the convectedcoordinates sand e (see Fig. 2) as

In the following sections of this paper the mainaspects of the mathematical problem formUlation aredescribed in some detail and the solution strategybriefly discussed. Thereafter, sample calculationsare presented to demonstrate the effects of the moreimportant parameters, followed by the conclusions.

In a separate paper [11], a design method ispresented based on the application of the UPBUCKcomputer program. The alternative concept ofintermittently dumping rock onto a trenched pipelineis discussed in another paper [12].

the safety against upheaval buckling of existingpipelines, a comprehensive research programme wasstarted early 1987 and has recently been completed[1]. This paper is one of a series describing theresults of that programme, and is concerned with thedevelopment of a dedicated model for analysing theupheaval response of submarine pipelines.

Analytical/numerical modelling of the upheavalbuckling response of offshore pipelines hasprogressed rapidly over the last few years, broadlyfrom the 'classical' analysis [2,3], to one coveringinitial imperfections [4,5], to one additionallycovering material non-linearity [6,7], to oneadditionally including large pipe displacement andassociated cover non-linearity [8]. An application ofthe more advanced models was presented at a recentconference [9], and case histories are reported aswell [10]. However, only idealised, symmetricfoundation imperfections were considered, thusrendering all these models unsuitable for fitness­for-purpose assessments of actual cases.

This paper describes a numerical model that, inaddition to the features mentioned above, can handlehighly irregular cover and pipe profiles (forevaluating in-situ conditions and using survey data),non-uniform finite foundation stiffness and differentcover response formulations (representing axial anduplift resistance of cohesive and cohesionless covermaterialS such as sand, rock or clay). Furthermore,the model deals with inClined buckling along a trenchslope as well as vertical buckling, this option beinguseful for analysing the 'alternative' concept oftrenching and dumping rock at intervals instead ofcontinuous cover.

The model is implemented in the form of a PC­based computer program, called UPBUCK, the results ofwhich are processed by an a~sociated program (REPORT)to give detailed information in an interactivemanner.

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OTe 6333 KLEVER, VAN HELVOIRT AND SLUVTERMAN 3

Constitutive relationshipsThe pipeline, responding to its loading, may

develop both axial stress and circumferential (hoop)stress. The hoop stress a is determined solely bythe pressure differentialhp of the line, which isassumed to be constant over the model length. Theaxial stress a is determined by the residual laytension NO' th~ temperature differential T and thepipe deformations.

In the pipe material a plane stress state willdevelop, and its elastic/plastic deformations aretaken to be based on J -flow theory of plasticity[13]. Thus, the strain2rate is expressed as

where the internal axial force N and the internalmoment Mare:

EquilibriumThe equilibrium equations are expressed in the

form of the following virtual work equation:

where 11 is to be seen as a function of thedisplacements in taking variations. The second termin (16) denotes the virtual work done by the externalloading, to be defined later. Introducing (8) into(16) gives:

(16)

(17)

o

o

bWextI I R t 0a b~ dB dsL B

I [N bE + M bK] dsL

(11)Cli °a + Cl2 0h + a T

where a superimposed dot denotes a rate or anincrement, where

+ -Y3E

Et (A11 - a AT) - Ao (1 - ..-!.)

°h *a 2 E

2v'3 0h Ao* arctan( a

2 ) 0 (15)30h + (20

a-Oh+2Ao

a) (20

a-Oh )

Here, ~O can be used to account for the stress-freepipe imperfections, and a can be used to accountfor initial stresses due tg, for example, a residuallay tension NO'

The stress/strain curve is taken to be tri-linearas shown in Fig. 3. For each part of constant E ,equation (11) can be integrated in the plastic ?angeto give. for constant 0h'

In these equations a is the thermal expansioncoefficient, E is Young's modulus, E is the tangentmodulus of the uniaxial stress/strai~ curve at stresslevel a, a is the von Mises' equivalent stress, a isthe current yield stress, v is Poisson's ratio, RYisthe mean steel pipe radius and t is the pipe wallthickness. For straining beyond yield, the yieldsurface is assumed to expand according to theisotropic hardening rule.

