Seismic Analysis of Suspended Submarine Pipelines

8/12/2019 Seismic Analysis of Suspended Submarine Pipelines

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The following assumptions are used in the tbrmulation of motionequations under seismic loading:

1) No coupl ing occurs be tween the ver t ica l and hor izonta lresponse o f the p ipe;

2) material prope rty is l inearly elastic;3) the d i rec tion o f wav e propagat ion is a long the p ipe axis, and

the two pr inc ipal d i rec tions of hor izonta l ground mot ion arecoincident respectively with the pipe axis and perpendicularto it;

4) seismic loadin g is exclusively considered, thu s the influencesof cur rent and w ave are ignored.

Equation of Mo tion of Free Spanning Subm arine Pipeline Lapidaire1985)

The m odel o f a s tra ight segment o f the f ree spanning submarinepipeline between tw o idealized end supp orts is i l lustrated in Fig. 1.

LI - - 1Idealizedend support

Seabed

Figure 1 . Ideal ized mod el of f ree spanning subma r ine p ipel ine

The horizontal tbrce perpendicular to the pipeline is described byMorison equat ion . Thus , hor izonta l force f ~ caused by w a v eand current i s g iven as :

1 ),TZ 2 ~ 72- - D pC ~ D~,OCA~4 a t 4in which: p is the density of water; is the outside diameter ofthe pipe; t t is the velocity of the water particles; W is thehor izonta l d isplacement of the p ipe; W is the hor izonta l ve loci ty ofthe pipe; C D is the drag coefficient; C M is the inertialcoefficient, C D a n d C M can be obta ined by experiments ; C Ais the added m ass coef f ic ient wh ich i s re la ted to C M t h r oughC A = C M - 1 o n l y val id for a p lane bed) 2)

The equat ion of mot ion Ibr hor izonta l v ibra tion of the f reespanning submarine pipeline becomes:

m ~ + CsW + k~w : f ~ , ~whe r e m , i s the mass of the p ipe per uni t of length; C i s structural dam ping coefficient; ks is the spring constant; ~) is hor izonta l accelera t ion of the p ipe . Af ter l ineariz ing f m a , follow ing definit ions are introduced:

m a = 4 D Z p C Ap C D l u [

u l : = R M S t ) )

4. = P C D D U I u I + D 2 / 3 C M Ot Then, Eq. 3 can be t ransforme d in to the fo l lowing form

(m~ + r na + ( c, . + Ch~ W + k ,w = f mi n wh i c h m a i s the added mass ; Chi is the horizontahydrody namic damping; f m is the hor izonta l ly hydrodyn amforce,. O-u i s t he r oo t me a n s qua re o f wa ter particle velocity U t )

Bo th C M and C D are re la ted to the rat io o f the s tee l oudiam eter D to the distance e from the seabed see Fig. 1). Fo r p ipes in the proximi ty o f the seabed, i t i s show n tha t the Cdecreases exponent ia l ly f rom a maxim um o f C M =3.29 for e /D to an asymptot ic minimum of C M =2.0 for e/D tending to infinTokuo 19 74) . Similarly, the drag coefficien t C D i s repor ted

vary with the e /D depextding on the environmental conditions, surfroughness o f the p ipe , ampl itude of p ipe line m ot ion and so on. large diameter pipes, damping is small due to the drag effeThere fore the drag effect is ignorable. M eanwhile, ign oring influence of hydrody nam ic force , namely, U = 0 , and denothor izonta l se ismic accelera tion as / /hg , then the equat ion of motfor f ree spanning of subm ar ine p ipeline only under hor izontse ismic load i s

m s + m.)fO + c,w + k, w = - m s + o ) i~hgA generally adm itted expression for the vertical forc e on a pipel

due to wave and current i s present ly lacking. In the moapproximate descriptions a re used for the vertical force due to wand current at a state-of-the-art level. The extra force co mp onen t to vibrating vertically can be expressed by the Morison equatiThus, the total vertical force fVM perpen dicular to the pipeline

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be descr ibed as :

i n w h i c h Z a n d I f a r e v e l o c it y a n d a c c e l e r at i o n e r p e nd i c u l a r t ot h e p i p e r e s p e ct i v e ly ; C L i s t h e l i f t c o e f f i c ie n t ; U i s t h er e s u l t a n t e l o c i t y f w a t e r p ar t i c l es .

