[Owen] Mechanical Behaviour of the.

7/27/2019 [Owen] Mechanical Behaviour of the.

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I~UTTERWQRTH"~ IE I N E M A N N

0141-0296(95)00023-2Engineering Structures, Vol. 17, No. 4, pp. 240-253, 1995

Copyright 1995 Elsevier Science LtdPrinted in Great Britain. All rights reserved

01414)296/95 $10.00 + 0.00

M e c h a n i c a l b e h a v i o u r o f t h em e t a l l i c e l e m e n t s o f s u b m a r i n ec a b l e s a s a f u n c t i o n o f c a b l el o a d i n gG . F e l d a n d D . G . O w e nDepartment o f Civil and Offshore Engineering, Heriot-Watt University, Edinburgh, UKR . L . R e u b e nDepartment o f Mechanical Engineering, Heriot-Watt University, Edinburgh, U KA . E . C r o c k e t tPirelli G eneral plc, E astleigh, Ha nts, UK

Subm ar ine pow er cab les a re w id e ly used fo r l oca l and n a t iona l d i s -t r i bu t ion and i t i s o f i n te res t to know the i r mechan ica l l im i ta t ionspar t i cu la r ly w i th regard to the loads exper ienced dur ing the ins ta l-la t ion process where s tat ic or low-cyc le fa t igue fa i lures are poss-ib le . A deta i led und ers tand ing of the g lobal and in ternal me chanicsof such cables is therefore of in teres t in order to achieve therequ i red per fo rmance a t reasonab le we igh t . Th is paper addressesthe re la t ionsh ip be tween app l ied loads and the responses o f themeta l l i c e lements w i th in t yp ica l subsea con f igu ra t ions . The ex is t -ing theory on mechanics of cables of th is type is ex tended to takein to accoun t s t ra in d is t r i bu t ions w i th in he l i ca l e lements and tomode l the in te rac t ions be tween load ing m odes as we l l as inc lud ingnonl inear e f fec ts assoc iated w i th p las t ic i ty in the meta l l ic e lementsand compl iance o f the po lym er ic ma te r ia l w i th in the comp os i tecons t ruc t ior t . The resul ts o f a nu m be r of mater ia l tes ts are incorpor -ated in to the new analys is and the predic t ions of the model are a lsocomp ared w i th mea sureme nts on the conduc to rs o f cab les undercond i t i ons o f ax ia l and bend ing load ing .Keywords: submarine cables, cable loading, mechanical l imitat ions

S u b s e a p o w e r c a b l e s p r o b a b l y e x p e r i e n c e t h e ir m o s t s e v e r eloa d ing du r ing l a y ing w he re s t a t i c o r a l t e rna t ing s t r e s s e sc a n be s u f f i c i e n t t o c a us e imme d ia t e o r l ow c yc le f a t iguefa i lu re in a l l o r , more l i ke ly , pa r t o f t he c ompos i t e s e c t ion .To m a x im iz e the de s ign e f f i c i e nc y w i th r e ga rd to s uc h loa d -ing i t i s ne c e s s a ry to ha ve a de t a i l e d know le dge o f t hes t re s s e s o r s t r a in s in the c ompo ne n t s o f t he c a b le a s a func -t ion o f t he g loba l ly a pp l i e d loa d .A typ ic a l c a b le c ro s s - s e c t ion migh t be a s s how n in F i g -u r e 1 w h e r e t h e m a i n m e t a l l i c e l e m e n t s a r e t h e a r m o u rw i re s a nd the c onduc to r s w h ic h a re u s ua l ly w ound in c on -c e n t r i c he l i c e s a bou t t he a x i s o f the c a b le . The p re s e nc eo f a n u m b e r o f p o l y m e r i c m a t e r i a l s a n d a l s o t h e p o s s i b l em a g n i t u d e o f s o m e o f t h e s t r e ss e s m e a n s t h a t d e f o r m a t i o n

i s no t ne c e s s a r i ly a lw a ys l i ne a r e l a s t i c a nd the c ons ide ra b led i f f e r e n c e s i n e l a s t i c m o d u l i b e t w e e n c o m p o n e n t s m e a n stha t s l i p be tw e e n d i f fe re n t ma te r i a l s i s a l s o pos s ib l e .The w ork re po r t e d he re c onc e rns the s t r a in s in the he l i -c a l ly w ound c onduc to r s a nd a rmour w i re s in s ubs e a c a b le sa n d t h e a p p r o a c h d e v e l o p s f r o m t h e w o r k o f Knapp t-3 a ndL u t c h a n s k y4 . A . b r i e f s um ma ry o f t he l i t e ra tu re on thed e f o r m a t i o n o f h e l i ca l e l e m e n t s o f f l ex i b le c o m p o s i t e c y l i n -de r s unde r t e ns ion a nd be nd ing i s g ive n be low .T e n s i o n a n d t o r s i o nT h e a p p r o a c h o f K n a p p 1 h a s b e e n t o c o n s i d e r t h e w a y i nw h i c h a h e l i x w i l l d e f o r m w h e n t h e i m a g i n a r y r e f e r e n c ec y l inde r a round w h ic h i t i s w ra ppe d s u f fe r s a g loba l t e ns ion

2 4 0


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Me chanics o f subma~dne cab les: G. Fe ld e t a l . 2 4 1

~~ Steel rmourwire

F i g u r e 1 T y p i c a l c a b l e c r o s s - s e c t io n

T a b l e 1 N o n l i n e a r h e l i x c l e fo r m a t i o n r e l a ti o n s a c c o r d i n g t o K n a p p ~

t he he l ix and the d iamete r o f the wi res themse lves a resmal l. I t i s a s sum ed tha t any a symm et r i e s in cab le s t ruc turea re no t l a rge enoug h to d i s rup t the he l i ca l fo rm of the w i rese i the r be fore or a f t e r de format ion .For the more compl i an t ma t r i ces involved in cab les , a sopposed to ropes, chang es in p i t ch c i rc l e rad ius can b ecomemo re s ignif icant and, for an incom press ible f i ller, K napphas g iven the l inea r he l ix re l a t ions shown in T able 2as suming tha t t he de format ions a re sma l l and l inea r e la s ti c .Fur the r , t he ove ra l l cab le load ings can be expres sed a s al inea r combina t ion of the de form at ions of the va r ious he l ixre fe rence cy l inde rs v i a a ma t r ix equa t ion