In the elastic range, equation (11) can beintegrated to give:

(18)

21T 21T 2I [f Rto dB] bE + [- I R to sinB dB] bK +L o a 0

a

b[N (E -du 1. (dv)2)] +ds 2 ds

2b[M (K - !LY)] } ds bw 0 (19)

ds2 ext

21T

N + I R t a dB0

a

21TR

2tM I a sinB dB

0 a

where a is to be seen as a function of E and K, andwhere v~riations are taken with respect to u, v, E,

K, Nand M. In the derivation of a finite element,this principle (19) allows functions to be chosen foru, v, E, K, Nand M independently. In addition, (19)allows functions for E, K, Nand M that may bediscontinuous over elements and this property will beused to eliminate the parameters of these fourfuncions at element level, thus rendering essentiallyan element with only displacement unknowns. Thisprocedure will be called the hybrid approach.

The superiority of the hybrid approach over thefull displacement approach will be demonstrated laterfor the case of the post-buckling behaviour of anaxially compressed beam.

The common way to proceed is to introduce (11)into (18) and both (10) and (18) into (17), thisresulting in a set of equations for the unknowndisplacements u and v. This procedure will be calledthe full displacement approach.

However, a finite element that is much moreeffective in terms of post-buckling behaviour can bederived by means of an extended variationalprinciple [14]. In this extended principle thecompatibility equations (10) are added to (17) bymeans of a 'Lagrange multiplier' method, for whichthe multipliers can be iden~ified as Nand M. Theresulting equation, which replaces (10), (17) and(18), is then

(14)o

R°h P - (12)

t

and where the compliance moduli Cll and C12 are:

Cll + 1. + ~ (L 1:) [1 ~0h 2

E Et

E 4 (~) ]

(L 1:) [1 _ ~a a

C12E. - ~ (--Lh) ]E E t E 2 4 2

a2 (a )2 2

a - aa °h + (Oh) (13)a

0 if a = a and a < 0 or a < a~

y y

1 if a = a and a ~ 0y

for arbitrary~ finite incre~ents AOa from a givenstress state (Oa,Oh) at the yield surface for whicha = a •

y

Pipe weight loadingThe virtual work done by the submerged weight

loading of the pipe is

bWw

I - W sinat

bv dsL sub

(20)

531

4 FINITE ELEMENT MODEL FOR ANALVSING UPHEAVAL BUCKLING RESPONSE OF SUBMARINE PIPELINES OTC 6333

where at

is the angle of the x–y-plane with the YHDOC for jSOhorizon al, i.e. 90” for vertical uplift or equal tothe trench angle for uplift along the trench wall.

P= = { ~HDoc [l-~] +~Pm(Hm) for O<jSVm

Pressure loading m m(26)

The pressure differential p (taken as positive Pm(Hm) for ~ > Vm

for internal pressure) haB two effects on the pipe.Firstly, it induces an axial strain and has its while P vanishes for ~ ~ H. In (26) ~ is theinfluence on the yielding of the steel. This is specifi~ submerged weight of the soil, H the distanceaccounted for via the constitutive relationships between the top of the pipe in its reference position(11). Secondly, the pressure exerts a distributed and the top of the cover, and D is the outerload on the pipe wall that is deformation-dependent. diameter of the pipe including %y coating. TheFor the situation analysed here this pressure loading maximum shear strength is mobilised at a displacementis conservative [15]. For the deformation defined by Vm, and P as a function of Hm denotes the maximum(3) the resulting force is a lateral, line load which uplift fo?ce.is proportional to the curvature of the pipe axis.The virtual work done by the pressure is then In (26) H = H-v for ~ ~ v , while H . H–j for

~ > Vm.6W

For c~ay cov~rs the mod~l (26) ismused with= f mR2 p K (sin~ 6U - COS(36V) ds

P(21) its submerged weight term neglected (y ❑ O).