T h e f ~ s t e r m o n t h e fi g h t s i d e o f E q . I 0 c a n b e l i n ea r i ze d i t h t h et h e o r y o f B o r g m a n . T h e n t h e e q u a t io n o f m o t i o n f o r v e rt i ca lvibra t ion i sm , + m , , ) i f + c , + c,,) ~ , + k , z = f r ~ 11 )

i n wh i c h Cu i s the vert ica lly hydrod ynam ic damping; re / i s thevertically hy drod yna mic force. Ignoring the influence ofhydro dyna mic force, and denoting vertical seismic acceleration asV v g , t h e equation o f mo tion for free spanning submarine pipelineonly under vertically seismic load is( m .~ + m , ~ ) if + c ,? , + k , z = - ( m ~ + m , ~ ) f ,, , (12)Discre tizat ion of the equat ion o f mo t ion

Th e pipeline is idealized to discrete one-dimensional bea melements in numerical analysis. T he Herm ite cubic polyn om ial isemployed as the in terpola tion f imction. The equat ion of mot ion ca nbe discretized

L o t - L jin which [ /~ ] i s the to ta l mass matr ix including added mass ;is the structural damping matrix; [ K ] is the stiffness matrix;is the input acceleration of earthquake. Ra yleig h's theor y (Kalliontzis1998) is adopted herein in ord er to estimate and to calculate the effecton damping. A pplication of the finite element principles o n there levant te rm in Eq. t 3 leads to

1 4 )The dam ping parameters can be evaluated according to

~colco 3 ~O t a - - - f l a - - - (15)CO + 093 (O1 + O93in which (O1 and 093 are the first- and third-mode frequencies,respectively; and ~ is an assu me d dam ping ratio.

The Wilson- 0 me thod is uti lized to solve equilibrium equation ofmotion, and 0 is equal to 1.4.

N U M E R I C A L E X A M P L E A N D R E S U L T S

To study the results obtained from the m odel, a practical exam pis selected. T he cro ss section of the free spanning subm arine pipelishows in Fig . 2 . Th e outer radius o f the s tee l p ipe R o is 30.50cmthe steel wall thickness t s 1.27cm, the concrete thickness t6.00cm, the e las tic mo dulus o f the s tee l E 2 .06 105MPa, Poissratio of steel /2 0.3, the density of steel 7.8 x 103kg/m3, the densiof concre te 2 .4 103kg/m3. The seismic input is El-centro earthquakwave , and the maxim um ho r izonta l accelera tion i s 0 .2g .

ConorFigure 2 . C ross sec t ion of s tee l p ipe

Effec t of free spanning lengthTh e seismic pipeline respo nses for different f lee spanning len g

(60m , 90m and 120m, respectively) in simply supported econd ition are calculated. T he respon se histories of displacement astress fo r different free spanning length in ce nter are displayed in F3 and Fig. 4 , respectively. I t is eviden t that the respon se increaswi th the increas ing of f ree spam tng length. The natura l f requencysubmarine pipeline in the span length of 90m is close to earthquawa ve frequency, thereby, the re spon se is amp lified distinctly.

0 . 40 F - I - ~- . . . . o . . . . - ~ - . . .. - ]o.30 - F ~: ...... ' i ..... i iF T a . . . .... h i . I

g 0 1 0

_020 L 2s, ,L :_0,0 - I : : : : l i i i l : l 1 7 1 I-0 .40 . . . . . . . . .

0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 0 4 5T u n e s )

Figure 3 . Displacem ent h is tory in centerfor d i f ferent f ree spanning length

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/4O i l . . .s o . . . . . . . . 1 . . . . .. . . . . I - - . . . . . . . . . . .. . . . . . T . . . . . . . . . . .. . . .. . r . . . . . . . . . . . . . . . . .2 o . . .. . . ~ i . . . . .. . . k + ~ - . . - - . . . - . . ~ , ' . . ~ - - : ' - / v - . ' - . . .. . .. . . . - , - . . .. . i ~ ' . i ' ~ I :1 0 - ~ ~ , . . . - ~ . , ~ . - ~ ' . . . . ~ ( . ~ . . . . . . . . . .. . . . .