[ N ] = I k H k ~ 2 / [ A n ] ( 1 )tk21 k22J A~b

where k l l a re the submat r i ces o f the s t if fnes s ma t r ix formed

H e l i x s t r a i nC h a n g e i n h e l i x l a y a n g l e r e l a t i o n s , i = [ (1 + c ) 2 cos 2 a i + (1 - e r ; ) 2 (1 + 7~ i ) 2 s in 2 (~ i] v2 - 1(1 + c )c o s a ; = c o s t a ( 1 + c ~ )( 1 + % i ) ( 1 - e , i )s i n a ' i = s i n a i ( 1 + c ~ i)(1 + %; ) (1 - ~ i )t a n a ' ~ = t a n a ~ ( 1 + c ~ )

or to rs iona l de format ion . T able 1 summ ar izes the re l a t ion-sh ips de r ived by K napp for the ax ia l s t ra in of the he l ix andfor the chan ge in he l ix h ty angle for an ov e ra l l cab le t ens i les t ra in, ~, and an over~dl cable ro ta t ion per pi tch length27r%~. Th e ini t ia l and de:formed hel ix param eters are show nschemat i ca l ly in F i g u r e 2 and th i s ana lys i s has been usedfor w i re ropes where changes in the p i t ch c i rc l e rad ius of

l~ ( = % )= i

z . ( t + 7 . ] , R I = v2 w ( l 7 d ) R ; ( I - ~ )

F i g u r e 2 H e l i x p a r a m e t e r s

p i =p , ( = % )

by succes s ive subs t i tu t ions of the s t ra in and de format ionequat ions into the bas ic cable equi l ibria .B e n d i n g a r o u n d a s h e a v eIn th i s work , deve lopm ent s for conduc tor s t ra ins under g lo-b a l b e n d in g h a v e b e e n b a s e d o n t h e m o d e l b y L u t c h a n s k ywhich desc r ibes the e f fec t s o f to ro ida l bending on he l i ca l lywound th in wi res . An impor tan t a spec t o f Lutchansky ' smode l conce rns f r i c t ion be tween l aye rs in the cab le con-s t ruct ion and i ts effect on the dis t r ibut ion a long the hel ixl ength of any changes in pa th l ength occas ioned by thebending . The two l imi t ing cases of ze ro and in f in i t e f r i c t ioncor respond to an evening ou t o f any ne t changes in pa thlength ove r the he l ix l eng th and the ' f reez ing ' o f loca ls t ra ins , respect ively, the la t ter leading to a s inusoidal vari -a t ion of axia l s t ra in a long each lay length.For cases of in t e rmedia t e f r i c t ion (no t cove red byK n a p p ' s m o d e l , a l th o u g h b a s ed o n t h e s a m e f u n d a m e n t a lT a b l e 2 L i n e a r h e l i x r e l a t i o n s f o r c a l c u l a t io n o f s ti f fn e s s m a t r i xH e l i x N i = n.,AiE,~ic o n s t i t u t i v ee q u a t i o n s Mt i = y~iE~I~

M b i = - - ~ ' b lYlH e l i x s t r a i n C a~ = e c COS 2 (X i+ 7 c # i n 2 a i - e , ~ s i n 2 air e l a t i o n s Y ; 2 [ s in z


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242

T - - - - - - JK (u - Up )ds

k-- - - T +dT

Figure3 Lutchansky hel ical e lement equi l ibr iumd s ,

Insulation /t

C o n d u c t o r / du r~,Lt~ Insulation

d, : : d sC a b l e c e n t r e - l i n e

Figure 4 Lutchansky element before and after deformation

as sum pt ions ) , quan t i f i ed by the in te r ac t ion shea r cons tan t ,K, i l lus tra ted in Figure 3, L u t c h a n s k y d e r i v e d t h e a x i a lhe l ix s t r a in f r om the cor r e sponding s t r e s s

o'o = A ( K ) [ a ( K ) [_ _ _ ~ [ I e ~ ' ~ " - ' / J cos0o + s in0]( 2 )

w h e r e A ( K ) a n d a(K) are f unc t ions o f the shea r s t i f f nes sa n d t h e c a b l e d i m e n s i o n s a n d g e o m e t r y a t m a n u f a c t u r e a n d0 and 0o ( f ina l and in i t i a l pos i t ions ) a r e m easur ed a r oundthe cab le c i r cum f e r ence f r om the cab le neu t r a l ax i s( d i f f e re n t f r o m t h e o r i g in a l f o r m u l a t i o n o f L n t c h a n s k y ) .S ince then , a nu m b er o f au thor s 5,6 have a l lowed f or s l ipp-age when in te r f ac ia l shea r r eaches a c r i t i ca l l eve l wi th theinev i tab le r e su l t o f hys te r es i s in the load ing /un load ing pr o-ces s .

Figure 5 Testinteraction

=

i i

sample for investigating conductor / insu la t ion

Mec han ics o f subm ar ine cab les : G . Fe ld e t a l .M o d e l l i n g o f d e fo r m a t i o n o f c a b le e l e m e n t sT h e d e v e l o p m e n t s t o t h e t w o a b o v e a p p r o a c h e s a d d r e s s e din th i s wor k a r e Em p i r ica l de te r m ina t ion of the m agni tude o f the in te r -

c o m p o n e n t f r i c t io n M e a s u r e m e n t o f t h e e f f e ct i v e c o m p r e s s i b i l it y o f th e p o l y -m er ic l aye r s C o m b i n a t i o n o f t h e c o m p o n e n t s o f s t ra i n d u e t o b e n d i n g ,tens ion and to r s iona l loads Deve lopm ent to a l low nonuni f or m s t r a in d i s t r ibu t ionsac r os s the m e ta l l i c c r os s - sec t ionsT h e s e d e v e l o p m e n t s a r e d i s c u s s e d b e l o w f o r e a c h o f t h ec a s e s o f a x i a l, b e n d i n g a n d c o m b i n e d l o a d i n g .

A x i a l l o a d i n gB a s e d o n t h e a x i a l l o a d i n g m o d e l d e s c r i b e d a b o v e , b u te m p l o y i n g a m o d i f i e d e q u i l i b r i u m e q u a t i o n , a n o n l i n e a rm o d e l h a s b e e n d e v e l o p e d e n c o m p a s s i n g n o n l i n e a r l o a ddef or m a t ion cha r ac te r i s t i c s o f the m a te r ia l s , l a r ge cab ler o ta t ions , l a rge r educ t ion in p i t ch c i r c le r ad ius and chang esin l ay ang le .T h e f o l l o w i n g a s s u m p t i o n s h a v e b e e n m a d e( 1) He l ices have c i r cu la r c r os s - sec t ion( 2) Wir e d iam ete r i s m uch le s s than he l ix p i t ch( 3 ) W i r e s a r e h o m o g e n e o u s a n d i s o t ro p i c( 4) He l ica l wi r es a r e equa l ly spaced a r ound the i r p i t ch c i r-c le( 5) Al l l aye r s a r e concen t r ic( 6 ) T h e t e n s i l e m o d u l u s o f a n y p o l y m e r i c m a t e r ia l i s n e g li -g ib le com par ed wi th the m e ta l l i c m a te r ia l sF or a cab le un der go ing tens i l e o r to r s iona l load ing the ax ia lh e l i x e q u i l i b r i u m c a n b e s u m m a r i z e d b y a g e n e r a l i z a t i o no f t h e t o p e q u a t i o n i n Table 2 to