L

The maximum force Pm is dependent on the current

Foundation ‘embedment ratio’ H /D and vanishes for H ~ O. For

The pipeline is laid on the seabed or on the cohesionless coversm(s%d, gravel or rock) The

bottom of a trench that usually is not perfectly following model is used:

straight. Three different types of imperfection are H Hconsidered. ~HmDoc [1 + fl ;] for 0<#Sf31

The first option is a sinusoidal shape, according ~to ={

Oc Ocm H H

(27)

~HmDoc [1 + fl~l + f2(#- ~1)] for +> f31~)+1] for -L~x~Ly = A $ [COS(U (22) Oc Oc

where A and L are the imperfection height and half- enabling the use of a hi–linear curve; or a linear

length, respectively. The second option is a ‘prop’ one when f = f2 1“

In the case of rock covers

shape, defined by ff . = O and only the weight term is retained.

y= A[4(~)3-3(~)4] for -LSXSL (23)

E&uati~n (27) is shown in Fig. 4a.For cohesive covers such as clay the maximum

force is defined in terms of the ‘undrained shear

The third option is an arbitrary profile, to be strength’ Cu a~d the ‘dimensionless maximum upliftdefined by a series of (x,y) points, such as survey resistance! P

cm:data.

Furthermore, these imperfection types can be H H P*positioned anywhere in the analysis model, since uocf#-CD for O<fi<y

asymmetrical problems can also be analysed. Pm={Oc Oc

When the pipe tries to penetrate the foundation,(28)

P*the latter’s resistance is modelled as a spring. If

HCUDOC P;m for ~>~

the penetration perpendicular to the pipe/foundationinterface is denoted by ~, the normal foundation

Oc

force per unit length is and this equation is shown in Fig. 4b.Finally, the virtual work done by the cover

for ;<0 loading (apart from friction) is then

{Cf Y

‘f ❑ o (24)for ;>0 &w = f - Pc sinat fIVdsc (29)

L

The virtual work done by the foundation (apart fromfriction) is then Frictional loading

($Wf = f p (- sinp NJ + COSB tiv) dsDisplacements of the pipe also mobilise

L’(25) frictional forces. The model acounts for axial

friction forces on the pipe due to the foundation,the trench wall (if present) and the cover. In

Cover loading addition, uplifting along a trench wall generates

If the pipeline is (partly) covered, this may lateral friction forces. All these four friction

involve sand, gravel, clay or rock. When the pipe is force components are defined on the basis of

uplifted, the cover uplift resistance P varies with Coulomb’s model, relating the interface shear force F

the uplift displacement as shown in Fig? 4a for sand, per unit length and the relative slip ~ as follows:

gravel or rock and in Fig. 4b for clay. Details of Fthis behaviour are reported in a separate paper [16]. ; . -=; for lFl<Fmor lFl=FmandF~>OHere the uplift with respect to a reference position u

is denoted by ~. The uplift resistance modelm

comprises three branches F= - sgn(;) Fm for IFI = Fm and F ~ ~(~o)

---5$L

OTC 6333 KLEVER , VAN HELVOIRT ANO SLUYTERMAN 5

where a superimposed dot denotes a rate or increment SOLUTION STSATEGYand sgn( ) denotes the sign function.

The positive quantity Fm is dependent on the The equations of the model described in theforce normal to the interface and on the friction preceding section are solved through a finite elementangle or friction coefficient, both functions of the method. In this section the discretisation isembedment ratio. For the frictional forces at the described, after which how the resulting non-linearfoundation and the trench wall the effect of the equations are solved with an arc-length method is(trench) angle a is accounted for appropriately. briefly discussed.

