otc-1071-ms offshore pipelines

8/10/2019 OTC-1071-MS Offshore Pipelines

1/14

OFFSHORE

TECHNOLOGY

CONFERENCE

6200

North

Central Expressway

D allas Texas 75206

THIS

IS

A PRE

PRINT

SUBJECT

TO

CORRECTION

R e

7

Stress nalysis

uring

of

Offshore

Pipelines

Installation

By

T. Powers Humble Pipe

Line Co.

an d L. D. Finn Esso Production Research Co.

Copyright 1969

Offshore TechnologyConfe-rence

orr behalf

o f

American

I ns t i t ut e o f Mining Metallur gical

and

Petroleum Engineers I n c . The American

Association

o f Petroleum

Geologists

American I ns t i t ut e of

Chemical Engineers American Society o f

Civil

Engineers The American Society o f

Mechanical

Engineers The

I ns t i t ut e

o f E l e c t r i c a l

an d E lectr onics Engtneers I n c .

Marine

Technology Society

Society

o f Expl orat i on G e ophys i c is t s

arid Socie-cy

o f Naval

Architects

Marine Engineers.

This paper

was

prepared fo r

presentation

a t the Firs t Annual Offshore Technology Conference

to

be

held

in Houston Tex.

on

May 18-21 1969. Permission to

copy i s

r e s t r i c t e d to an a bstra c t o f

no t

more than 300

words.

I l l us t r a t i ons may not

be copied.

Such use o f

an

a bstra c t should contain

conspicuous

acknowledgment o f where

an d

by_whom

the paper

i s

presented.

ABSTRACT

A finite-beam-element

i n i t i l ~ v l u e

analysis

procedure d e t e = i n e s s tr es se s i n

a

subsea

pipeline

suspended between

th e

ocean

fl o o r an d a laybarge or--stinger. The b as i c

theory i t s a dv an ta ge s o ve r

other

the orie s an d

a comparison o f r e s u l t s

with

th e

r e s u l t s

o f

a na lytic a l

p r oc ed u re s b as ed

on

other

t h eo ri es

are

included. The finite-element

theory is

applicable over a wide

range

o f

marine

pipelay

in g problems an d compares

favorably

with

other

accepted theories

in

the-

ranges

o f thei r a p p ~ i ~

c a b i l i t y .

INTRODUCTION

Thick-wall piRe an d o n r e t e ~ t i n g s axe_

used to weight marine p i p el i n es to insure t h a t

the pipeline wil1.remain

in

place a ft er i ns ta l

l a t i o n . During inst i l lat ion i n deepwater this

weight

imposes

high stre sse s i n

th e

pipe

sus-

pended

from th e laybarge and may

cause

t h e i l l ~

to fai l Deepwater

re pa ir

may

nof

be

possible;

as a r e s ul t a f a i l u r e may r e ~ u i r e

the

entire

l i n e to

be replaced. o ~ e l e c t the

proper

pipe weight an d

grade anclt;o prevent

c os t l y

f a i l u r e s

th e

stre sse s

imposed

i n

th e

pipe

during laying must be defined an d construction

procedures

an d

e9.uipment must be accurately

analyzed.

This paper

presents a

procedure

fo r

making t he s e a n al ys e s. The c a lc u la ti o n t ec h

nique

may

be used to

analyze

an y construction

method which ins ta l l s

th e

l i n e from

th e

surface

References

an d

i ll u st ra ti o ns a t

en d

o f paper.

in a continuous s t r i n g .

This

paper describes

the

a na lytic a l procedure

specifically

in terms

o f t h e s ti n ge r -I a yb a rg e c o ns tr uc t io n method.

The b as i c theory

used in

the an al y s i s i t s

advantages

over other

the orie s an d

a

compari

so n of r e s u l t s

with results using other

the orie s

a re p re se nt ed .

THE

MARINF.

PIPELINE

SUSPENSION

PROBLEM

The

problem

o f p r ed ic t in g the

suspended

geometl.,Y c f

marine

pipelines

in

deep water i s

one o f nonJi::. 3ar, large-angle bending.

This

problem

may be

s o lv e d n u m er i ca ll y through

th e

us e

of a f iro.i.te-beam

element i n i t i a l - v a l u e

approach

L l l ~ t r e a t s

th e

to ta l

beam

as

a

se rie s

o f sm -ll

hemna,

each

o f

which

is

analyzed with

l .

.inear

theor.i. Combining

a l l

th e

small-angle

bending so]U: ;iOl.S produces

the

large-angle

bending marj :Qc p i pe li n e s o lu t io n . In the

in itial-valtl -;--a.pproacn.;imknown

boundary

condi

t ions the

ocean

fl o o r are assumed

an d

then

s Y f l t e m a t i \ a l l ~ _

changed

unt i l

th e

p r o f i l e which

has been bui l t up

segment

by segment s a t i s f i e s

boundary conditions

s p eci fi ed a t th e

u pp er e nd .

The appendiX to th is paper

contains

th e b as ic

e 9. ua ti on s a nd ~ h e i r

derivation.

The t h e o l ~

accounts

fo r applied tensions

ex t ern al fluid pressures va ria tions

i n

water

currents

With

depth pipe s t i f f n e s s va ria tions

due

to

weakness o f

the weight coating

a t

th e

f ield j oi nt s

an d

support buoys [ i f used]. The

approach

is l uite general an d is applicable to


2/14

I I lO

STRESS ANALYSIS

OF

OFFSHORE

PIPELINES

DURING INSTALLATION

OTC 10

both two-and t h re e -d i m en s io n a l f o r ce s an d de

flections;

crosscurrents an d

la tera l barge

movement ca n

be considered.