N = ~ n i A i E i c a i 7i= l CO S O / i

(3 )or , wi th a cen t r a l cor e

N = AdEcec + ~ n i A i E i c a ii = 1 C O S t a l

( 4 )

The to r s iona l equ i l ib r ium f or a cab le wi th a cor e can bes u m m a r i z e d a s

T GcJ~ ~n i GiJi= - - Tc + T,cosa~rc i=1 L ~ - /E i I i }+ - - ebi s ina i + AiEiRjeais ina iYi (5 )I nse r t ion o f the r e la t ionsh ips g iven in Table 2 i n t o e q u a -t ions ( 3 ) , ( 4 ) and ( 5 ) g ives the s t i f f nes s m a t r ix desc r ibedin equa t ion ( 1 ) .Mate r ia l non l inea r i ty in the m e ta l l i c m a te r ia l s can bei n t r o d u c e d b y m e a n s o f a s e c a n t m o d u l u s.


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Mech anics of submatqne cables: G. Feld e t a l . 2 4 3Bending load ingT h e i m p o s i t i o n o f a b e n d r a d i u s r e s u l t s i n t h e f o l l o w i n ge l e m e n t s o f d e f o r m a t i o n o f t h e h e l i c a l c o m p o n e n t s An ax ia l s t r a in due to the loca l change in pa th l eng th A bend ing s t r a in due to the loca l chang e in rad ius o f cur -va tur e A to r s iona l s tr a in due to the loca l change in r ad ius o f to r -s ionU s i n g t h e a s s u m p t i o n s( 1 ) W i r e s a r e r o u n d( 2 ) T h e g e o m e t r y o f d e f o r m a t i o n o f a w i r e c a n b edes cr ib ed b y i ts cenlLroidal axis( 3 ) W i r e d e f o r m a t i o n i s li n e ar , h o m o g e n e o u s a n d i s o tr o p i c

t h e s e t h r e e c o m p o n e n t s ca n b e e v a l u a t e d a s f o l l o w s

Axial strainA t y p i c a l c o n d u c t o r s u r r o u n d e d b y E P R w i l l d e f o r m a ss h o w n i n Figure 4 a n d t h e e x a c t b e h a v i o u r w i l l d e p e n d o nt h e s h e a r m o d u l u s o f t h e E P R ( w h i c h i n t u rn w i l l b e p a r t l yd e p e n d e n t u p o n r a d i a l p r e s s u r e ) a n d t h e m a x i m u m s u s -ta inab le in te r f ac ia l shea r f o r ce be tween the EP R and theinsu la t ion ( a l so pa r t ly dependent upon r ad ia l p r es sur e ) .Assum ing the r e to be no s l ippage and tha t the insu la t ioni s f ixed , the Lutchansky K- f ac tor can be expr es sed inm e a s u r a b l e p h y s i c a l t e r m s a s

K = G i . z r dl (6 )t i n

w h e r e G ~ . i s the shea r m odulus o f the insu la t ion m a te r ia lo f t h i c k n es s ti. (Figure 4 ) a n d t h e L u t c h a n s k y4 e q u a t i o ncan be used to ca lcu la te the ax ia l s t r a in away f r om f ixede n d s

STOP

Figure 6

~ START3f] ABDAT

m , , a l d = M = f ~ mm r - d m l M f ik

OUTCABI d l = m l d e , ~ , i =

= t ~ t r d =

ty 2D A .

LOADDATim,-4~md ro d fih-+ .

i II_ ..C o m p u t e r m o d e l f l o w d i a g r a m

AXSUMNm.-lhu~ ita-tti~

*ml mlutim@ I"CAI~+

Tl~

AXSU~Llium ind

0 0 r s Lm'ite eumm l l l y mgt~tomtput~le /


5/14

UaKA I + A E ]

UaG~r dit i n

s i n O = ) t s i n Oa 2 /+ A E Jw h e r e

sinO

(7 )

p.

a = [R 2 + (p/2 7r)2 ] 1/2U = [R(p/27r)2]/[pa]

a nd R i s t he p i t c h c i r c l e r a d iu s (pc r ) o f t he he l ix , t he a ng le0 a ga in be ing t a ke n a s z e ro on the c a b le ne u t ra l a x i s i nc on t ra s t t o Lu tc ha ns ky ' s o r ig ina l fo rmu la t ion .Bending s trainFo l low ing the a pp roa c h o f K n a pp L-3 the be nd ing s t r a in fo rth in he l i c a l w i re s unde r g loba l be nd ing loa d ing c a n be c a l -

OUTFOSwrite rmults to outpu t filef or u c h ~ c m ~ s t i m

BENDBEND~,],temdiug,tnin~totmlqorm~1 , d c u r r m t 0

POSDATread next 0 studm K i a t e d

C s f r ~ p m t i ou f il e

244 Mec han ics o f sub m ar ine cab les : G . Fe ld e t a l .

YES

bniml slam =aadd drdu

=0

AXCIR N o~ c u l a t e b e a d i agd r a i n d w

BI~NDAXLca lcu late am] s t ra indue t~ beadings t v u m m t 0

BENDTORi t a m m t 8@.

NO

Figure 6 Cont inued


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Mechanics of subm~,rine cables: G. Feld e t a l . 2 4 5cu la ted . The pr esen t t r ea tm ent uses an a l t e r ed t r ans f or m -a t ion m a t r ix which i t i s f e l t m or e cor r ec t ly ca lcu la te s thea n g l e t h r o u g h w h i c h t h e r a d i u s v e c t o r o f a s e c t io n o f h e l i xr o ta te s when the cab le : i s ben t . Also , in v iew of the l a r ged i a m e t e r o f t h e w i r e s o f i n t e r e s t h e r e, t h e f o r m u l a t i o n r e c o -gn izes tha t the bending s t r a in wi l l va r y a r ound the sur f aceo f a h e l i c a l e l e m e n t7.The to ta l long i tud ina l s t r a in ( m easur ab le us ing s t r a ing a u g e s ) c a n t h e r e f o r e b e g i v e n b y

E tA( K) s in0

E ,psrwsin~b - (sinth co sto - cos~b s i n t o ) p r w+ ( 8 )p r w ( s i n 4 ~ costo - cos~b s into) + p p s

