influence of lay barge motion on pipelay

7/28/2019 Influence of Lay Barge Motion on Pipelay

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W. V. BREWERUn i ve rs i t y o f Tu lsa ,

T u l s a , O k l a .

0 . A. DIXONP an A m e r i c a n P e t r o l e u m C o r p o r a t i o n

Research Cen te r ,T u i s a , O k f a .

I n f l u e n c e of Lay l a r g e M o t i o n s i n aD e e p w a t e r P i p e l i n e L a i l U n l e r T e n s i o nTensioning is a prime requirement for laying deepwater pipelines. Of the several newor improved pipe laying methods which have been proposed, or even utilized, all requiretensioning of the pipeline to minimize stress in the critical area. In these methods theuse of a stinger is optional. Dixon and Rutledge produced charts by which the minimum tension and angle of inclination could be determined for laying pipelines in intermediate and deep waters. The pipeline is freely suspended from the sea floor up to thestinger, or to the lay barge if no stinger is used. It takes a shape over its unsupportedlength which differs from a natural catenary at its ends due to the pipeline stiffness. Inthe present work the authors h ave employed the same mathematical technique to studythe sensitivity of a tensioned pipeline to lay barge motions, i.e., surge, heave, and pitch.Two general trends emerge from the results which are presented both in the form ofgraphs for several sizes of the pipeline and for w ater depths up to 1000 ft, and in theform of dimensioidess charts. Surge, and to a lesser degree heave, is influential inshallow or intermediate water depths. Pitch becomes a critical parameter in deepwaters where, the tension is large. The influence of a sloping rather than horizontal seafloor is also studied in this ivork. It is shown that for a practical range of slopes theprior analysis gives a satisfactory description of the configuration. However, additional curves 'which have been provided are required to make application of the previous results.

L IntroductionO F F K H O U K production moves to ever increasingwater de pths, there arises a growing dem and for mean s of layingdeeper and larger pipelines. M an y new techni ques have beenproposed for meeting this demand; some have become practicewhile others are s t i l l in the developm ent s tages. Mo st of thesemethods have in common one impo r tan t fe a tu re : the use oftensioning in order to increase the unsupported length of pipelineand shorte n (or e lim inate ) the conve ntional s t inger. As opera tions move to deeper water i t seems reasonable to expect the unsupported segment of the pipe to become much greater .

Recen tly , the Lavan Pipelin e, "one of the world 's deepest ,longes t and la rges t underwa te r p ipe l ines" [ l ] , 1 was laid in 290 ftof wa te r us ing an "abbrev ia ted s t inge r" and "cons tan t tens ion ing," Fig . 1(a) . Th e sam e organiza tion which la id this pipelineis currently building the world 's largest pipe laying barge, whichwill be equipped with a means for applying tension and will becapable of laying 48-in , pipe [2]. An other contra ctor, know n forlaying pipe from reels , found i t advantageous to develop a

1 Numbers in brackets designate References at end of paper.Contributed by the Petroleum Division and presented at theASM E Co-sponsored O ffshore Technology Conference, H ouston ,Texas, M ay 18-21, 1909, of TH E AMERICAN SOCIETY O P MECHANICALENGINEERS. Manus cript received at ASM E H eadquarters , January22, 1970.

" to rque conver te r" by wh ich a cons tan t amoun t o f tens ion cou ldbe ma in ta ine d [3 | . I t i s imp or tan t to po in t ou t tha t "co ns tan ttensioning" helps to reduce the influence of lay barge motions,which is the subject of this paper.

Sti l l o thers have proposed total e l imin ation of s t ingers . O nesuch suggested technique was to lay the pipeline from the verticalderrick of a dril l ing vessel , s imilar to runn ing dril l p ipe [4]. Th epipeline would be run vertically downward through the center ofthe vessel while maintained under tension, making a large 90 degbend to lay on the sea door, Fig . 1(c) . A slightly different schem ewhich was pa ten ted by Pos t lewa i t and L udwig showed th e p ipe line made-up in an angular posit ion and then la id under tension,sti l l witho ut a s t inger, Fig . 1(6) [5]. Th e benefit of the s lopingderrick would be to reduce the bend at the lay barge end of thep ipe l ine .