Although th e

develoIJed

procedure

can>

be

used

to analyze

the stresses in th e

suspended

portion of a p i p e l i n e u r J ~ e r an y

support

con

d i t i o n i t is discussed>here in

terms

o f using

t h e s t in g er -l ay b a rg e

construction

method [Fig.

1]. The stresses calculated

account

for

s ta t ic

loads

only

considering

ocean

currents as

s tatic

loads. The e f f e c t s

o f

dynamic loads an d ex

t e r n a l hydrostatic

pressure

on th e actual long

i td u in a l s tr es s

must

be considered

in th e selec

t io n of allowable s tr es s levels used in con

junction with this procedure.

USE OF

FINITE-ELEMENT

CALCULATION PROCEDURE

The

laborious

an d r ep etit iv e

nature

of th e

finite-elem ent procedure

makes

i t necessary to

us e a computer. The

p r oc ed u re h a s

been

p r o ~

grammed fo r the

IBM 360.

In a dd it io n

to

the

boundary-condition assumptions

the

procedure

requires th at w a te r d e pt h p i pe properties

applied tension height

to upper

support point

an d eith er moment or slope

a t

th e

upper support

point be given. When

analyzing

th e stinger

laybarge

method the

upper support

point w i l l

be

the

l i f t o f f point on th e stinger [ i .e . th e

las t p o i n t

t ha t t he

pipE l is in cgnt.?-ct with.j:;he

s tin g er ] . I f

buoys

or

currents

are being

included in the

analysis

th e p oin ts of

appli

cat ion

a nd m a gn it ud e

o f th ese forces

i s

al80

re luired.

To

adequately analyze th e stresses imposed

on a

pipeline

when using the s t i n g ~ r l a y b a r g e

~ o n s t r u c t i o n

method

some means

must

be

found

to

simultaneously

describe

the

s t i n g e r

and the

suspended pipeline geometry as an y change in

stinger

position

w i l l

r e s u l t in a change in

the

geometry of

the

suspended

l i n e .

This

ca n

be

accomplished

by

u t i l i z i n g th e p l o t s

shown

in

Figs. 2 an d 3.

Fig.

2

is

a p lo t

of

the maximum

s t r e s s in

the

suspended portion under various

a pp l ie d t en s io n s as

a

function of the >l:i.ft-off

angle. Fig. 3 is the corresponding p lo t of

height

to

l i f t o f f point

fo r variQus

tens tons

as

a function of th e l i f t o f f angle. The

a p p l i e d t e n s i o n s a ~ ~ ~ ~ ~ a x i a l

tensions

applied a t th e barge an d do not allow

fo r

fr ic ti on i n

the

r o ller s

o f

the

stinger. How-

ever the calculation procedure considers the

tension reqUired

to

hold

the

wei ght

o f

th e pipe

on the stinger between th e barge an d the l i f t -

o f f point. The

l i f t o f f angle is

the

angle

be

tween th e

horizontal

an d a tangent

to

th e p ip e

l in e a t th e point where leaves the

s t i n g e r .

The water

depth

pipe

size

an d over-all pipe

s p ec i fi c g ra v it y

for this s e t

o f curves

are

fixed.

Two p o t e n t i a l problems

ex is t

in

the

sus

pended portion

of

th e pipeline. The

pipeline

may bend excessively an d

buckle

in th e

suspende

p or ti on o r

i t may

f a i l

to l i f t o ff th e s t i n g e r

an d

b uc kle o ve r th e

en d

of th e s t i n g e r . Both

p ro bl em s c an be

lessened

by

increasing

tension

decreasing the specific

gravity or

lengthening

th e s t i n g e r . Fo r

a

given

tension an d

specific

gravity

th e maximum s tr es s in th e suspended

portion

is reduced

when the

l i f t o f f

angle

is

reduced

[Fig.

2 ] .

However th e submerged en d

o f the

stinger m us t h ave

s u f f i c i e n t

slope an d

depth

to

permit the pipe

to l i f t

o f f t he s ti ng en

Fo r a given s t i n g e r length the maximum pipe

l in e s tr es s is m i ~ i m i z e d when the l i f t o f f angl

is the minimum obtainable.

To

achieve l i f t o f f

when laying in deep water i t is desirable t h a t

the

stinger be curv ed to lim it s t i n g e r length.

Just how stinger curvature

is

obtained is un

important to the following

discussion. optimum

stinger

deployment is

defined

as

th at

which

yields the

minimum

obtainable l i f t o f f angle

an d minimum flooding-schedule s tr es s or curva

ture

with

a

given

amount

of tension.

optimum

deployment a ls o re qu ire s th e pipe

to l i f t o ff

near the en d

o f

th e

s t i n g e r . This

analysis

.assumes optimum l i f t o f f

to be

20

f t

from the

end.

Obviously the stinger shape or geometry

must

be

known. The

stinger geometry curves are

obtained

by

eith er

choosing varying

amounts

of

bui l t in curvature or by making a s tr es s anal

y s is o f

a

p a r t i c u l a r

s t i n g e r ~ u si ng f lo o di ng

sc he dul e s c orre spondi ng to different

s tr es s

l e v e l s .

These

curves

are

then

superimposed on

on

the

height

to l i f t o f f point vs l i f t o f f

angle

p l o t as shown on Fig. 3.

I f

a s tr aig h t

s t i n g e r

is flooded

to

obtain

a

given

curvature

th e flooding schedule

is

usually

selected

b y

t r i a l

an d e r r o r .

Since

s ti ng e rs c ur re n tl y being

used are hinged

o r

pinned

a t

th e barge each

t r i a l

flooding

arrangement

must

be

chosen

so

the stinger w i l l have j u st enough buoyance

to

support the pipe r e s t ~ n g on the stinger plUS a

p or ti on o f

th e

suspended

pipe weight. The

suspended pipe weight

to

be

supported

can be

determined

from t h e f in i te - el e m en t calculation

procedure.