T o r s i o n a l s t r a i nTh e to r s iona l s t r a in 2 is g iven by

e,i = y - ( 9 )

wh er e ~" and z ' a r e the r ad i i o f to r s ion o f the he l ix in thed e f o r m e d a n d u n d e f o r m e d s t a te s , r e s p e c t iv e l y , a n d y i s t h ed i s tance f r om the neu t r a l ax i s . The r ad ius o f to r s ion of as p a c e c u r v e i s g i v e n b y L i p s h u t z 81

X ' Y ' Z " + X ' Y " Z ' + X ~ " Y ' Z - X ' Y " Z ' - ) g ' Y ' Z " - X " Y ' Z 'A 2 + B 2 + C 2

( 1 0 )O n c e a g a i n , h o w e v e r , t h e c o m p l i a n c e o f t h e i n s u l a t i n gm a t e r i a l d r a s t i c a l l y r e d u c e s t h e m a x i m u m p o s s i b l e s h e a rs t r a in due to twis t ing .W i t z a n d T a n 6 have sugges ted tha t to r s iona l s l ippage i sposs ib le a s we l l a s ax ial [ s l ippage and the exp e r im en ta l ev i -dence i s cons i s ten t wida e i the r th i s in te r pr e ta t ion and /ora t t r ibu t ing l a r ge ( bu t no t i r r eve r s ib le ) de f or m a t ions to thep o l y m e r m a t e r ia l .

C o m b i n e d b e n d i n g a n d a x i a l l o a d i n gW her eas the ben ding s l~ :a in in wi r es du e to app l ied bendin gi s d e t e r m i n e d s o l e l y b y t h e g e o m e t r y o f th e c a b l e a n d o fthe im posed bend r ad ius , the ax ia l o r to r s iona l s t r a ins a r ed e p e n d e n t u p o n c on d uc , t o r - i n s u l a t i o n i n t e r ac t i o n w h i c h i s,in tu r n , in f luenced by any g loba l ax ia l load ing . I t i s the r e -f or e neces sa r y to cons ide r th i s in te r ac t ion i f the r e i s com -b i n e d g l o b a l b e n d i n g a n d a x i a l l o a d i n g .T h u s , t h e e f f e c t s f r o m t e n s i le l o a d i n g w h i c h n e e d t o b ec a r r ie d o v e r t o t h e b e n d i n g a n a l y s is a r e( 1 ) R a d i a l p r e s s u re e x e r t e d b y e x t e r n a l l a y e rs( 2 ) C h a n g e i n h e l ic a l p e r d u e t o t e n s i o n( 3) Cha nge in l ay ang]Le due to t ens ionA s i n d i c a t e d i n e q u a t i o n ( 6 ) , t h e L u t c h a n s k y K - c o n s t a n tc a n b e e x p r e s s e d i n t e r m s o f t h e s h e a r m o d u l u s o f t h e i n s u -

la t ion . I f a t e s t spec im en i s loaded as in F i g u r e 5 , t h e n t h es h e a r m o d u l u s , G i , , c a n b e e x p r e s s e d a s

r L ( ~ d ~ )G in - - - - - -7 ~ l t l ,

= ~ \ T r d i ] ( 1 1 )Exper im enta l r e su l t s , d i s cus sed la te r , g ive

L- - ~ a + b A r ( 1 2 )6t swhe r e , Ar i s the dec r ea se in s am ple r ad ius due to a r ad ia l lyappl ied com pr es s ion f o r ce , i . e . the shea r m odulus wi l l no tr e m a i n c o n s t a n t w h e n s u r r o u n d i n g c a b l e m a t e r i a l c o m -pr es ses the insu la t ion under ax ia l cab le load ing .S ubs t i tu t ing equa t ion ( 12) in to equa t ions ( 6 ) and ( 11) ,t h e r e f o r e , g i v e s a n e x p r e s s i o n f o r t h e L u t c h a n s k y K - c o n -s tan t a s

K = a + b a r ( 1 3 )T h i s c a n b e m e a s u r e d e x p e r i m e n t a l l y w i t h a s a m p l e s u c has tha t shown in F i g u r e 5 .Thus , equa t ion ( 7 ) can be used to ca lcu la te ax ia l s t r a ind u e t o c a b l e b e n d i n g w h e r e s l ip p a g e h a s n o t o c c u r r e d .T h e t o t a l lo n g i t u d i n a l s tr a in u n d e r c o m b i n e d l o a d i n g c a nt h e r e f o r e b e f o u n d b y a d d i n g t h e f o u r c o m p o n e n t s

E t = Eat + E b t + E a b + E b b ( 1 4 )and th i s s t r a in can b e m easur ed u s ing g auges , to t e s t thep r e d i c ti o n s o f t h e m o d e l .The ca lcu la t ion of the to ta l long i tud ina l s t r a in i s an i t e r a t-i v e p r o c e d u r e w h o s e o v e r a l l s t r u c t u r e i s s h o w n i n F i g u r e6 . A p r o g r a m b a s e d o n t h i s p r o c e d u r e w a s w r i tt e n t o c a lc u -la te cab le s t r a in and to r que and laye r load sha r ing undertens ion w i th the end s r o ta t ionaUy f r ee o r r o ta t iona l ly f ixed .Also , long i tud ina l and shea r s t r ains can be ca lcu la ted in theh e l i ca l c o m p o n e n t s f o r c o m b i n e d t e n s i o n a n d a x i a l l o ad i n gaga in in the r o ta t iona l ly f r ee o r r o ta t iona l ly f ixed condi t ion .

E x p e r i m e n t a l w o r kT w o t y p e s o f e x p e r i m e n t h a v e b e e n c a r r i e d o u t i n s u p p o r to f t h e m o d e l d e s c r i b e d a b o v e . T h e f i r st o f t h e s e a r e e x p e r -im ents des igned to a s ses s f undam enta l m a te r ia l p r ope r t i e s

s t r e s s( N l m m 2 )

Figure 7

3OO200

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stee l J

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strain (rnicrostraln )R e p r e s e n t a ti v e m a t e r ia l R a m b e r g - O s g o o d c u r v e s

Y 1000


7/14

2 4 6 M e c h a n i c s o f s u b m a r i n e c a b l e s : G. r e i d e t a l .Table 3 T e s t r e g i m e f o r c a b l e s w i t h 7 0 m m 2 c o n d u c t o r c r o s s - s e c t i o n sS a m p l e n o . C a b l e t y p e T e s t r e g i m eI 7 0 m m 2