In the follo;ing, the virtual work done by allthe distributed friction force contributions will be Finite element discretisationdenoted by bWfr. With principle (19) as the basis for the finite

element formulation, a linear field for u and a cubicEnd loading field for v is assumed within each element between

The horizontal pipe sections adjacent to the nodes i and i+l. If the discretisation proposed by

characteristic part modelled are assumed to be Besseling [17] is followed and the necessaryeffectively straight. The axial behaviour of these restrictions (6) accounted for, u and v take the formsections, including axial friction, is then solved inclosed form, resulting in a non-linear axial force/ u = $(LL-L)

displacement relationship. If the relative axial (36)

~isplacement of the end point at s = L is denoted by v= [(i-2~2+i3)(di-~) + (-t2+:3)(@i+@)l Lu, the result is:

where

NO-EAaT+vAoh-;~[EAFm/um] for IGI < uFx= { (31?

s-s.

NO-EAaT+vAoh-sgn(~)~[EAFm(2l~l-um)] for 1~1 ~ urnt =+

where ~ is the end force in the x–direction, N isL = Si+l - Si (37)

the res~dual lay tension and A is the steel cro~s-sectional area.

LL = ~[ (Si+l+Ui+l-Si-Ui)2 + (vi+l-vi)z ]

For symmetric problems ~ = O at s = O, but forasymmetric problems an analogue condition such as

and in this way the deformed position, i.e. equations

(31) is set. The two additional boundary conditions(2)-(6), is now fully defined in terms of the

at each model end point are:horizontal (U,) and vertical (V.) displacements andthe rotation &ngle @i of each n~de i:

F =ii=oY

(32) +a= {ui, vi,@i} (38)

stating that the shear force and the moment shouldvanish.

Implementation of (36) in (17) would lead to the fulldisplacement approach. However, here the hybrid

The solution (31) is derived for infinitely longapproach is adopted~ in which principle (19) is used

pipe sections adjacent to the part modelled. It alsoand c, K, N and M are taken as

assumes the cover height to be constant, leading to aconstant maximum frictional force F .

e . ~1

The computer program, however, !lSO features the(39)

option of analysing a part of a pipeline betweenu= (1-E)~2+~’y3

intermittently spaced rock dumps (see Fig. 5). Inthis case, the part between x = O and x = L is

and

modelled with finite elements, while the behaviour ofthe adjacent sections is solved analytically in

N= PI

~losed :orm. This results in a relationship between(40)

Fx and u that is much more elaborate than (31) andM= (1 - Z) P2+$(33

takes the effect of a rock dump with a finite lengthLd and a height Hd fully into account.

For elastic materials, choosing the strain componentsin the form (39) would effectively lead to internal

For both options the virtual work done by the endloads exactly in the form (40). However, for

loading can be denoted byelastic/plastic materials, defining (39) and (40)independently is relevant.

8W = Fx fiu (33)Equation (39) assumes the axial strain e to be

e s=L constant over the element, whereas in the full

under the condition u = O for symmetric problems, ordisplacement approach (in which (36) is used in (10)

o directly) e would appear as a quartic function in g.

6W = 5X &u + Gx w S=L (34)The dramatic difference in performance between these

e S=o. two approaches is demonstrated for the post-buckling

for asymmetric problems.behaviour of an axially compressed ‘Euler’ beam (seeFig. 6). The response of the beam as given by the

Total external virtual workexact ‘elastica’ solution [18] is followed very well

In terms of the notations defined in thisusing only two hybrid elements, whereas even eight

section, the virtual work of the external loading tofull displacement elements respond much too stiffly.

be used in (19) is:The strain functions (39) and the internal force

functions (40) are chosen independently for each

8W = 8WW+ bwp+ hwf + 8WC4 6wfr + 8W (35)element. Implementing these into prin~iple [19)

ext e enables sol~ing for ~. and ~. in terms of the nodalparameters ai at elemknt lev$l, and in this way an

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6 FINITE ELEMENT MOOEL FOR ANALVSING UPHEAVAL BUCKLING RESPONSE OF SUBMARINE PIPELINES OTC 6333

element is derived with only the displacementparameters (38) as unknowns.