To

o bt ai n t he stinger geometry

curves

in

Fig. 3 a 550-ft

s t r a i g h t

stinger was theore

t ica l ly

flooded

to

cause

maximum

s t r e s s

[due

to

flooding schedule]

o f

0 lO l5 20 an d 25 k s i

in the stinger members. These same curves

could have

been

generated

by

assuming 0

0

4

0

9

10

0

an d

14.5

0

b u i l t - i n

curvature.

The

s t i n g e r

is assumed

to

have

an

in i t ia l slope of 13

0

a t

th e water surface. The stinger geometry curves

are formed

by

p l o t t i n g th e slope of th e s t i n g e r

as a function of height above th e ocean floor

for the four stinger s tr e ss l ev e ls . A

curve i s

then drawn connecting a p o i n t 20 f t from

the


3/14

GTC 1071

J . T. POWERS and L. D FINN

I I 11

en d

of the

s tin ge r f or eaCh s t i n g e r s tr es s

lev el. This

i s

r f r ~

to

as

th e

l i f t - o f f

l i n e . The

intersection

o f each s tin g er geometr

curve w ith th e a pp lied tension curves i s

then

transferred

to

F ig . 2. Also transferred

to Fig.

2 i s the i nt er s ec t io n o f

the

applied tension

curves with

th e

l i f t - o f f l i n e s. This

l i f t - o f f

l i n e on Fig. 2

i s

a l i m i t of

stinger

p l o y m n ~

in

other words for

t h i s

p a r t i c u l a r

s tin g er

there

i s

no

solution

to

the

l e f t

o f

th e l i f t

o ff lin e

Fig. 2. I f the s t i n g e r geometry

s tr es s i s limited

to

25 000

p s i

then

there

i s

no solution to

the r ig h t

of

t he 2 5-k si l i n e .

The p lo tted output shown in Figs. 2

an d

3

ca n

be used to determine [1] th e

tension

re

quired

t o p re ve nt overstressing the

pipeline

with a given s t i n g e r

curvature

[2 ]

the

tension

required to l i f t o f f the s tin g er

an d

[3] the

e f f e c t of changing the curvature of the stinger.

The output ca n

also

be used

to

d e J ~ e r w i n e the

e f f e c t of

using

d i f f e r e n t equipment

p p

s i z e s

specific g r av ities e tc .

Examples of Finite-Element

Calculations

The following examples i l l u s t r a t e the uses

o f

th is an aly tical procedure.

Determination

of

T ension Re qui re d to

Prevent Overstressing

th e

Pipeline

for

a Given Maximum Stinger Curvature

Using Figs. 2 an d ~ p r e p r e s p ecif ically

20-in. pipe i n 200 water l e t

us

assume t h a t th e following

conditions

have been

se t :

Maximum

allowable

s tin g er

s tr es s l5 OOO

p si

Maximum allowable s tr es s

in

pipeline

36 000 p si

Enter

Fig. 2 with a 36-ksimaximum combined

s t r e s s

proceed

h oriz on ta lly to th e 1 5-k si

s t i n g e r

st r e ss

l in e an d observe t h a t 40 OOO lb

tension i s

required.

In

th is case

th e

pipe

would l i f t o ff

the s t i n g e r

more than 20 f t from

the

end.

T e nsion Re qui re d to Li f t

Of f

Stinger

Assuming t h a t i t

i s

desired

to l i f t

o ff

20

f t from the en d o f the

s tin g er

an d s t i n g e r

s tr es s i s limited to 10 000 ps Fig. 3 in d i

cates t h a t 60 000

Ib

of tension i s required.

This

i s

determined b y f ol lo wi ng

the lO OOO psi

s t i n g e r

st r e ss

l in e

to

i t s intersection

with

the l i f t - o f f

l in e [ignore the

dashed

l in es

which are used

in

Ex. 3 ]. This intersection

occurs a t

the 60 OOO lb applied tension

curve.

[An

applied te ns io n o f les s

than

60 000

Ib

would

allow

the

pipe

to droop

over

the en d o f

th e s tin g er . ] The l i f t - o f f angle would be

about 17

0

occurring

a t a point about 64 f t

above

th e

ocean

floor. Fig.

2

indicates

t h a t

th e maximum combined s tr es s

in

the suspended

portion

o f the

pipeline

for t h i s condition

would

be

27 000 p s i ..

Effect o f Changing the

Curvature

o f the

Stinger

F i r s t

assuming

40 OOO lb

applied

tension

an d a stinger c u r va t ur e p r ov i de d

by

15 000 p s i

flooding

schedule

s t r e s s

Fig.

3 shows

[see

dashed l in es ] t h a t the p p w i l l l i f t

o ff

68

f t

above th e ocean floor

a t

a l i f t - o f f

angle

o f

about 19.8

0

Referring

to F ig.

2

the s tr es s

in th e

pipeline i s found

to be 36 000 p s i fo r

the

19.8

0

l i f t - o f f

angle

an d

applied

tension

o f

40 000 I b s . However

the

f l oo d in g s ch ed ul e

ca n b e r ed uce d to a bo ut 1 2 5 00

p s i

an d

s t i l l

allow

the

pipe to l i f t o f f

20

f t

from the

en d o f

the s tin g er . This

can be determined

in

Fig. 3

by

moving

down

along th e 40 000-10

tension curve from th e 15 OOO psi s t i n g e r s tr es s

lin e

to the

l i f t - o f f

l i n e

an d interpolating

be

tween the 10 OOO and 15 OOO psi s t i n g e r

st r e ss

l i n e s. The

pipeline

w i l l

now

l i f t

o f f

59

f t

0

above the ocean f l o o r a t an

angle

of about 18

Fig.