7 0 m m 2 n o a r m o u r

II 7 0 m m 27 0 m m 2 n o a r m o u r

T e n s i o n 1 , 2 , 3 , 4 , 5 , 6 T : r o t . f i x e d: r o t . f r e e

T e n s i o n 0 . 5 , 1 , 1 . 5 , 2 , 2 . 5 , 3 T : r o t . f i x e d: ro t . f reeB e n d i n g 6 , 5 , 4 , 3 , 2 . 3 5 , 2 m e t r e s r a d i iB e n d i n g 6 , 5 , 4 , 3 , 2 .3 5 , 2 m e t r e s r a d i i

for input into the model and the second are tests on com-plete cables or subassemblies designed to validate themodel output.The first category included tension tests carded out onsamples of the metallic materials involved in cable con-struction to obtain a Ramberg-Osgood description of thematerial deformation characteristics. Representative stress-strain curves for the copper conductors and steel armourwire are given in Figure 7. In addition, experiments werecarded out to determine the shear and compression moduliof the insulation material and also to determine themaximum sustainable interfacial shear. Figure 8 illustratesthe dependence of the Lutchansky K-constant on thedecrease in sample radius due to a radially applied com-pression force.Cable tests were carried out on 7 m length samples undertension in both rotationally fixed and rotationally free con-ditions and under pure bending and combined bending andtension in the rotationally fixed condition. Some secondary

2 5 0

tests were also performed on cables from which the armourhad been removed and on an assemblage of three conduc-tors from one of the cable samples. The cable tests weredesigned to ascertain The overall cable deformations under axial tension in therotationally fixed and rotationally free conditions The strains suffered by helical components under axialloading and bending The movements of helical conductors within cablesunder axial loading and bending The effects of repeated loadings on the behaviour of boththe cable and its internal metallic elements

In all, four cable samples were tested, two involving70 mm 2 conductors and two involving 80 mm 2 conductors.The cross-sections of the two types of cable were very simi-lar, both consisting of three helically wound conductors anda contra-helically wound armour layer. The structure is

O-OOI,

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00 . 0 0 0 . 2 5 0 . 5 0 0 . 7 5 1 . 0 0 1 . 2 5 1 . 5 0 1 . 7 5DELTA R (ram)

F i g u r e 8 V a r i a t i o n o f L u t c h a n s k y K - c o n s t a n t w i t h r a d i a l c o m p r e s s i o n o f s a m p l e . ( ) l i n e o f b e s t f it ; ( A ) s a m p l e s 1 - 4 m e a n ; ( O )s a m p l e 5 ; ( F I) s a m p l e 6 ; ( ~ ) s a m p l e 7 ; ( A ) s a m p l e 8


8/14

M e c h a n i c s o f s u b m a r i n e c a b l e s: G . / re i d e t a l .Table 4 T e s t r e g i m e f o r c a b l e s w i th 8 0 m m 2 c o n d u c t o r cross-sect ions

2 4 7

S a m p l e n o . C a b l e t y p e T e s t r e g i m eI II 8 0 m m 2

I V 8 0 m m 2

IV 80 mm 2 no a rmour80 mm 2 no armo ur or o. f .8 0 m m 2 o n e c o n d .

T e n s i o n 1 , 2 , 3 , 4 , 5 , 6 T : ro t . f i xed9 c y c l e s 1 - 6 T : r o t . f i xedT e n s i o n 1 , 2 , 3 , 4 , 5 , 6T : r o t . f i xedF l e x e d t o 6 m b . r . @ : 1 T r o t . f i xe d: 2 T r o t . f i xed: 3 T r o t . f i xed: 4 T r o t . f i xed: 5 T r o t . f i x e d: 6 T r o t . f i xedT e n s i o n 1 , 2 , 3 , 4 , 5 , 6T : r o t . f i xedF l e x e d t o 6 m b . r . @ : 1 , 2 , 3 , 4 , 5 , 6 T : r o t . f i xed4 m b. r . @ : 1 , 2 , 3 , 4 , 5 , 6T: rot . f ixed3 m b . r . @ : 1 , 2 , 3 , 4 , 5 , 6T : r o t . f i x e d2 m b . r . @ : 1 , 2 , 3 , 4 , 5 , 6T : r o t . f i xedT en s i o n 1 , 2 , 3 , 4 , 5 , 6T : r o t . f i xed7 5 0 0 c y c l e s f l e x e d t o 2 m b . r . @ : 6 T ro t . f i xedT e n s i o n 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 : r o t f i x ed2 0 c y c l e s 1 - 1 0 T : r o t. f i xed6 5 0 cyc les fl exed to 2 m b. r. @ : 2 T r o t . f i x e d

5 m b . r . @ : 2 T r o t . f i x e d8 m b . r . @ : 2 T r o t . f i xed9 m b . r . @ : 2 T r o t . f i xedT e n s i o n 1 , 5 , 10 , 11 , 12 , 13 , 14 , 15 : r o t . f i x e d2 c y c l e s 1 - 1 5 T : r o t . f i xedT e n s i o n 1 , 2 , 3 , 4 , 5 T : ro t . f i xedT e n s i o n 1, 2, 3 , 4 , 5" I ": ro t . f i xedT e n s i o n 1, 1.2, 1.4, 1.6T: rot . f i xed: r o t . f r ee

a lmos t symme t r i c a l , t he on ly d i s rup t ion to t h i s be ing asma l l bund le o f op t i c a l f i b re s a s show n in Figure 1. Tables3 a n d 4 s u m m a r i z e t h e t e s t r e g i m e s f o r e a c h o f t h e f o u rs a m p l e s . N u m e r o u s s i n g l e l e g s t r a i n g a u g e s ( a c t i v e g r i dl e n g th 3 . 18 m m a n d w i d t h 2 . 5 4 m m ) a n d 4 5 ro s e t t e g a u g e s( a c t i v e g r id l e n g t h 2 m r n a n d 0 . 9 m m w i d e ) w e r e a t t a c h eda round the c a b le a nd ind iv idua l w i re se c t ions . A n e xa mpleo f t h e c o v e r a g e i s g i v e n i n Figure 9.