While technology has grown rapidly, so has understanding ofthe meclianics of tensioned pipelines. At f irs t, Postlew ait andLudwig suggested that an unsupported, tensioned pipeline wouldassume the shape o f a na tu ra l ca tena ry ; thus they neg lec tedbend ing stiffness of the pipeline. Such an assum ption is valid indeep wa te rs excep t a t the ends. Subsequen t theo re t ica l worksexamined the shape of the tensioned pipeline including the influence of pipe st iffness. This includes the s tudie s by G arcia ,Wilhoit , and Merwin , using a f inite e leme nt ap proa ch [6], andDixon and Rutle dge, using a s t iffened cate nary tech niqu e [7].

At the same time .Stewart and F raze r [8] were cond ucting experimental , s tra in gage measurements of pipelines la id with and

Journal o! Engineering for Industry AUGUST 1 9 7 0 / 59 5Copyright 1970 by ASME


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F I G . IA PARTIAL STINGER

FIG. I B - I N C L I N E D D E R R I C K

FIG. ID-VERTICAL LAYINGWITH "OVER STINGER"F i g . 1 Me th o d s for l a y in g d e e p wa te r p ip e l in e s

without ten sioning in wate r dep ths up to 295 ft . I t was interesting to note their observation that " in the regions of highest betiding stresses, dynamic effects accounted for as much as 10 percen t o f the max imum s t re ss , desp i te nea r ly smoo th wa te r du r inglay ing . " Cer ta in ly , the unsuppo r ted and tens ioned p ipel ine i squite vulnerable to the influences of hydrodyiiamic forces;S tewar t and F raze r have emphas ized the apparen t seve r i ty ofsea influences on pipelines.

An understanding of this entire technology will not be comp le te un t i l an unders tand ing o f i t s in te rac t ion wi th the hydrodyiiamic medi um has been mastere d. The se influences affect th epipeline in two way s: the f irst is by direct ac tion of drag and in

ertia! forces on the pipeline, and the second is through wave orcu r ren t induced lay ba rge mot ions . l i t t l e in fo rmat ion i s ava i l able concerning the interaction of pipeline with wave forces dueto the complex natu re of the problem . Stew art and F raze r werenot able to arrive at any correlation between bending stresses andwave action, despite s imultaneous recording of lay barge motions.T heore t ica l work by G a rc ia , Wi lho i t , and Merwin [9 ] , examinedinfluences of curr ent on pipelin e, but thi s is still an initial endeavor .

In order to add further information to this diverse and complexarea of s tudy, this paper presents the results of a quasi-sta ticstud y of pipeline response to lay barge motion s. The re sultsmay be used to determine the motions for which the pipeline willbe s tressed to yielding and, subse quent ly , will be dam aged . Th estu dy is purely theor etical and uti l izes a "stiffened ca ten ary " desc r ip t ion o f the unsuppo r ted p ipe l ine . T he o r ig ina l ma them at i cal deve lopm ent was by Dixon and H ut ledge [7], who used anasymp to t ic expans ion techn ique p resen ted by P lunke t t [10 ] toinclude the influence of pipeline stiffness.

Lay barge motions to be considered quantita tively are those inthe vertical plane of the pipeline: surge, heave, and pitc h.Maximum bending stresses in the pipeline are most sensit ive tothese three motions. The remain ing three motion s, sway, yaw,and roll , are less s ignificant and will be treated accordingly.H owever, in very deep w ater, t he effect of roll becomes of thesame order of magnitude as pilch; th is point will be discussed atthe approp riate place in the analysis . Torsion of the pipeline isignored; this could be justif ied through the observations ofS tewar t and F raze r .

A brief description of the influences of a sloping sea floor hasbeen included.

Definition of Applicability of the ModelT he leng thy segmen t of unsuppor ted p ipe takes a shape ov e r

mos t of i t s l eng th wh ich approx im a tes a na tu ra l ca tena ry . Nea reither end the shape diverts from that of a natural catenary dueto bending stiffness and boundary conditions which are not com-

Nomenclature-

u

EIW

A s =

= outer radius of pipe crosssection (f t)

= wa te r dep th a t the po in t oftangency with sea f loormeasu red f rom the upperend o f the unsuppor tedsegmen t (po in t P, in F igs ,land 2 ) ( f t )

= wa te r dep th under the layba rge (benea th po in t P,or benea th s t inge r ) in theease of sloping sea floor(ft)

= e la s t ic modu lus ( lb / in . 2 )= mom ent of inertia ( in .