Automatic load steppingThe principle (19), after the above finite

element discretisation has been implemented, leads toa set of non-line$r equations for the total number ofnodal parameters a. These equations are dependent onthe temperature differential T as well (see eq.(11)), so the problem is to solve

+*F(a,T) = O (41)

to find the temperature/displacement response of thepipeline.

+ An arc-length method is used [191, in which botha and T are treated as unknowns and an additionalequation is required to define the solution path interms of an arc-length parameter, called s. Theprocess is shown in Fig. 7: given the solution pointsup to number n, a next solution point n+l is to befound. The following equation is used:

l:l-l:nl+~in$ T-TnH(;,T) = COS$n

Y nTmax max

where the norm of ~, denoted as 1~1,maximum vertical nodal displacement.S-sriis set such that the angle $0+1between point n and the new solutlon

(S-sn) = o (42)

is chosen as theThe step lengthof the chordpoint n+l is not

‘too much different’ from @ (see Fig. 7). Ifnecessary, the step length ?s iteratively reduced sothat very sharp bends in the solution temperature/uplift curve are also followed closely.

Equations (41) and (42) are, for each step,solved following a full Newton-Raphson scheme: atrial solution is iteratively improved untilconvergence criteria are met, using

(43)

where superimposed dots denote iteration differences.

SAMPLE RESULTS

A number of calculations have been performed withUPBUCK to investigate the effect of various modelassumptions. For actual design examples the reader isreferred to a separate paper [11].

The results presented in this paper concern an 8in. pipeline of X-52 steel covered by one metre ofmedium dense sand. All the relevant data are found inTable 1. Figures 8-12 show resulte for a pipe on anassumed prop imperfection, while Fig. 13 shows thebehaviour of the same pipe on an irregular surveyprofile. Finally, the alternative of preventingupheaval buckling by intermittent rock dumps insteadof deep burial is shown in Fig. 14.

In the ‘effective weight’ approach, the maximumuplift force of the cover is added to the submergedweight of the pipe and thus an analysis is performedof a heavy beam on a foundation without separatecover loading. In Fig. 8 it is shown thatincorporating realistic non-linear cover behaviourmay give significantly different results, and this is

:onsistent with earlier findings [8]. In general, the‘effective weight’ method is not conservative. Local?hanges in the shape of the,pipe at the crown of theimperfection before the uplift mobilisation~isplacement v is reached seem to play a major role,since additional calculations in which v was5ecreased to almost zero reeulted in a t~mperature/~plift response much closer to the response found#ith the ‘effective weightl model. Only in the farpost-buckling range does the non-linear soil modelgive, as expected, lower temperatures.

For higher temperatures and pressures, yielding(locally) of the pipe under bending will beinevitable, although this does not usually threatenthe integrity of the pipeline. The effect ofplasticity on the pipe response (see Fig. 9) dependson a number of factors, but it is clear that anelastic approach generally is not conservative.

Internal pressure acts as much as an ‘upheavaldriving force’ as thermal expansion does. Inaddition, the pressure affects the plastic behaviourof the pipe. The effect of pressure on thetemperature/uplift response is given in Fig. 10 foran elastic/plastic pipe.

Stress-free pipe imperfections commonly occur,although these are very small under normalcircumstances. However, if the pipe is lowered onto arock, plastic bending deformation can result insubstantial residual strains. For the extreme case ofpipe imperfections as large as the foundationimperfection the effect is important: see Fig. 11.This figure also shows the results from the‘effective weight’ model, both with and withoutstress-free pipe imperfection. Again, it isdemonstrated that it is essential to incorporaterealistic, non-linear uplift resistance behaviour ofthe soil cover: the ‘effective weight’ approach isnot reliable.