A s m e n t i o n e d a b o v e , a n u m b e r o f t e st s w e r e c a r r ie d o u tw i t h t h e a r m o u r i n g r e m o v e d f o r t h e p u r p o s e o f a s s e s s i n gt h e r e l a t i v e e f f e c t s o f t h e t w o p o l y m e r i c l a y e r s , L a y e r 1be ing tha t ma te r i a l i n s ide the c onduc to r he l i c e s a nd La ye r2 be ing tha t be tw e e n the a rmour he l i c e s a nd the c onduc to rhe l i c e s . B e nd ing t e s t s w e re ge ne ra l ly c a r r i e d ou t i n a num-ber o f d i f fe rent c ircumfe~rentia l or ienta t ions the ob jec t be ingt o o b s e r v e t h e s y m m e t r y o f t h e m e a s u r e m e n t s y s t e m a n dh e n c e t h e a c c u r a c y w i t h w h i c h g a u g e p o s i t i o n i n g w a sk n o w n .T h e c o m b i n e d b e n d i n g a n d a x i a l l o a d in g t e s t s w e r e c a r-de d ou t u s ing the t e s t r i g i l l u s t r a t e d in Figure 10. T h e t e s tr i g w a s m o v e d l a t e r a l l y t o d e f o r m t h e c a b l e ' s c e n t r a l s e c -t i o n t o c o n f o r m t o i t s o w n b e n d r a d i us . T h e e n d s o f t h ec a b le w e re a b l e t o sw ive l a s i nd i c a t e d in t he f i gu re , a nd a no i l r e se rvo i r e na b le d the a x i a l hyd ra u l i c r a m to ma in t a in ac ons t a n t a x i a l l oa d du r ing the l oa d ing p roc e ss .The c a b le r e su l t s w e re r e c o rde d a s t o t a l l ong i tud ina ls t r a in a s a func t ion o f l oa d ing c ond i t i on a nd po s i t i on in t hec a b le . Figure 11 show s a t yp i c a l se t o f r e su l t s fo r tw o o ft h e g a u g e p o s i t io n s i n S a m p l e I V w h e n b e n t t o 2 m b e n dra d ius fo r va r ious l e ve l s o f t e ns ion . The ( r e l a t i ve ly sma l l )hys t e re t i c e f fe c t i s i l l u s t r a t e d in Figure 12 f o r t h e s a m ec a b le f l e xe d re pe a t e d ly to a 2 m be n d ra d iu s w i th a n a pp l i e dt e ns ion o f 9 t .

D i s c u s s i o nA nu mb e r o f i n te re s t i ng po in t s r e ga rd ing the su i t a b i l it y o ft h e m o d e l a n d a l s o s o m e i s s u e s w h i c h m i g h t i m p i n g e u p o nt h e d e s i g n o f s u b m a r i n e c a b l e s a r i s e f r o m t h e a b o v e e x p e r -ime n t s . O n ly a f e w o f the se w i l l be r a i se d he re t o i l l u s t ra t ethe de s i r a b i l i t y o f a n a na ly t i c a l , r a the r t ha n a nume r i c a l ,so lu t ion to t he s t r e ss a na ly s i s p rob le m.Figure 13 show s some c onduc to r s t r a in s fo r t e ns i l e l oa d -i n g o f c a b l e s a m p l e s w i t h 8 0 m m 2 c onduc to r s a long w i ththe c a l c u l a t e d s t r a in s fo r a r a nge o f va lue s o f t he c ompre ss -i b il i ty o f p o l y m e r L a y e r 2 a s i d e nt i fi e d a b o v e ( o n t h ea s s u m p t i o n t h a t t h e c o m p r e s s i b i li t y o f L a y e r 1 is u n a l t e r edb y t h e r e m o v a l o f t h e a r m o u r ) . O n e i m m e d i a t e l y a p p a r e n tf e a t u re i s t h at t h e c a b l e b e h a v i o u r c h a n g e s w i t h t h e n u m b e ro f t e ns i l e c yc l e s w h ic h ha ve be e n a pp l i e d a nd a l so tha t t h i si s d i f f e r e n t b e t w e e n t h e t w o s a m p l e s o f t h e s a m e t y p e o fc a b le . The re i s ne ve r the l e ss a c ons i s t e n t t r e nd tow a rds al o w v a l u e o f L a y e r 2 c o m p r e s s i b i li t y a n d t h is t r e n d a p p e a r st o b e c y c l e - d e p e n d e n t r a t h e r t h a n t i m e d e p e n d e n t . T h i s t y p eo f be ha v io u r i s c ons i s t e n t w i th t he sugge s t ion tha t vo id sinhe re n t i n t he c a b le s t ruc tu re a t ma nufa c tu re a re g ra dua l lyf i l l e d w i th po lyme r i c ma te r i a l i n a n i r r e ve r s ib l e f a sh ion sot h a t t h e c a b l e e v e n t u a l l y b e c o m e s m e c h a n i c a l l y ' c o n -d i t i one d ' . A s th i s p roc e ss oc c u rs , t he A p i va lue s t e ndt o w a r d s a v a l u e o f 0 . 5 ( c o r r e s p o n d i n g t o a n i n c o m p r e s s i b l ema te r i a l ) f rom the h ighe r , more a rb i t r a ry , va lue s w h ic hre f l e c t t he p re se nc e o f a n inde t e rmina te numbe r o f vo id s i nthe in i t i a l s t ruc tu re . A l though no t i l l u s t r a t e d in Figure 13,t h e s t r a in s m e a s u r e d o n t h e c o n d u c t o r s t h e m s e l v e s d e c r e a s ew i th t ime , t he imp l i c a t ion be ing tha t t he l oa d - sha r ingb e t w e e n t h e a r m o u r a n d t h e c o n d u c t o r s a l s o v a r i e s a s t h ec a b le ' se t t l e s i n ' . In t h i s pa r t i c u l a r c a se , t he a rmour t a ke s


9/14

2 4 8 Mec han ics o f subm ar ine cab les : G . Fe ld e t a l .

View ow =~ re= e=l

Figure9 C i r c um f e ren t i a l gauge m e as u re m e n t po s i t ions f o r S am p l e I V . (@ ) s ing l e gaug es ; (41~ ) gauge ros e t t es

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Figure 10 A x i a l / b e n d i n g t e s t r ig


10/14

Me chanics o f sub ma r ine cab les: G. Fe ld e t a l . 2 4 9GAUGE E82000 .V-W-W-T-~

1750 "/ -- \1 50 0 1 ~ f * -V - V - V i t ~ '

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- 1000 0 5 10 15 20 25T I M E ( S I : ' C S )