4)= buoya n t we igh t pe r un i t

leng th ( lb / f t )= sl ope of sea floorpositive

if pipe is laid from shallow to deep waters

surgehorizontal f inite inc remen ta l d i sp lacemen tof the upper end of unsupported pipeline (f t) ,p o s i t i v e c o n v e n t i o nshown in F ig . 2

heave ver t ica l f in i te inc re men ta l d i sp lacemen t ( f t)

4 ,7/

T.s. 7//

D

A (r s , Ao"n = change in max im um s ta t icbend ing s t re ss , a, due tosurge and heave, respec- at ively h

tb = maxim um s ta t ic bend ing ' jstrain , found at point P '/ zo( i n / i n )

bend ing s t ra in a t po in t P tp i tchfin i te inc remen ta l

rotation of upper end ofpipeline, point P fromposit ion of zero moment

rotations of upper end required to relieve bending Ae bstresses at point P du e eto surge and heave motions, respectively


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pa l ib le wi th those i the na tu ia l ( a ten a i \ T he shape of the un -siippoite d pipeline, wheie influence of s t i l lness 1- included , is

ailed a "stiffened ca teu anIn the disc ussion that follows, the unsu ppo ited pipeline segm ent

i s sumes the shape ot a s t i l l ened ca le nan wi th a s ing le uuderbend ,1'ig 2 The uppet end / ' , is maintained at the lequned angle fori / em s ta tu ( (a im sea l bend ing moment T he s ta tu dep th is

measuied t iom the sea ffooi to the uppei end ol the unsupportedpipeline Th e uppei end leceives the motions suige, heave, andpitch as small (mite displac emen ts horn the s ta tu posit ion Th epipeline motion is assum ed to be qua si- sta tn , th us lgnonn g itschnamic iespouse This s implif ied configuration con esp ond s tohe la un g me thod p ioposed I n Pos t lew ail e and Ludw ig [~i] (F ig.1(6)) I t is umle istood that when constan t tensioning, s t ingers ,n gimbaling devices aie used, the uppei end ot the unsupported

pipe does not icceive the motio ns of the la \ b aige hull When,toi exam ple, i pai t ia l s tmgei is used (Fig !()) the upper mostunsu ppoi ted point is just below th e last beaiin g point on thestmgei i t will expe nenc e sin i lle i motions th an those ot the layba ige

With the use ot a pait ia l s tmgei, the s lope ot the pipe at theuppermost support will not correspond to the s lope tor zero bending moment: . Therefo re, s ta tic bending stresses will exist in thepipeline at th is point. I f the s ta ti c moment: is small , then the results in this paper for pitch may be used to determine i ts value aswell as that du e to motio n. I f, however, the s ta ti c mo me nt at thelast bearing point is large (on the order of the maximum staticbending stress) , the system is beyond the range of validity of thepresen t results ; but i t could be exam ined by repenting th e analys isusing a modified boundary condition to include the large non-zeros ta t ic momen t .

Points of SensitivityThe effect of the motions on the pipeline can become crit ical a t

two po in ts in the unsuppor ted segmen t :1 F irst is the poi nt of ma xim um flexure of the stiffened

catenary, near the sea f loor P f, where max imum sialic b e n d i n gstress a,, occurs.2 Second is the upper support point P, at the top of the un

suppor ted segmen t .To simplify the analysis the three motions have been isolated fromeach other and their influences at P f an d P, were s tud ied sepa ra te ly .

Consider first the effect of the motions on the point P ; near thesea floor. P itch has a negligible influence. Surg e and he avehave significant influences tha t decrease as dep th increases. Th e

stresses due to heave and surge are not additive because of thephas ing di f fe rence be tween these mot ions . Ma k ing rea sonab leassumpt ions rega rd ing the phas ing and l inea r i ty o f the cont r ibu t ing te rms , the max imum s t re ss va r ia t ion can be approx i mated by the following expression:

Ac | m a : = l(Ao-s)2+ (MufV'-' ( l )

sma l l i . e . ,where it . is assum ed that the surge and heave per tur bat ion s ar eAcrs. , Acr,, ~ 1 and 1

The effect of the motions on the point P, at the top of the unsupp orted pipe is more involved. Th e largest and most obviousinfluence is pilc h. Pitch rota tes the pipe an am oun t yP a w a yfrom the posit ion for zero mo me nt. Th e bend ing stress whichresults is a linear function of y,, for small motions and increasesas the dep th increases. This is the tensioned beam (dog-leg)effect produced by the increased weight of unsupported pipe.

