otc-1071-ms offshore pipelines
TRANSCRIPT
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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 deep- water 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
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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
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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
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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
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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
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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
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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
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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.
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8/10/2019 OTC-1071-MS Offshore Pipelines
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
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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:
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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
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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
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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
,
-
8/10/2019 OTC-1071-MS Offshore Pipelines
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.