Axial friction can most conveniently beattributed to those pipe sections adjacent to thepart modelled: this is done by means of the boundaryconditions (31). UPBUCK has the option of includingaxial friction along the finite element model, andthe effect of this is shown in Fig. 12.: neglectingfriction along the finite element model is aconservative approach.

The capability of UPBUCK to analyse the pipebehaviour on an arbitrary survey profile isdemonstrated in Fig. 13, in which the uplifted shapefor increasing temperature loading is given. For anasymmetric problem such as this, a non-zero value forthe axial friction should be taken to make theproblem well-posed!!

Finally, if a design against upheaval buckling isbased on intermittently spaced rock dumps in a trenchrather than continuous covering, UPBUCK can be usedto design the size of the rock dumps. The same pipeon the same imperfection is now situated in a opentrench with an assumed trench angle of 30 degrees.For these situations the temperature/uplift curvedoes not usually show a sharply defined maximum, so adifferent criterion has to be used. Two possiblecriteria to use are (1) axial feed-in slip should notreach the centre of the rock dump, or (2) the pipeshould not move out of the trench as it slidesupwards along the trench wall. Results are presentedin Fig. 14 for both criteria. It can be seen that fora given distance between the dumps (Lbd) of 100 m.,the length Ld of the rock dumps has a significantimpact on the critical temperature.

OTC 6333 KLEVER, VAN HELVOIRT ANO SLUYTERMAN 7

CONCLUSIONS 3. Boer, S., Hulsbergen, C.H., Richards, D.M., Klok,A. and Biaggi, J.P.: ,,Bu~klingconsiderations in

Upheaval buckling is a phenomenon that is the design of the gravel cover for a high-Joverned by a complex interaction of the - temperature oil line”, paper 0TC5294 presented atessentially non-linear - mechanics of the pipeline, the 1986 Offshore Technology Conference, Houston,~he underlying foundation and the covering material. Texas, USA, 1986.

In this paper a model is presented that takes all 4. Richards, D.F!.and Andronicou, A.: “Seabed:he relevant non-linear parameters into account, such irregularity effects on the buckling of heatedIS elastic/plastic material behaviour, axial friction submarine pipelines”, Holland Offshore, 1986.md sand/clay/rock cover uplift resistance. 5. Taylor, N. and Gan, A.B.: “Submarine pipelinefurthermore, both idealised imperfections and buckling - imperfection studies”, Thin-Walledirbitrary foundation profiles are possible. The Structures, Vol. 4, 1986, pp. 295-323.;ituation of a pipeline that is trenched and 6. Ju, G.T. and Kyriakides, S.: “Thermal Buckling ofintermittently rock-dumped can also be analysed. The Offshore Pipelines”, J. Offshore Mech. and Arcticnodel makes, where possible, use of analytical Eng., Vol. 110, 1988, pp. 355-364.:losed-form solutions, for instance, for the 7. Pedersen, P.T. and Jens&, J.J.: “Upheaval creep:onstitutive equations and the boundary conditions. of buried heated pipelines with initial[n addition, the finite beam element that is imperfections’i,Marine Structures, Design,incorporated is shown to be very efficient. Finally, Construction and Safety, Elsevier Applied Science~ dedicated arc–length method is used to find the Publishers, 1988.temperature/uplift reponse of the pipeline 8. Pedersen, P.T. and Michelsen, J.; ‘!Largeautomatically and quickly. deflection upheaval buckling of marine