H eave or surge motio ns will produce similar bend ing stresses.These motions change the pipeline tension, and therefore i ts entireconfiguration is a ltered . The new configuration has associatedwith i t a new slope required for zero sta tic bending stress a t theupper en d. Since the pipeline is s t i l l a t i ts original s lope, i t hasexperienced an effective rotation away from the zero staticposit ion . Th e effective rota tion influences the pipeline in thesame way as pitc h. I f the upper end of the pipe were gimballed,then the pitch motion of the lay barge would not be applied tothe pipeline, and the end would be free to rotate when subjectedto heave or surge . Theref ore, bendin g stresses at the top wouldnot be encountered and only axial s tress would exist .

Assuming pitch and surge are in phase with each other andboth are about 90 deg out of phase with respect, to heave, thenthe max imum dynamic s t re ss a t the top may be approx ima ted by

= [(o> + asy- + (< rHy} l/* mwhere each of these individual s tresses must be small (must beless than ab). This must be added to the axial s tress , which isself-evident. H oweve r, for the range of examp les in this s tudy ,the axial s tress a t the top is s t i l l small and m ay be ignored compared with the influences of lay barge motions.

Sway, yaw, and roll motions, not in the plane of the s ta tic pipeline configuration, are beyond the scope of the direct , quantita tivestud y. I t is the presu mpt ion of the authors tha t sway, yaw , androll will be rather less important in their influence on the point P fnear the sea floor as they appear to influence to a lesser extentthose areas already subjected to high static (calm sea) s tresses.In general these motions would apply a twisting as well as a bending moment, but. in most cases twist should be small in relation tothe length over which it is d is tr ibut ed. At greate r dep ths wherethe angle sett ing for zero sta tic moment at the upper point / ' , isclose to the vert ical , it is expe cted that the bend ing influence ofroll is nearly the same as for pitch. Since mag nitud es of vesselroll motions are usually greater than pitch for common length tobeam ratios, roll would become the more significant factor in deepwate r ope ra t ions . T he quan t i ta t iv e re su l t s in th i s pape r for bending at P t due to pitch are applicable to roll in deep waters .

Influence of Sloping Sea Floor

F i g . 2 Con vent i on for posit i ve moti ons and posit i ve s lope of sea f loor

Re sult s are stat ed for a hor izo ntal sea floor. Sloping sea floorswere also inves tigate d. F or a practic al range of small , s lowlychang ing bo tto m slopes (where 5 < 4 deg), the results of Dixonand R utledg e [7] and those to be prese nted are an adequ ate description of the pipeline configurations. Slopes of more than 4deg are infrequent in most, current areas of operational interest .D e p t h "D," used to presen t the results for a horizo ntal sea f loor,is th at de pth a t the point of pipeline tangeney with th e sea f loor.Fo r p rac t ica l app l ica t ion when (5 ^ 0, this depth is not easily determ ined ; therefore, a mea ns is provide d (F ig . 3) to convert d ept hread ing D r under the lay barge (or under the upper end of the un-

Journal of Engineering fo r Industry A U G U S T 1 9 7 0 / 597


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Fig, 3 Depth adjustment for sloping sea f loor

M O M E N TUSING NATURALCATE NARYRE P RE SE NTAT I ON MAX I MUMM O M E N TPf P I P E L I NE

M A X I M U M M O M E N TFR OM N A T U R A L -CATENARYRE P RE SE NTAT I ON

M A X I M U MM O M E N TF R O M S T I F F E N E DCATE NARYREPRESENTATIONHORI ZONTAL SE A FL OOR

NE G AT I VE SLOP EFig. 4 Inf luence of sloping f loor on max imum b ending

suppor ted segmen t ) to dep th D used in this analys is . F or anegative sloping sea f loor, the abscissa in F ig . 3 is s imilar ly negat i ve .

T wo genera l s ta temen ts can be made abou t the s lop ing seafloor prob lem for all slowly chang ing slopes sm all or lar ge: if the

solpe is negative ( laying from deep to shallow water) , the maximum static bending stress will be less than for a horizontal s lopea t the same dep th D; for all positive slopes (shallow to deep), themaximum static bending stress will be close to the value for ahorizontal sea f loor when the depth I) i s t aken to be the m ax imu mdepth experienced by the unsupported length of pipeline (not thedepth under the barge or a t the point of tangency with the seafloor) . F ig . 4 shows distr ibution of bend ing mom ents for thethree cases discussed above.