~ o t , . .~: : . . : ~ . ~- ~ 0 . . . . . f - - ( ; - "- 2 0so ............................... . . . . .. + . - . ~ L . i i - . . . . 4 . . . . . . .

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T i m e ( s )

F i g u r e 4 . S t r e s s h is t o r y i n c e n t e r f o r d i f f e re n t f r e e s p a n n i n g l e n g t hE f f e c t o f a d d e d m a s s

T h e c o n c e p t o f a d d e d m a s s c a n r e a l l y s i m u l a t e t h e e f f e c t o f l i q u i da r o m ~ d t h e s u b m a r i n e p i p e l i n e . A d d e d m a s s a f f e c t s t h e r e s p o n s e i nt w o w a y s . F i r s t l y , th e s u b m e r g e d n a t u r a l f r e q u e n c y o f t h e p i p eu n d e r g o e s a c h a n g e , a n d s e c o n d l y , t h e s e i s m i c e x c i t a t io n f o r c e i sa l t e r e d d u e t o t h e c h a n g e i n t h e a d d e d m a s s . F i g . 5 a n d F i g . 6 s h o wt h a t t h e d i s p l a c e m e n t a n d s t r e s s r e s p o n s e s c o n s i d e r i n g a d d e d m a s sw i t h t h e f r e e s p a n n i n g l e n g t h o f 6 0 m i n c e n t e r i s m u c h l a r g e r t h a nt h a t o f i g n o r i n g o n e .

0 . 0 2

, ~ O . O l0 . 0 10.00

- 0 . 0 1- 0 . 0 1- 0 . 0 1- 0 . 0 2

0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 0 4 5T i m e ( s )

F i g u r e 5 . D i s p l a c e m e n t h i s t o r y i n t h e e f f e c t o f a d d e d m a s s15tO

0

- 1 0- 1 5

: i : : i ~ i : , : , ii ~ i ~ ~ ~ . . . . . , ~ d m , ~

" = " 1 . ~ . . . . . ' . . . . . . ' . . . . . . .. . ' . . .. . .. . .. . . ." ' l . . . . . . . . I . . ........... +..+:....., ..........~........... ...~......~ .........~......... ........

a~:..~[ .... ...1........ -~-.4 .. . :. ----- -L. -~ ....... t ......i i i i. . . i . . . . i . . . . ~ . . . . i . . . . i . . . . i . . . . . . . .

0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 0 4 5T i m e ( s )

F i g u r e 6 . S t r e s s h i s t o r y i n t h e e f f e c t o f a d d e d m a s sE f f e c t o f s e a b e d p r o x i m i t y

I n e r t ia l c o e f f i c i e n t C ~ v a r i e s w i t h e / D r a t i o r a n g i n g f r o m 2 , 0 t o3 . 2 9 , w h i c h r e s u l t s i n t h e c h a n g e o f a d d e d m a s s . F i g . 7 a n d F i g . 8s h o w t h e v a r i a t io n o f re s p o n s e w i t h e /D . T h e m a x i m u m r e s p o n s ea p p e a r s a t th e m o m e n t t h e b u r i e d f r e e s p a n n i n g p i p e l i n e j u s t h a p p e n s

i n t h e m a x i m u m v a l u e o f C M .) i i i A - *. 0 ~ . . . . . . . . . . . . . . . . .. . . .. . . .. . . . .. . . . .. . . . .. . . . . . . .. . . . .. ' ] : ' : ' : ' : : ' : : T . . . . . . . . i . . . .. . . . :"a' o.os ............. .............., . . . . . . . f . . , , ; , . ~ - - . : . : _ . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . i . . . . . . . . . . . . . .: A . o ~ : ~ i: . l . . . . . . . i .~ i io . 0 2 F " ' ' : " ' ) . .. .. .. .. .. .. .. ; . .. .. .. .. .. .. . ? I . .. .. .. .. .. .. a . . . . .~ n x e d ~ p * ~ I

: i : I / - @ - ~ l e ~ l , o ~ J_~ o . o l i I i" :.......... . .............. .........."t ............." - I I "'~0 . 0 1

o . o , Z . . . .. . . .. . . .. . . : ; - t . . . .. . . .. . . .. .~ . ~ . . . . . .. . . . ~ . .. .. .