Figure 11 V a r i a t io n i n s tr a i n f o r t w o o f t h e g a u g e s o n S a m p l eI V du r i ng f l ex i ng t o 2 m w i t h d i f f e ren t ax i a l tens i ons . (O ) 2 t ;( e ) 5 t ; ( A ) 8 t ; ( V ) 9 ta g r ea te r pe r cen tage o f the load a s the cab le ages bu t thee f f ec t o f th i s on conduc tor s t r a ins i s r a the r l e s s than the' c o n d i t i o n i n g ' e f f e c t d e s c ri b e d a b o v e .Figure 14 shows the m easur ed to ta l s t r a ins in thes a m p l e s w i t h 7 0 m m 2 c o n d u c t o r s u n d e r p u r e b e n d i n g , s t ra i nb e i n g r e c o r d e d i n t h e c o n d u c t o r w h i c h i s o r i e n t e d a t t h e9 0 p o s i t i o n ( o u t s i d e t h e b e n d ) . T h e d o t t e d l i n e s s h o w t h ec a l c u l a t ed b e n d i n g c o m p o n e n t o f t h e s t r ai n . D e s p i t e s o m esca t te r ( due p r inc ipa l ly to the d i f f i cu l ty o f accu r a te ly iden t -i f y i n g t h e l o c a t i o n o f t h e g a u g e s i n t h e c o m p l i c a t e d t h r e e -d i m e n s i o n a l g e o m e t r y ) , t h e o v e r a l l p a t t e r n o f m e a s u r e ds t r a i n a r o u n d a s e c t i o n o f t h e c o n d u c t o r c i r c u m f e r e n c e i sv e r y s i m i l a r t o t h a t c a l c u la t e d f o r t h e b e n d i n g c o m p o n e n t .T h e v e r t i c a l d i f f e r e n c e b e t w e e n t h e t w o s e ts o f c u r v e s c a nbe a t t r ibu ted to the ac tua l ax ia l s t r a in due to bend ing exp er i -e n c e d b y t h e c o n d u c t o r a t t h e o u t s i d e o f t h e c a b l e b e n d .T h e m a g n i t u d e o f t h i s d i s p l a c e m e n t i n d i ca t e s t h a t t h e b e n d -ing s t r a in i s o f cons ide r ab ly m o r e s ign i f i cance than the ax ia ls t r a in wi th th i s type o f cab le and load ing condi t ion . Theor -e t i c a l v a lu e s v a r y f r o m a b o u t 3 0 / ~ E a t 6 m b e n d r a d iu s t oa b o u t 9 0 / x c a t 2 m a n d t h e s e a r e c e r t a i n ly o f th e s a m e o r d e ras those ind ica ted in Figure 14.Figure 15 i l lus tr a te s ~ae e f f ec t o f supe r im pos ing b endingonto an ax ia l ly loaded cab le and the inc r ease in s t r a in dur -

ing the f l ex ing ope r a t ion i s shown p lo t ted aga ins t the ax ia lload . Al thoug h gauges a t d i f f e r en t pos i t ions r ecor d d i f f e r en ts t r ains , the inc r em ents d ue to f l ex ing r em ain f a i r ly cons tan tw i t h a x i a l lo a d s u p t o 9 t a n d b e n d r a d i i d o w n t o 2 m . T h etheor e t i ca l va lues o f such inc r em ents a r e ca lcu la ted to ben o m o r e t h a n 1 5 0 / z e , a g a i n h i g h li g h t in g t h e r e l a t iv e u n i m -p o r t a n c e o f t h e a x i a l c o m p o n e n t w i t h r e g a r d t o t h e b e n d i n gc o m p o n e n t o f t h e c o n d u c t o r s t r a i n d u e t o c a b l e b e n d i n g ,even in cases w her e a subs tan t ia l ax ia l load i s p r esen t . Thetheor e t i ca l r ange o f va lues f o r the 6 m b end r ad ius i s_ _ _8 5 0/ xc a n d 2 3 0 0 / . ~ c f o r t h e 2 m b e n d r a d i u s . T h es t r ains r ecor de d a r e wi th in the theor e t i ca l un i t s a s r e f e r enceto Figure 15 r evea l s .Figure 11 a l s o s h o w s c o m b i n e d b e n d i n g a n d t e n s i o n d u r -ing f l exur e cyc le s wi th d i f f e r en t ax ia l t ens ions . The absenceo f a n y s u d d e n c h a n g e s w h i c h m i g h t b e i n d i c a t i v e o f s li p p -a g e i m p l i e s t h a t t h e m a x i m u m s h e a r i n t e ra c t i o n f o r c e h a sn o t b e e n o v e r c o m e d u r i n g t h e s e e x p e r i m e n t s a n d h e n c ehys te r es i s o f the na tur e desc r ibed pr ev iou s ly i s no t ev iden t .C o n c l u s i o n sA n a n a l y t i c a l ( a l t h o u g h i t e r a t i v e ) m e t h o d h a s b e e n p r e -sen ted f o r the ca lcu la t ion of s t r a ins in the he l ica l e lem entso f s u b m a r i n e c a b l e s w h i c h t a k e s i n t o a c c o u n t m a t e r ia l a n dcab le non l inea r i t i e s and incor por a te s a m easur ed pa r am ete rf or in te r f ac ia l e f f ec t s be tween conduc tor s and insu la t ion .T h i s m o d e l h a s b e e n v a l i d a t e d a g a i n st m e a s u r e m e n t s o nw h o l e c a b l e s u n d e r t e n si o n , b e n d i n g a n d c o m b i n e d b e n d i n gand tens ion .U s e f u l p r e d i c t i o n s c a n b e m a d e o v e r t h e c a b l e l o a d i n gr a n g e 0 - i 0 t a n d d o w n t o 2 m b e n d r a d i u s a l t h o u g h ' th em o de l a s i t s t ands does n o t adeq ua te ly ca te r f o r ax ia l s t ra insdue to bending f o r ex te r na l e lem ents , cab les wi th subs tan-t i a l ly a sym m et r ic c r os s - sec t ions o r cab les wi th cy l indr ica le lem ents .T h e p r e s e n c e o f v o i d s w i t h i n t h e c a b l e r e s u l t i n g f r o mt h e m a n u f a c t u r i n g p r o c e s s i s t h o u g h t t o b e r e s p o n s ib l e f o rs ign i f i can t and unpr ed ic tab le changes in cab le behav ioura n d l o a d s h a ri n g . T h e p r e s e n c e o f s u c h v o i d s c a n b e c a t e r e df o r i n t h e d e s i g n o f c a b l e s b y i n d e p e n d e n t v a r i a t io n o f t h et w o p o l y m e r l a y e r c o m p r e ss i b i li t ie s a n d t h e e x t r e m e s w h i c ha n y g i v e n c a b l e i s l i k e l y t o e x p e r i e n c e c a n b e c a l c u l a t e d .Chang es in the com p l iance o f the insu la t ion m a te r ia l a r ea l so l ike ly to a f f ec t cab le behav iour , p r im ar i ly dur ing bend -ing . I n cab les o f the type t e s ted he r e , the low shea r m odu lusof EP R causes a cons ide r ab le r educ t ion in the s t r a inse x p e r i e n c e d b y c o n d u c t o r s c o m p a r e d w i t h t h o s e w h i c hm i g h t b e e x p e c t e d f o r a c o n d u c t o r ' f r o z e n ' i n t o p o s i t i o ndur ing cab le f l ex ing .N o t a t i o na