2 indicates t h a t th e maximum pipeline

s tr es s i s reduced to 34 000 p s i .

Other Uses

of

Finite-Element

Calculations

New Stinger Design

Fo r a specific

i n st a l l a t i o n

where the water

depth pipe s i z e

e t c .

are known curves simi

l a r to those in

F igs.

2

an d

3

ca n be

used to

s elect the

proper

s te e l grade and s t i n g e r

leng th. S tinger geometry curves

could

be drawn

corresponding

to

available

grades

o f

s te e l.

Kilowing the maximum value o f te ns io n which

c ou ld b e

applied in

the

p a r ti c u la r s i tu a t io n

th e

s tin g er

length

could be

determined from th e

i nt e rs e ct io n o f the stinger s tr es s curve

with

th e maximum available

applied

tension

curve.

S el ec ti on o f Pipe

Wall Thickness

Grade an d

s p ecif ic

Gravity

In a s itu atio n where

the

p p size has

been

selected an d the lay barge i s equipped

with a s p ec if ic s t in g er an d a

tensioning device

o f known

capacity the proper

selection of

p p

wall thickness an d grade ca n be made by pre

paring

a

se t

o f

curves

similar

to

F igs.

2

an d

3

fo r each wall thickness and each specific grav

i t y .

COMP RISON WITH

OTHER

THEORIES

Results obtained

with

th e

finite-element

approach were compared with r es u lts from sever

a l existing

deflection

t h e o r i e s . The f i r s t

comparison was made

with

the simple beam theory

t h a t

does not consider ax ial tension. Thus

for

a f a i r comparison

i t

was

necessary to

apply


4/14

1112

TRESS ANALYSIS OF OFFSHORE PIPELINESDURING INW .MXATION

cTc lo~

approximately14 kips tension at the barge to

have zero effectivetension at the ocean floor.

It was found that the simple beam and finite-

element solutions compare very well for lift-

off angles as large as 55 [Fig. 4]. Because

the simplebeam solutioncannot account for

axial tension, it has limited application.

Abeam column solutionwas developedby

solving the small-mgle bending, linear differ-

ential equation for a singlebeem element in

tension. In water depths to at least 250 ft

with a relativelystiff pipe the beam column

and finite-elementsolutionsagree very well

for lift-off mgles as large as 30 [Fig. 4].

It is evident from Fig. 5 that the sus-

pended length is considerablylonger and flatte

for 60 kips applied tension thaa.for.20kips,

so the small-auglebending theory accuracy

improves for higher tensions.

The beam columu

solutiongenerallypredicts stresseswhich are

higher than those predictedby the finite-

element solution [Fig.6]. IU deeperwater

[1,000ft] with a relativelyflexiblepipe the

bean-column solutionand the finite-element

solution do not comparenearly as well [Figs.

7and8]. Althoughthe.bendingstress diagrsm

[Fig.81 is significantlydifferent,especially

near the upper end, the maximum stress differs

by only 13 percent.

With the natural catenary,it was found

that for a constantapplied tension,the pro-

file describedby the suspendedportton was

independentOf the lift-offheight.

Thi.s same

trend was observedwith three of the other

theories for deep water with relativelylarge

tensions am.dflexiblepipelines. Although the

profile describedby the natural catenaryin

deep water differs from the others [Fig. 7],

the slope at the up$er end comparesvery well

with the slopes of the finite-elementand stiff

ened catenarysolutions.

As the natural cate-

nary solutionignorespipe stiffness,the pipe

can bend more freely at the ocean floor, so the

horizontal distanceis shorter than obtained

with solutionswhich considerstiffness. The

maximum bending stresspredicted by the natural

catenarystress diagram follows the finite-

element and stiffenedcatenarydiagrsms through

the middle part of the suspendedpipeline,but

the boundary conditionsat the end are not

satisfied.

In shallowerwater with relativelysmall.

tensions and stiff pipes, the natural catenaxy

solutionis grossly in error [Figs.4, 5 and 6]

Fig. 5 illustratesthat the error in the profil

decreasesas the tension increases. The maxtiw

stress in the suspendedpipelineprgdicted by

the finite-elementmethod asymptotically

approachesthe maxim~ stresspredictedby the

natural catenaryas the lift-offangle in-

-~

manner that the results couldb= cmpa:redto

those from tl:efi.nite-e].ementme~hod.

In deeg

water with re?&%i.ve.Qflexiblepipes and Ioug

suspendedlengths,

the resu.1.tsbtr,inedwere

identicalwith ~,hosefrcm ttief%.i+;e-element

method [Figs.7 ant 8].

The stiffei~.e


5/14

OTC 1071

~_.--J...PQWjlRSand L. D. FINN

11 13

the finite-elementsolutionsagree in deep

forces at n along X, Y and

water, the former can be used in deep water to

Z axes, respectively

obtain a two-dimensionalsoultion.

However,

Fq Fq Fq

XiI)Yn) Zn

= summationbetween poin-t1

ocesn currentsand nonzero moments at the lift-

and.n of current drag

off point cannotbe considered,and the resul*-

forces along X, Y and Z

ing suspendedgeometrycannotbe made compatible

axes

with a curved stinger.

F

xopFYopFzop

= componentsof the outside

fluid pressure acting

CONCLUSIONS

normal to the pipe wall

F

Xip> Yip) Zip

= componentsof the inside

The calculationtechniquepresented is

fluid pressure acting

quite general and is applicableand accurate

normal to the pipe wall

for a wide range of OffshO.repipelayingprob-

g

= gravitationalconstant

lerns.