2 2 . 2 2 . 4 2 . 6 2 . 8 3 3 . 2 3 . 4InertialCoefficientCMF i g u r e 7 . V a r i a ti o n o f d i s p l a c e m e n t w i t h e / D r a t i o a n d s u p p o r t

2 4 . 0 . . . . . . . r . . . . . . ~ . . . . t ' | " "

o . o . . . . .. . . .. . . . . . . . . . .. . . . .. . . . . . . . .. . . . . . . . . . . . . . . . . . . . . i . . . . . . . . . . : . : - - . : . i : . . . . . . . . .. . . .~ s . o . . . . . . . . . . . .. . . . . . . . . . . .. . . . . i . . . . . . . . . . . . .. i . . . . . . . . . . ~ . . [ . . .. :. :. : . . . . . . . . . . . . . . . . . i .. . . .. . . .. . . .

. o . . .. . . .. . .. . .. ) - . : . , , ; - L i : : . . . . . . . . . . . . . . . . . . . . i . . . . . . . . . . . . . i . . . . . . . . . . . ., , . o . . . . . . b : : - i . . . . . .. . . . . .. . . . .. . . .. . . .. . i r .10. 0 . . . . . .. . . . . .. . ~: . . . . . .. . . . . .. . * . . . . . .. . . . . .. . . . . . . . . . . . . . . 1 . . . . . . . . . . . . I. . . . . . . . . . . . . . . . . . . . . . . . . . . .

8 0 . . . . . .. . . . . .. . -7 . . . . . .. . . . . .. . :" . . . . . .. . . . . . 'T . . . . . . . . . . . . . ~ . . . . . . . . . . . . ~ . . . . . . . . . . . . . ~..............6 . 0 . . . . . . . . . . . . . . - . . . . . . . . . . . . . . . i . . . . . . . . . . . . . . . - . . . . . . . . . . . . . . i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ~ . . . . . . . . . . . .' . . . . . i : . . .2 . 0 _

2 0 2 . 2 2 . 4 2 6 2 8 3 0 3 . 2 3 . 4L ~ti al Co e~ n t CM

F i g u r e 8 . V a r i a ti o n o f s t re s s w i t h e / D r a t i o a n d s u p p o r t

E f f e c t o f e n d s u p p o r t sS u p p o r t c o n d i t i o n o n e a c h e n d f o r f r e e s p a n n i n g s u b m a r i

p i p e l i n e h a s s i g n i f i c a n t i n f lu e n c e o n r e s p o n s e . F r o m F i g . 7 a n d F i gt h e r e s p o n s e f o r f i x e d s u p p o r t e d c o n d i t i o n i s m u c h l e s s t h a n t h a t s i m p l e s u p p o r t e d c o n d i t i o n . I t i s a f e a s i b l e w a y t o s t r e n g t h s u p p o r t e a c h p i p e l i n e e n d i n o r d e r to d e c r e a s e r e s p o n s e .

C O N C L U S I O N S

F r o m t h e p r e c e d i n g d i s c u s s i o n s , t h e f o l l o w i n g c o n c l u s i o n s c a n d r a w n :

( 1 ) T h e f r e e s p a r m i n g l e n g t h o f s u b m a r i n e p i p e l i n e i s t h e kf a c t o r a f f e c t i n g t h e s e i s m i c r e s p o n s e , e s p e c i a l l y w hf r e q u e n c y o f p i p e l in e a p p r o a c h e s t o t h e o n e o f e a r t h q u aw a v e , r e s o n a n c e t a k e s p la c e .

( 2 ) P r a c t i c a l s u p p o r t e d c o n d i t i o n f o r f r e e s p a n n i n g s u b m a rp i p e l i n e s i s b e t w e e n f i x e d s u p p o r t a n d s i m p l e s u p p oR a t i o n a l j u d g e m e n t a n d d e a l i n g w i t h s u p p o r t c o n d i t io n s h ob e d o n e i n d e s i g n a n d c o m p u t a t i o n .

( 3 ) I t i s d i f f ic u l t t o c o n f u ' m t h e p r o x i m i t y o f p i p e l i n e t o t h e s e a bh o w e v e r , t h e i n f l u e nc e o f a d d e d m a s s m u s t b e t a k e n ia c c o u n t .