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inc r em en ta l pa th l eng th Rinc r em en ta l chang e in wi r e t ens ion RcR a m b e r g - O s g o o d s t r a i n Rc cw i r e m o d u l u s o f e l a s ti c i ty sf o r c e a p p l i e d t o h e l i x / c a b l e / c o m p r e s s i o n ts am ple Tto ta l r ad ia l com pr es s ion f o r ce pe r un i t l eng th T Fo f h e l i x ur a d ia l c o m p r e s s io n f o r c e e x e r t e d b y a h e l i xs h e a r m o d u l u s upf l exur a l coupleto r s iona l coup le Uw i r e s e c o n d m o m e n t o f a r e a x ,yw i r e p o l a r m o m e n t o f a r e as t i f f nes s m a t r ix e lem ents X , Y , ZLutchansky in te r ac t ion shea r s t i f f nes sR a m b e r g - O s g o o d s h a p e p a r a m e t e r X ' , Y ' , Z 'l eng th o f wi r e in one p i t ch l eng thsam ple l eng th X" , e tc .t e s t loadcab le s am ple l eng th X" , e tc .m a te r ia l spec im en sam ple l eng thn u m b e r o f h e l i c al l a y e r s i n c a b l e yw i r e t w i s t i n g o r b e n d i n g m o m e n tR a m b e r g - O s g o o d s h a p e p a r a m e t e rn u m b e r o f w i r e s i n a l a y e rax ia l loadshea r ing f o r ce /3ax ia l load f unc t ion 7p i tch l eng th 7r ad ia l p r es sur e on cor e %;r ad ius o f wi r e 6

wi r e p i t ch c i r c le r ad ius ( pe r )r a d iu s o f p o l y m e r c o r er ad ius o f r ig id cor eR a m b e r g - O s g o o d s t r e s sth icknessax ia l to r queax ia l to r que f unc t ionloca l d i sp lacem ent o f a he l i ca l wi r e a longits pathc o r e d i s p l a c e m e n t d u e t o p l a n e s e c t io n b e n d -in g[R (p /2 ~r )2]paCar te s ian co- or d ina te s o f a po in t wi th in aw i r eCar te s ian co- or d ina te s o f a po in t wi th in acab lef ir s t dif ferent ia l of X , Y , Z w.r . t , angular pos-i t ionsecond d i f f e r en t ia l o f X , Y , Z w.r . t , angularpos i t ionth i r d d i f f e r en t ia l o f X , Y , Z w.r . t , angular pos-i t ionr ad ius o f wi r e

G r e e k s y m b o l sa w i r e l a y a n g l e m e a s u r e d f r o m l o n g i tu d i n a laxisr a t io o f f ina l wi r e pe r to in i t i a l wi r e pe rwi r e r o ta t iona l s t r a inin te r f ace shea r s t r a inc a b l e r o t a t io n p e r p i t c h l e n g t h o f h e l i x it e s t s a m p l e e n d d i s p l a c e m e n t


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Figure 13 M eas u red and c a l c u la ted s tr a ins f o r Sam p les I II andIV in r o ta t i ona l l y f i x ed c on d i t i on us ing a r ange o f v a lues f o rc om p res s ib i l i t y o f ou te r po l y m er l ay e r. Theo re t i c a l )tp=: (O ) 0 .5 ;( r- I) 5 .0; (~ ) 10.0; (V) 15.(3; (A) 20.0. Expe r ime nta l : (&) S am pleI II f i rs t pu l l (0 m in) ; (11) Sa mp le I II 11th pul l (0 m in) ; (O) Sa mp leIV f i rs t pu l l (0 m ins ) ; (V) Sam ple IV f ina l pu l l (0 m in )

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Subscriptsa a x i a lb b e n d i n gc c o r e , c a b l ei, k w i r e n u m b e rin i n s u l a t i o np p o l y m e rr r a d i a ls s t r a i g h t c a b l et t o r s i o n a lw w i r eSuperscriptss s e c a n t m o d u l u s' d i m e n s i o n a f te r d e f o r m a t i o n' f i r s t d i f f e r e n t i a l" s e c o n d d i f f e r e n t i a l" t h i r d d i f f e r e n t i a l' a f t e r l o a d i n g

R e f e r e n c e s1 K n a p p , R . H . ' D e r i v a t i o n o f a n e w s t i f fn e s s m a t r i x fo r h e li c a l ly a r m -o r e d c a b l e s ' , Int. J. Num. Meth. Engng 1 9 7 9 , 1 4 , 5 1 5 - 5 2 92 K n a p p , R . H . ' S i m p l e b e n d i n g m o d e l s f o r h e l ic a l l y a r m o r e d c a b l e s ' ,Proc. Second In t . Of f shore M echanics and Arc t ic Engineering Symp.A S M E , N e w Y o r k, 3 6 0 - 3 6 4 , 1 9 833 K n ap p , R . H . 'H e l i ca l w i r e s t r e s s e s i n b en t cab l e s ' , ASME J . Of f shoreMech. Arct ic Engng 1 9 8 8 , 1 1 0 , 5 7 1 -5 7 74 L u t c h a n s k y , M . ' A x i a l s t r e s s e s i n a r m o u r w i r e s o f b e n t s u b m a r i n e c a b -

l e s ' . AS M E J . En g n g f o r In d u s t ry 1 9 6 9 , 9 1 , 6 8 7 - 6 9 35 H a l e , A . L . 'T h e i n e l a s t ica - t h e e f f ec t o f in t e rn a l f ri c t i o n o n t h e s h ap ea n d t e n s i o n o f a b e n t c a b l e ' , Proc. 33rd In t . Wire and Cable Symp.E l e c t ro n i c s C o m m a n d , R e n D , N e v a d a , 1 9 8 4, p p 2 3 7 - 2 4 36 Wi t z , J . A . an d T an , Z . 'O n t h e f l ex u ra l s t r u c t u r a l b eh av i o u r o f f l ex ib l ep i p e s , u m b i l i c a l s a n d m a r i n e c a b l e s ' Marine S truct . 1 9 9 2 , 5 (2 an d3 ) , 2 2 9 - 2 4 9

7 F e l d , G . , R e u b e n , R . L . , O w e n , D . G . a n d C r o c k e tt , A . E . ' P o w e r c a b l ea n d u m b i l i c a l s - c o n d u c t o r s t r a in s u n d e r p u r e b e n d i n g ' Proc. Firs tInt. Offshore an d Po lar Engineering Conf. E d i n b u rg h , V o l . 2 1 9 9 1 ,p p 2 2 8 - 2 3 58 L i p s h u t z , M . M . Dif feren t ia l geometry M c G r a w - H i l l, N e w Y o r k , 1 9 6 9


13/14

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14/14

M e c h a n i c s o f s u b m a r i n e c ab l es : G . F e ld e t a l . 2 5 3

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[Owen] Mechanical Behaviour of the.

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