It is shown that elastic/plastic material pipelines”, paper presented at BOSS 1988.~ehaviour and realistic non-linear uplift resistance 9. Nielsen, N.J.R., Pedersen, P.T., Grundy, A.K. and~ehaviour of the cover are essential for a reliable Lyngberg, B.S., “New design Criteria for upheavalnodel. The ‘effective weight’ approach, in which the creep of buried subsea pipelines”, paper 0MAE-88-naximum uplift force of the cover is added to the 861 presented at the 1988 Offshore Mechanics and;ubmerged weight of the pipe, is not conservative. Arctic Engineering Conference, Houston, Texas,Inclusion of stress-free pipe imperfections leads to USA, 1988.nore conservative results, but the effect is only 10. Nielsen, N.J.R., Lyngberg, B.S. and Pedersen,relevant for excessive imperfection levels. P.T.: ItUpheavalbuckling failures of insulated

buried pipelines - case stories”, paper 0TC6488The computer model can be used for assessing the presented at the 1990 Offshore Technology

risk of upheaval buckling of pipelines and flowlines Conference, Houston, Texas, USA, 1990.that are (to be) operated at high temperature and 11. Palmer, A.C., Ellinas, C.P., Richards, D.M. andpressure. It thus forms an essential tool for Guijt, J.: ‘“Design of submarine pipelines a9ainst~esigning safe, cost-effective and non-conservative upheaval buckling”, paper 0TC6335 presented atprotective measures, and also for investigating the the 1990 Offshore Technology Conference, Houston,relative merits of different concepts, for example, Texas, USA, 1990.continuous sand\clay covers or intermittent rock 12. Ellinas, C.P., Supple, W.J. and Vastenholt, H.:dumps. Furthermore, it forms an equally valuable “Prevention of upheaval buckling of hot submarinebasis for assessing actual pipe fitness-for-purpose/ pipelines by means of intermittent rock-dumping”,integrity after installation, using survey data and paper 0TC6332 presented at the 1990 Offshorecondition assessment techniques. Technology Conference, Houston, Texas, USA, 1990.

This new PC-based computer program, called 13. Hill, R.: “The Mathematical Theory ofUPBUCK, enables the engineer to analyse the buckling Plasticity”, Clarendon Press, Oxford, 1950.response of a pipeline accurately and fairly quickly. 14. Washizu, K.:’’VariationalMethods in Elasticity

and Plasticity”, Pergamon Press, 1975.15. Sewell, M.J.:”On the calculation of potential

Acknowledgement functions defined on curved boundaries”, Proc.Roy. SOC. A., Vol. 286, 1965, pp. 402-411.

The authors wish to thank Shell Internationale 16. Schaminee, P.E.L., Zorn, N.F. and Schotman,Research Maatschappij BV for their permission to G.J.M. :”Soil response for pipeline upheavalpublish this paper and the project sponsors Maersk buckling analysis: full~scale laboratory testsOil og Gas A/S, Nederlandse Aardolie Maatschappij BV, and modeling”, paper 0TC6486 presented at theShell UK Exploration and Production, A/S Norske 1990 Offshore Technology Conference, Houston,Shell, BP Exploration, UK Department of Energy, Texas, USA, 1990.Marathon Oil UK Ltd, Elf/Petroland BV, Occidental 17. Besseling, J.F., ,,Non-lineartheory for elaStiC

Petroleum (Caledonia), Saga Petroleum A/S, Sun Oil beams and rods and its finite elementBritain Ltd and Total Oil Marine Plc for their representation”, Comp. Meth. Applied Mech. andcooperation and permission to publish this paper. Eng., Vol. 31, 1982, pp. 205-220.

18. Frisch-Fay, R., “Flexible bars”, Butterworth,London, 1962.

REFERENCES 19. Duffett, G.A. and Reddy, B.D., “The solution ofmulti-parameter systems of equations with

1. Guijt, J.: “Upheaval buckling of offshore application to problems in nonlinear elasticity”,pipelines, overview and introduction”, paper Comp. Meth. Applied Mech. and Eng., Vol. 59,0TC6487 presented at the 1990 Offshore Technology 1986, pp. 179-213.

Conference, Houston, Texas, USA, 1990.2. Hobbs, R.E.: ,,In-servicebuckling of heated

pipelines”, J. Transp. Eng., Vol. 110, No. 2,1984, pp. 175-189.