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V I B R A TI O N C O N T R O L M E T H O D S

Based on the above analys is , the response of f ree spanningsubmarine pipelines under strong earthquake is large and maybeco me the contro l factor of pipeline strength design. Thereby, i t isan importa nt and beneficial stud y to increase pipeline safety with thehelp o f the theory and techn ology of v ibra t ion contro l. The o pt imalspan len gth is a first priority in design.

Al though in most cases of fshore p ipel ines are bur ied in thetrenches w hich are back.fil led after lay ing the pipelines, there sti l lexists the possibil i ty that buried pipelines happens to suspend in theaction of incessant w ater scou ring and erosion, soil liquefaction, orbuoyancy. Th e free spanning is inevitable for pipelines, th us i t isfeas ible to ac t ive ly cont ro l f ree spanning of p ipe l ine in order toensure the safety.

There are many methods to mainta in submar ine p ipel ines (Liu1977, M o 199 8) . How ever , expensive cos t and grea t working

quant i t ies a re the problems for these methods . Two methods arerecomm ended here in to so lve the ab ove problems.

Firstly, the me thod o f segm ented burying, laye red fil l ing an dcov ering (Sekiguchi 1996 ) is suitable fo r the marin e pipelines that areburied in tren ch or directly laid o n seabed, a nd artif icially backfilled.Th e critical free spanning length o f pipe hne is com puted , consideringstrength an d fatigue failure due to earthquake, wa ve and current. Thelength of should er span is also calculated accor ding to soil propertyand current characterist ics. San d is pav ed along assu me d crit ical span,and gravel is f il led on sho ulder span. A certain depth of gravel iscovered on them at las t . Gravel on the sur face layer has goodcapabili ty to resist scourin g dependen t of environmental conditions.And gravel on the shoulder span a long the bur ied depth can cont ro lf ree spanning i f the sur face layer i s w ashed out .

Secondly, the pile-support method is originally uti l ized to solvefree spanning problem for pipelines crossing moderate and smallr ivers tha t water scour ing i s ser ious . Prac t ices have provedthemethod i s feas ib le . The m ethod can a lso be emp loyed to contro l f reespanning submarine pipeline in shallow sea. Do nble piles conn ectedby a beam are adopted to re inforce p ipel ine according to cr i tical f reespanning length. The m ethod can bo th suppor t p ipe l ine and decreasepipeline vibration. Alth oug h construction techniqu e is complicated,the wo rking quantities are small .

Offshore Pipel ines to Random Grou nd Mo t ion, Ear thquaEng ineering and Structural Dy nam ics, Vol. 19(2), pp. 217-228.Hou, ZL (1990) . Aseism ic Research on Underg round PipelinesAcademic B ooks a nd Periodicals Press.Kalliontzis, C (1998). Nu me rical Simulation o f Sub marin e Pipelini n Dyna mi c Con t a c t w i t h a Mov i ng Se a be d , EarthquaEngineering and Structural Dynamics, Vol. 27, pp. 456-486.Lapidai re , PJM (1985) . Sta t ics and Dynam ics of Pipel ine SpanBehavior of Offshore Structure, Elsevier Science Publishers B.VAmsterdam_Liu, XZ (1977). Risk Analys is and Countermeasure in HuiniDo uble O il Pipelines Traversing Yellow R iver , PetroleEngineering Constructz on,Vol. 1, pp. 10-12.Mo , Q (1998). Mainta in ing Me thod Imp rovem ent for the Segmeof Pet ro leum Pipel ines Cross ing R iver , Petroleum EngineeriConstruction, Vol. 5, pp. 28-30.Sekiguchi, K. , Matsuda, S and Adac hi, H (1996). ' 'Num erica l Stuon the Effectiveness of Stabilizing Techniques of Offsho re Pipelinagains t L iquefac t ion , Proceedings of the Eleventh World Confereon Earthquake Engineering.Tokuo, Y. , Nat_h, JI-I and Smith, C E (1974), W ave Force s Cyl inders near the Oc ean Bot tom, J. Waterways Harbour D iv. A SVol. 100, pp. 34-50.Wang, JY and Zhao, DY (1992). Present Si tua t ion and Problem sSubmarine Pipelines in Baohai Sea, Chinese Offshore Oil-GEngineering, Vol. 4(1 ), pp. 1-6.

R E F E R E N C E SDat ta , TK and Ma shaly EA (1990) . Transverse Response of

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Seismic Analysis of Suspended Submarine Pipelines