Becausethetechniqueis very flexible,

H=

total water depth

it can accurately considervarious boundary

moment of inertiaof pipe

conditionssmd, as is shown in the appendix,

~= length of beam segnent

currentprofiles,weight coatings>awiliary

%> Yn~ %

= moments requiredat point u

supportbuoys, crosscurrents>and lateral barge

for equilibriwnabout X,

drift.

Y and Z axes

% ~ zn

moments at point n about X,

The calculationprocedure can be used to

y and z axes

evaluateadequacy of constructionequipment,

% ~n %

= moments at point n resultin~

design new equipment,and.selectPiPe character-

from currentdrag on sec-

istics for a.proposed.pi~ellne Szn

= forces at point n along x,

ACKNOWLEDGMENT y and z axes

T

The authors wish to acknowledgethe contri- en

= effectivetension at n

.

horizontal currentvelocity

butions of D. W. Dareing, U. of Arkansas, to the

J

on jth

segment

calculationprocedure describedin this paper.

v

They also wish to thank the management of Esso

yj zj

= current velocityalong y

and z axes

productionResearch Co. for their permissionto

Wy? Wz

=

uniformly distributedload

publish this paper.

per unit length on beam i~

REFERENCE

y and z directions

wb

.

= buoyancyper unit length

.

e

= effectiveweight per unit..

1.

Dixon, D. A. and Rutledge,D. R.: Stiffene

d

length of pipe in fluid

CatenaryCalculationsin Pipellne Laying

P

= weight per unit length of

problem Traus. ASME, Petroleum Division, coated pipe in air

paper 67-=[Sept., I-9671.

x> Y> z

= local coordinates

Yn? n

= deflectionof point n rela-

NOMENCLATURE tive to n-1 along y and z

[forAppendix.]

x Yn, Zn =

)

global coordinatesat point

n

Ai =

cross-sectionalsrea inside pipe

X:g, Zcg =

coordinatesof center of

Ao

= total area enclosedwithin the

n

gravity of portion sus-

outermostdiameter

CD

= drag coefficient

pended between point 1 an(

n

Do =

outermostdiameter of piPe line

en

= angle in X-Z pleae between

E=

modulus of elasticity

Z and z axes

Fn)Fyn)FZn

= forces required at point n for

Yn

~ angle in x-y plane between

equilibriumalong X, Y and Z

Y and y axes

~b Fb ~b

axes, respectively

Y@ ezn

change in angle between

Xn Yn) Zn

= componentsof capped end fluid

point n-l and n in y-x


6/14

STRESS ANALYSIS OF OFFSHORE PIPELINESDURING INSTALLATION

OTC 10?1

1

.i

and z-x planes, respectively

Yf~~ Yfi

= specificweight of fluid outside

and insidepipe, respectively

P

= mass density of fluid outside

pipe

@ = angle horizontal currentmakes

with X axis

APPENDIX

Equations for Finite-Be&mElement,

Initial-ValueApproach

;eneralApproach

A pipeline suspendedbetween the ocean ___

floor and stingeror laybargebehaves as a long

beam smd deflectsaccording to the forces actine

m it.

This nonlinear,large-anglebending

problemmay be solved-throughthe use of a

finite-beam-element,initial-valueapproach

whichtreats the suspendedpipe as a series of

small beams each of which is solved by linear

theory. In the initial value approach, the

unknownboundary conditionsat the ocean floor

are initiallyassumed and then systematically

changeciuntil the profile which has been built

up segmentby segment satisfiesthe boundary

conditionsat the upper end.

The origin of the coordinateaxes is tsken

at the point where the pipe first touches the

ocean floor.

For two-dimensionalproblems, the

~mentj tension,position, and slope are how

~t the origin.

The trial solution is beg& by

assumingthe unknown shear force at this point.

Ihe shear, moment md tension at the other end

[point2] of a short-segment are calculated

N

Cos 43XCos Q

Y

-Cos 9

x

sin 0

Y

Y= sin @

Y

Cos 9

Y

z

sin (2Xcos Q

Y

-sin 6X sin Q

L

Y

from equilibriumequations. These internal

forces along with the external forces are then

used to determinethe incrementaldisplacement

and slope at tiat point.

The internal forces

at Point 2 could be used to obtain the internal

forces at Point 3, but for computational-error

reasons the forces are obtained from equilibria

of the segnent suspendedbetween Point 3 and

the origin. This procedure is repeateduntil

the pipe reaches a designatedheight above the

ocean floor. If the specifiedboundary condi-

tions at that point are not satisfied,a new

and improved shear force is assumed until they

are satisfied.

For three-dimensionalproblems,

the same approach is taken. In this case more

than one boundary conditionmust be assumed at

the origin. These unknown boundary conditions

are convergedupon separatelyuntil the tbree-

dimensionalboundary conditionsspecifiedat the

upper end are satisfied.

CoordinateSYstem

The global coordinatesystem [X, Y, Z] will

be defined with the X axis the direction of lay,

the Y axis vertical, and the Z axis the lateral

direction forming a right-hand system. The

local coordinatesystem [x, y, z] will be de-

fined with the x axis along the pipe axis, the

z axis perpendicularto x and in the X - Z

plane, snd the y axis normal to the pipe and

upward forming an orthogonalright-hand system

[Fig.9].

The transformation of displacement, force,

or moment vectors from local coordinatesto

global coordinatescan be accomplishedby the

transformationmatrix equation:

-sin Q.

x

K]

o

Y

[1]

Cos Q

x

z

and the reverse transformationby the transpositionof the above,

[H

cos Q

x

Cos Q

Y

sin Q

Y

sin

fax Cos

Qy

10

Y=

-Cos Q

x

sin 9

Y

eos Q

Y

-sin QX sin 0

Y

Y . . . . .+. . . . . .