ExamplesExplicit expressions for quantities of interest are difficult toob ta in . Nond imens iona l cu rves , F ig . 3 and F igs . 14 -18, were

selected as the most convenient way to represent results in acompact general form applicable to any problem within the scopeof s tud y. Thes e were prepa red with the aid of a digita l com puter . The Append ix contain s the nondim ensiona l curves together with the equations used to obtain them.

While nondimensional curves are more versati le , s ignificanttrend s are obscured by the genera li ty . To i l lustrate trends of interest , d imensional curves have been prepared for two of the pipeline s izes used by Dixon and l lutled ge [7 | . F or each of the twopipe sizes, examples are worked for both air-filled and water-filledpipe. I t is assum ed in these exam ples that t he pipe is being la idfrom an inclined derrick without a s t inger.

O u t s id e d i a m e t e rWall th icknessCorrosion coatConcre te wrapperSubmerged un i t we igh t

air-filledwater-filled

Yielding stress (a t/)Moment of inertia

S-in.Schedule 1008.625 in .0 .594 in .3 lb/f t1 in.44 .9 lb / f t63 .7 lb / f t35 ksi121.49 in .4

30-in .X-5230 in.0 .625 in .(negligible)3 in .183 lb/f t471 lb / f t52 ksi6224 .01 in

The start ing point for each example is the choice of a maximumstatic (calm sea) bending stress ab. For the 8-in . p ipe, two bending stress levels are chosen as examples and for the 30-in. pipe onecase is consid ered. Below are listed the levels of ub:

8 in. Schedule 100: ab8 in. Schedule 100:


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LUUJU-Xr -Q_UJQ(TUJ

1002 0 0 -3 0 0 -4 0 0 -

5 0 0 -6 0 0 -7 0 08 0 09 0 0

1 0 0 0 .

S U R G E -S I N G L E A M P L I Tt^ S ( F T . )

- 2- 4-

3

-1 0-12

|t

JD E

-yAi /Ayr l

zS

TOTAL B E NDI NG/ S T R E S S E Q U A L SYI E LD I NG STRE SS

0 2CHANGE

5.2 6BENDING

( K S I )8 10

S T R E S S - A C ^

010 0

2 0 03 0 04 0 0

5 0 06 0 07 0 08 0 0

nnn

8 " - c r b =3 l . 5 KS I ,A c r b = 3 . 5 K S I ^. | C3 0 " A c r b = 5 . 2 rA I R F I L L E D - :s i .

i3"-


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is influenced most by pitc h motions. F ig . 8 shows the tota l s tressversus pitch for a ir-f i l led 30-in . pipe; bending stress due to pitchhas been superimposed on axial s tress to show the relative influence of pitc h. I t is app are nt tha t the s tress due to pitch increases moderately with water depth, as opposed to the influenceof surge and heave at the bottom which decrease with increasingw a t e r d e p t h .

The bending stress is a l inear function of pitch at each waterdep th fo r sma l l ro ta t ions , say yP ^ 10 deg. Th is is easily concluded from the formu lation in the App endix. Ther efore, i t is inorder to represent the pitc h influence by a s ingle curve for eac hpipe size as is done in F ig . 9. F ig . 9 gives pitc h versus dep th

10 020 0

P 3 0 0UJUJu- 4 0 0CLU JQCCUJI - and F igs . 14 -18 .A comple te de r iva t ion of these formulas can be obtained from th eau tho rs upon reques t .

The calculations have been greatly s implif ied by using an empirical result from the Dixon and H ut ledge pape r (reference [7),p. 155): the required horizontal force II app l ied to the pipeline isvery little affected by bend ing stiffness. Therefo re, th is forceca n becom pute d with sufficient accurac y by the na tu ra l ca tena rymethod ,

II

h

cW6 ()

(4 )This result is used ex tens ive ly th roughou t to avoid the s imultaneous so lu t ion of two t r anscenden ta l equa t ions for the locationand magn i tude of maximum s ta t ic bend ing s t re ss eh. T h i s empirical re lationship isvalid for i n t e r m e d i a l e and deep wa te rs , say100-ft or grea te r for the 8-in. p ipe l ine and 200-ft org r e a t e r for the30-in . pipeline.