---

TASLE 1 PIPI3,FOUNDATIONAND COVER DATA

Pipe parametersPipe steel OD (m) ------------------

Pipe outer OD (m) ------------------

Steel wall thickness (m) -----------

Submerged weight (kN/m) ------------

Poisson’s ratio (-) ----------------

Thermal expansion coefficient (-) --Modulus of elasticity (GPa) --------

(API) yield stress (MPa) -----------

Yield strain (-) -------------------

Hardening coefficient (-) ----------

Foundation parametersFoundation geometry prop shapeHeight (m) -----------------------

(Half) length (m) --------------_-

Linear foundation stiffness (MN/m2)-Axial friction coefficient (-) -----

Axial frict. mobil. displ (m) ------

Cover parametersCover height above x-axis (m)-------Cohesionless, linear cover modelLoad factor f (-) ------------------

Submerged spec. weight (kN/m3) -----Mobil. displ./peak uplift (m) ------Friction angle pipe/cover (deg) ----

Loading conditionsResidual lay tension (kN) ----------Pressure difference (MPa) ----------

Rock dump parametersTrench angle (deg) -----------------

Axial friction coefficient (-) -----Axial frict. mobil. displ (m) ------Lateral friction coefficient (-) ---Lateral frict. mobil. displ (m) ----Rock dump interval (m) -------------Rock dump length (m) ---------------

Rock dump height (m) ---------------

Submerged spec. weight (kN/m3)-----Friction angle (deg) ---------------

0.2190.231

0.01200.2650.300

0.000011207358

0.00150.100

0.30028.979

100.5000.005

1.300

0.4008

0.01020

020

301.0000.0051.0000.005

100150

1.5001030

.

J!!J’k(a) Cohesionless materials(sand,gravel,rock)

—

Buckle

~~

7///// %&a

Fig.1 Upheaval buckling phenomenon

v, y

=##P-x’”●

L

Fig.2 Undeformedpipe anddeformed centreline

A

~o

b&y &Apl

Strain Z

Fig.3 Uniaxial stressktraincurve

Hm

G

(b) Cohesive material (clay)

Fig.4 Cover uplift resistance Pc andmaximum upliftforce Pm

536

I Fig. 7 Arc-length method with iterationson a normal Plane

-.0 0:5 i

Dimensionless axial (~) and lateral (~) chplacement

Fig. 6 Buckling of an Euler beam:Hybrid element, full displacementelement and exact elastica solution.

120I Elastic

j 60- ‘s- ~hy-ti-e-yjqh~ --..-”-

[do

# Non-linear soil

20-

-,0 0.25 0.50 0.;5 I.bo 1.25 1.50

Uplift (m)

Rg. 8 Difference between non-linear soil modeland ‘effective weight’ model

537

Uplift (m)

Fig. 9 Effect of plasticity

—

120.Eiastic/Plastic

?\

l\

L\\\\\p.o--ti,_ -l-

‘\p =20 MPa

-~. —--p =30 MPa

0.25 0.50 0.75 l.ilo 1.25 1.Uplift (m)

Fig. 10 Effect of pressure

120I Elastidpfestic

LAdjaumt sed”ons w-II]

-o 0.25 0.50 0.75 I.& 1.% 1,Uplift (m)

Fig. 12 Effect of full friction along the model

lZV

100

Gso

$’

~w

!! 40P

2U

o0 0.25 0.50 0.75 1.00 1.25 1.50

Uplift (m)

Fig. 11 Effect of pipe imperfectionas large as foundation imperfection

0.6] ElastiI.Yplastic

o.5- A

0.4-

~ 0“3-g$0.2-

0.1-

O.o-

-o.1~-75 -50 -25 0 25 50 75 100 125 1

Fig. 13 Response from arbitrary foundation

10

0 o~2(

Ftockdump length (m))

Fig. 14 Effect of rock-dump length Ldon allowable tempe rature

538