[2]

z

-sin Q

x

o

cos Q

x

z

where @y is the angle between the Y s.udy sxes, and ~ is the angle between the X axis and the

projection of the x ~es.oq the X-Z plane_____

Forces on Pipeline

The effect of the fluid acting around the outside of the pipe segment can be derived by

consideringa section of pipe immersedin fluid with both ends capped [Fig. I-O]. Archimedes

principle states thgt the total ~orce or buoyancy acts in a vertical directionand is equal to

the weight of displacetlwater, ~yfo~, where & is the area of the end plate, yfo is the

specificweight of the fluid and A is the segment length.

This total buoyancy can be divided


7/14

JTf -071

J. T. POWERS and L. D. FINN

11 15

into three parts, the two end forces Pn.I.and Pn, and the net fluid force per unit length qw,

where qw is obtainedby integratingthe fluid pressure aro~d the circumference.

The summation

of forces in the vertical direction equals the total buoys.ncy:

n

Aoyfo .$ . Pn ~ sin Qti-l - Pn sinQyn +

J

~cos Qyds . . . . . . . . . . . . . .[3]

n-1

and in the horizontaldirectionsequals zero,

n

O = Pn ~ cos Qm ~ cos Qyn ~ - Pn cos Qh cos Qyc -

[

~ cos Q

x

SiZ10 ds. . . . . . .

Y

[4]

n-1

The two end forces can be defined as

Pn

=Aoyfo( H- Yn), . ..o o..... . . . . . . . . . . . . ..- . . . .. [61

P

n-1

=Aoyfo( H- Yn-l ), . . . . . . . . . . . . . . . . . . . . . . . . . . . .. [7]

where H is the total flutd depth and Yn sad Yn-~ are the heights from the bottom to point n and

n-l, respectively.

In an actual, suspendedpipeline the ends are not capped. The vertical and horizontal sums

of the q forces are t~e terms of interest.

Since the deflected shape of the segment is not

known, q cannotbe defined in terms which can be integrated.

Since the above equationsare not

restricted to short segments,

they can be applied to the section suspended between Point 1 and

Point n by substituting1 for n-1 and s for ~, where s is the suspendedlength from the origin.

Expressionsfor the vertical snd horizontal sums [Fxop,Fyop sad Fzop] are derived from Eqs. 3

through 7:

n

Xop =

J

- COS Qx Sin Qy ds = A y

[

-H cos Qn

o fo

1

COS Q~+(H-Yn) COS ~fi COS Qyn . . . [8]

T

L

n.

F

Yop =

J

[

~cos Qyds =Ao~fo

S- HSin QU+(H-Yn)Si.n Om

. . . . . . . . .. [9]

1

-1

-L

n

F

J

op = -

[

si nxfi Oy

ds = A. Yfo -H s~ Qu cos Qn+(H-yn) sfi @wcos fi . . . [IO]

1

These equations show that the total effect of the fluid pressure along the pipe can be repre-

sentedby the weight of water displaced acting verticallythrough the centroidof the suspended

portion minus the capped end forces acting at each end.

If the pipeline is

fille_d_wit_hfluid__Q~_inghe lay operation,the effect of this fluid

acting inside the pipe that extends from the ocean floor [Point1] to Point n can be similarly

derived.

F

Xip

= Ai Yfi

[

cos(3n cosQ~-(H-Yn) COS@mCOS~h .. o.OOO. .o*. [8A1


8/14

11 16

STRESS ANALYSISF OFF~ORE PIPFLINESDURINGINSTALLATION

OTC 10

F

Yip

= Ai yfi

c

-s+ Hsin Q

-( H-Yn)sin Oyn -[9A]

F

Zip

[

Aiyfi Hsfi Q~cos Qu - (H-yn)sin Qfi.cos@yn . . . . . . . . . . . .[1OA]

where Ai is the inside cross-sectionalarea and yfi is the specificweight of the inside fluid.

me weight of a pipe line suspendedbetween Point 1 aud n is Wps, where Wp iS the weight

per unit length of the coated pipe in air.

___Thebuoyancy per unit length cm be defined by

%AoYfo

-Alyfi, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

[11]

and the effectiveunit weight of pipe in water by

we

W

-Wb. . . . . .. . , . . . . . . . . . . . . . . . . . . . . . . . . . .

[12]

P

The forces acting on the pipe suspendedbetween the origin and Point n are shown in Fig. 11.

The effectivetotal suspendedweight of the segment.oflength s is WeS. me componentsof the

capped end forces w~H and ~[H-Yn] that must be subtractedat Point 1 SUd n are defined as:

b

W =

Wb (H - Yn) COS Qfi

COS ~h

b

Yn =

Wb (H - Yn) SillQM

1

.

b

Zzl =

Wb (H - Yn) sin Qfi cos Qvn

b

/

. . . . . . . . . . . . . . . . . . . .

[131

i l =

Wb H COS Q

xl

cos Q

Xl

~b

n=

Wb H sin Q

n

I

b

21 =

Wb H Sin QD COS Qn

J

Other forces acting on the pipe section 1 to n could includebuoy forces

Q . The= forces

would always act verticallyand could be randomly spaced, but for ease of formulationthey will

be assumed to act only at node points.

The current forces per unit length acting on the jth segmentwill be designatedqj. This

force is the drag which acts on a body innnersedin flowing fluid and is expressedby

=1/2PAC V?

j

D

J ~. . . . .. . . . . . . . .. . . . . . . . .

.