The following expressions from reference [7] were used toder ive the formulas from which thenond imens iona l cu rves havebeen p repa red . At the point of t angency wi th the sea floor th ep ipe l ine makes anangle 0 with respect to the ver t ica l g iven by

Q{z) = tan- 1 ah. (o )

The expression used in prepa r ing Fig. 3 isI),. = I) + StSh'^Ul')1 + '2D)l'"-\ + S' = + o> i l l i

I t isob ta ined s t r ic t ly on the basis of geometry , see Fig. 2, wheree q u a t i o n s (8) and (!)) were subs t i tu ted forI) andA.

PitchFig. 14. The expressions in reference [7] were developedfor a ze ro bend ing moment at the top. The angle of the p i p e line necessary to satisfy this condition is given by (7). Theexpression describing the s h a p e of the unsuppor ted p ipe l ine wasrede r ived , chang ing the boundary cond i t ion at theupper end toconform to a prescribed slope

10.05 . 0

1.000 . 5 0 -

B- D D0.10

0 . 0 5

0 . 01 ,

.10.c

/ = v

i4 /

.0 2 /.01

Iffb) 3 .00'

/X

r

//

00l |0 . 2 0 . 3 0.4 0 . 5 0 . 6

J_fbt _[m./nr:r\Yp b l l N / D E 6 - /where za is a p p r o x i m a t e d by F i g . 14 Nondimensional curve for strain at top due to pitch

60 2 / A U G U S T 19 7 0 Transactions of the ASME


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Fig . 15 N ond imen siona i curve for change in strain at bottom due tosurge

0- eb D

F i g . 16 N ond imens ionai curve for change in slope at top for zero ben ding due to surge

0( 1) = 9( 1 ) | ,_ + 7p ( 12 )where y,, is the angula r deviatio n from the slope requ ired forzero mom ent. I f the pipe is c lamped at the lay barge, yP equa lsthe pitch of the barg e. Th e mom ent in the pipeline at the top isit linear function of y,,. U s ing th is app roach , the bend ing s t ra inat the top e , , as a fraction of the maximum static bending strainat (he bottom e, is given by

J D + l (13F ig. 14 shows a family of nondimen sionai curv es developed fromthis equatio n, where the abscissa is the fraction I J per degreeof pitch y,,.

SurgeFigs. 15 and 16. Pos itiv e surg e A s will increase thehorizontal component of force and increase the unsupportedlength of the pipeline. Theref ore, the ma xim um static bendingstres s 6,, is decre ased an am ou nt Ae 6. Conv ersely, negativ esurge will increase th by an amoun t Ae b. T here mus t be no heaveor pilch so as to isolate the surge influences.

F ig . 15 was developed by first apply ing a small fractional changeAe i/e b to the s ta tic bending stress; the configuration of the unsupp orted pipe is a ltered. The n the surge AA. was equ ated to theresult ing incremented change in X (9) while s lope (12) and depthI) (


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Since no slope change is permitted during surge, a moment is applied at the to p of the pipe which is equi valen t to applyin g a pitch7 ; , = ys. The result ing bending strain at the top is foundente ring F ig . 14, for pitch, wi th the angle 7 , s .H eaveF igs. 17 and 18. Th e influence of hea ve is anal ogou s toth at of surge. While ma intai ning the horizon tal posit ion andslope of the pipe constant a t the top , a small vertica l displacem entupwards will increase the horizontal component of force and theunsu ppor ted length. Th e steps for calculat ing heave are s imilarto those for surge:

Ae,,(1) G iven a val ue of ', Li is found from ! 11).(2) By replacing 7.s. with y in (1(5), L2 and y are solvedsimultaneously from (16) and

(L , - l ) l / s = c(L, - l)1 '-- + S/i-|(L 2 - i) 1 '-']- c S / r 'K L , - l )''''= ]

"S r !(L . - 11u 2 "+ (7

S(L, - l) 1 -'I V

Sc (L . - 1 C I S )

Fig. 18 No ndim ens ional curve for change in slope at top for zero bending due to heave (o) F inally , , shown in F ig . 17, is found by sub sti tu tion into

' t U - l )Dc

+ (7.s)

(L , - n ' A iD

' \Dr ) "S r ' ( LL . - D " (17)T he computa t iona l pa ramete r 7 ,s. , shown in F ig . 16, representsthe change in the required slope for zero moment at the lop.

I)+ &

|L 2 ] 'A - (l + Sc=)'A

(1 + Scs)''1 - (7 r s

influence of lay barge motion on pipelay

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