[14

where the mass density P is 7fo/g2 A is the area exposed per ~it length [outsidedismeter

DO in

this case], CD is the dimensionlessdrag coefficient~d Vj is the comPonentof velocitYno~al

to the pipe on the jth segment. The assumed horizontal current acts at a arbitrary angle 0 wit

X axes, as shown in Fig. 12.

The assumed constantcurrent velocity acting on the jth segment

[Vj] is first resolved into componentsalong the X and Z axes [Vjcose, Vjsin o], and then

resolved into local coordinatecomponentsnormal to the pipe using Eq. 2.


9/14

-r-r. n

,-i 1 n-l

.T W

.lL -L{

XHJE R.~ and T.. T). 17TNN

I -Lu J-

. ~. .,,

u,w_ - .,., . --- . . . .. .

v

= Vj Cos @ -COS 0

sin QYj)+ Vj sin @ (-sinQ

sin 0

)

yj

Xj

Xj

Yj

)

. . . . . . . . . .

[15]

v C o.g@ -sin Qxj) + Vj sin @ ( Cos Qxj)

V=j

zj

J

The

drag forcesper unit length are obtained from Eq. 14:

%j =

I/2p D CD~j (sign Vyj)

o

1

[16]

. .. *O. .. .

. . .

q

zj =

1/2 p Do CD ~j (Si.@ Vzj)

These unit forces are then resolvedinto componentsalong the global coordinateaxis using

Eq. 1:

qxj = qyj

(-COS Gxj .cJ~Q

~j) + qzj (-sin Oxj)

q~j

= S ( Cos YJ )

1

.

. . . . . . . . . . . . . . .

.

[17]

q~j

= ~j (-sin X,j sin Qyj) + qzj ( Cos Xj )

The global coordinateforces at n are obtainedby applying equilibriumto the free body

diagrsm in Fig. 11.

n-1

ml

=F~+F~-F&-

2

1.qxj

j=l

n

n-1

Yn

= Fm+Fb~-Fb +Wes-

Xn

z

J -

z

1

~qyj . . . . . . . . . . . . .

.[I-81

j = j a

n-1

h

= Fm+F:l-F:n -

Z

g q~j

j =

These force componentsare transformedinto local coordinatecomponents

[Sm, SW>

Szn] using

Eq. 2.

The moment at n resultingfrom the current is obtainedby knowing the current forces and

moments at n-1 [Fig. 12] Wd then applying equilibriumto the n-1 segment:

% = .%-1+ n-l 4 Os Yn-l h Xn-l - %-l 2 b yn-~

4 yn-l

,4/2cos Qyn , sin QX_ ~ +4 qti ~ Q/2sin gti-l

--

?

= %-1 -1-;n-l ~ COS @fi-l SiilQm-l - F~-l ~ s~ gyn-l

1


10/14

n 18

ST@EE-ANKLYSTS OF OFFSHOREPTPELTNT SllTTRTNGNSTALLATION OTc 10;

.

.-.

-.. ..- --.-

),

.$2/2(~j sfi ~-l - zj

Os m-l in %-l

&q

--1

-F&4 cos Qmlsin Qh1 +F&l Aces Qyn&os Qh-l

[191

= ~n-~ -

- 22/2 (qzj

Cos Yn-l

),

%

= ~-~ + ~-~ L in Yn-~ - ~n-~ 2 Os fi-~ Os yn-~

+ .42/2(qzj sin ~-l

h Xn-l + %d Os fi-l)

where the force componentsat n-l resulting from current forces are:

n-2

1 .

I

1

A Xj

j=l

n-2

~~1 .

z

4qYj + . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .[20]

j=l

n-2

~q.l .

Z

A q~j

j=l

The total moment componentsin global coordinatesare obtained from applyingmoment equilibrium

about n [Fig. 11]:

% = ~-yn(Fzl

b,

FL) + Zn (F~ + F~

n-1

c~)+~Qj zn

+Wes zn-zn

-zJ)+M ,

j=l

l =

~+xn ~zl+~~)-zn Fn+F~)+qn ,

=

~1 - Xn ~n + FL) + Yn Fn + FL)

}

. . . . . . . . . . . . .

[21]

.

n-1

x,) +%

g)+ ~Qj(Xn-J

we s (Xn - ~

j=l

These moment componentsare transformedinto local coordinatecomponents [Mn,

Y

, Mzn] using

Eq. 2.

The coordinatesof.the ceriter-ofravity for point n are obtainedfrom t e center of

gravity of n-l:


11/14

Tfl1071

-J. T..PQWERS anl_L. D. FINN __

II 19

x;~ =

[

&

~-~ -A)+ Xn+x

J

4/2]

s

)

....................

[

.[22

~:g . ~::1

1

s-2)+ n+znJ4/2 /s

In Eq.s.

13 through 22 the coordinatesXn, Yn and ~ and the sloPe ~ and @yn are not known

until the forces and moments have been calculated. If the segment length is short, the change in

slope and deflectionof n-relative.ton-l is small. Very little error in the csJ-culatedforces

and moments will result if it is assumed that the slope

}

Yn = Yn-l , snd

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

[23

in = Xl-1

and the coordinates

}

.

.[24

z = Znl

n

4 tiom-l Cos Yn-l

J

After the forces have been obtained,the deflectionand change in slope at Point n can be calcu-

lated and an improved slope and coordinatesof Point n obtained.

This cycle can be repeated

se~eral times for each segment, if necessary.

The effectivetension at Point n is defined as the axial force at n which has an effect on

the moment, shear and deflectionsof the suspendedportion,

en = xn

tWb[H-yn] . . . . . . .:. . . . . . . . . . . . . . . . . . . . . . . .

.[25

1

1

1

1

It is apparentfrom Fig.

11 that the term wb[H-Yn] could have been left out of the force summa-

tions F~, Fyn and FZn, and we

still would

have obtained the value for the shears ~ and Szn,

since this term is first broken into vertical and horizontal componentssad then transformedback

to a single axial component.

The axial componentthus obtainedwould equal the effective tension

defined aboye. Although the pipe is not actually capped at Point n, the capped end axial force

is present at that point. The sxial stress should include this force and hence should be based

on s=

as previously defined.

The uniformly distributedloads normal to the pipe acting on segment n-1 are:

Y =

p-l -

e

Os Yn-l>

)

..............................

[26]

w=

z

q

zn-1

Deflectionsand Rotations

The coordinatesof Point n are determinedfrom the deflectionof that point relative to the

tangents at n-l by imaginingthis sectionas a csatileveredbeam fixed at Point N-1 and loaded

by shear forces Sp and Sznj moments M

and &n, effectivetension Ten) ~iformly distributed

forcesw

f

and Wz, snd buoy force Qn.

~e local coordinatedeflectioncomponentsfor element n-l

as calcu ated by small angle bending formulas are:

r

>-n = l/EI Mzn ~,2/2+ (S

+ Qn COS 0

yn

yn-~ ) 43/3 + T?yIkp

1

j

)

...........

[27]

z

.

l/EI -M

42/2 + Szn

,43/3 +WzA4/8

n

w


12/14

w. -. -- . . . . . . . . . . ..- --

--- - .. ~_ .__.

Similarly,the angle changes are:

m. =

Yn

@I

[

Mznj + (Syn +

~ c Qyn.l)A2/2 + Wy 43/6

Qzn/EI -M

~ + Szn

P

&2/2 +

WZ A

3/6

1 f

.

. . . . . . . . . . .

[28]

The

new coordinatesfor Pointn are:

D

n-1

~

n-1

z

n-1

+

[

Cs h- l CsYn-l

I

h l

i n XI I - lsYn-l

=Os

xi v- li n h- l

Cos%-l-l

sin Xl&l in %-l-l

-i

si n L-l

o

cos Q

Xn

. [29]

11 20

-:C ITREW

ANAT:YSTti m? OliFSHOREPIPELI~S DURING INSTALLATION

OTC 1071

and the new angles are

9

Yn

=Q

ynl+AQ

Yn

[ 1)

in AQ

Xn

= %-l-l

+ arc tan

zn

.

. . . . . . . . . . . . . . . . . . . . . .

[30]

Cos Q

Yn-1


13/14

=

_ PORTION SUPPORTED

g

LIFT-OFF POINT

%

e

*

n

%

SUSPENDED

=

s

z

g

.

Fi g. 1 - Layi ng pi pe by sti ng er- l aybarge method. 3

0

\ 19.8

04812162024 2832364044

ANGLE AT LIFT-OFF POINT DEG)

Fig.

2 - Pipe stress vs lift-off angle in 200 ft.

of water 2o in. 0.0., .625 in. wall thickness,

1.3 empty specific gravity pipe).

200

180

160

140

120

100

80

60

40

20

00 ~ ~ ,2 ,6 Z. 24 28 ~z 36 ~. 44, -

A NGL E A T L IF T- OF F PO IN T ( DE G)

Fig. 3 - Lift-off height vs lift-off ang

200 ft. of water 20 in. O.D., .625

i n,

thickness,

1.3 empty specific gravity p

F1g

100

[

FINITE ELEMENT

------STIFFENED CATENARY

80

- NATURAL CATENARY

,4 K

BEAM COLUMN

60 -

40

---- ~ >_-

_-

~~oK

/

/

r

I

00 816 24

32

40

48

ANGLE

AT LIFT-OFFPOINT DEG)

4 -

Comparison of maximum stress vs lift-off

angl e for various tensions in 250 ft. of water

e in

as predicted by various theDries 24 in. 0.0.

wall

.625 in. wall thickness,

1.2 empty specific

pe).

gravity pipe).

250

20K

1

TENSIO~

d-

20 K 60K

60K

200

, F NITE ELEMENT

~ - -- -- - ST lFFE N6~ CAT EN ARY

/

/

150 -

/

-

NAIURA: CATENARY

,


14/14

FINITE ELEMENT

~

5TIFFENED CATENARY

z

o

- NATURAL CATENARY

g

BEAM COLUMN

g 500 -

~

~

9

0

0

500 1500

HORIZONTAL POSITION FT)

Fig. 7 - Comparison of deflected shapes for 60

kips tension in 1,000 ft. of water, as predicted

by various theoriss 16 in. O.D., .50 in- wall

thickness,

1.2 smpty specific gravity pipe).

Y

K

I

20 ,

I

I

I

1

I

I I

\

\-

\

\

\

10

\

FINITE ELEMENT

.-----sTIFFENED CATENARY

-NATURAL CATENARY

BEAM COLUMN

2000

LOCATION ALONG PIPE FT)

Fig. 8 - Comparison of bending stress diagrams

for 60 kips tension in 1,000 ft. of water, as

predicted by various theories 16 in. 0.0.,

.50 in. wall thickness, 1.2 empty specif i c

gravi ty pi pe).

x

Fig.

10 - Fluid pressure on beam element

/

z

Fi g. 9 - Global- and local coordinates.

nds capped).

I

MY.;

/,

Fig. 11 - Three-dimensional fores diagram

Y

M :n

1/

M ;n

M;n.I

M ~n

M;n-l Fq

Xn-1

4

\

I

/

X

r

/

N

/

~q x,

I

/

Yn-1 ~,

1

/1

I ,/

\

, 1,

z

w

M :n-1

Fig. 12 - Current-induced forces.

otc-1071-ms offshore pipelines

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