THE DANISH CENTER FORAPPLIED MATHEMATICS AND MECHANICS

Søren Christiansen

Frank A. Engelund

Svend Gravesen

Erik Hansen

¡var G. Jonsson

P. Scheel Larsen

Hugo Møltmann

Frithiol L Niordson

Pauli Pedersen

Kristian Refslund

Scientific Council

Laboratory of Applied Mathematical Physics

Institute of Hydrodynamics and Hydraulic Engineering

Structural Research Laboratory

Laboratory of Applied Mathematical Physics

Institute of Hydrodynamics and Hydraulic Engineering

Department of Fluid Mechanics

Structural Research Laboratory

Department of Solid Mechanics

Department of Solid Mechanics

Department of Fluid Mechanics

Secretary

Frithiof I. Niordson, Professor, Ph. D.

Department of Solid Mechanics, Building 404

Technical University of Denmark

2800 Lyngby, Denmark

TC*CHE Uft' '

boratorluSc .- ..'

Archiefe 2, 2628 ' DG!ft

O1 .y35

EQUILIBRIUM OF OFFSHORE CABLES

AND PIPELINES DURING LAYING

by

P. Terndrup Pedersen

DEPART ME NT

OF

OCEAN ENGINEERING

THETECHNICAL UNIVERSITY OFDENP.IARK

A

Cn

Ct

D

EI

F fn' n

f,,g.J)H

Hb.hbH1 ,h.

M

N,n

px r

T

Tb

V

Vb, Vb

V V.i' i

Wb

wt

NOMENCLATURE

Distance between barge deck and water surface

Normal drag coefficient

Tangential drag coefficient

Outer diameter of pipe or cable

Bending stiffness of pipe

Normal drag force per unit length

Tangential drag force per unit length

Auxiliary functions

Water depth

Horizontal force component at the ocean floor

Horizontal force component at upper end ofsuspended length

Suspended length

Bending moment

Axial tension

Components of load per unit length

Arc length

Shear force

Horizontal anchor force

Current velocity

Vertical force component at ocean floor

Vertical force component at upper end ofsuspended length

Buoyancy per unit length

Weight per unit length

X,Y,x,y Rectangular coordinates3 Tangent angle

Mass density of water

EQUILIBRIUM OF OFFSHORE CABLES AND PIPELINES DURING LAYING

P. Terndrup Pedersen

Department of Ocean Engineering

The Technical University of Denmark, Lyngby, Denmark

ABSTRACT - An efficient solutin technique for determination of the staticequilibrium form of a cable or a pipeline suspended between the ocean floorand a laying barge or a stinger is presented. Variations in bending stiff-ness, weight and buoyancy and forces due to ocean current are taken intoaccount. The governing non-linear boundary value problem is transformedinto a non-dimensional form such that the a priori unknown suspended lengthof the pipeline or cable acts as a scaling parameter. The numerical solu-tion is then based on successive integrations. The ajìication of themethod is illustrated by the analysis of equilibrium curves and stressesin pieeiines laid with or without the use of stingers and during abandonand recovery operations. -

1. INTRODUCTION

The object of this paper is to present a technique that

will provide an efficient and direct solution to the static

equilibrium form of a cable or a pipeline suspended between

the ocean floor and a laying barge in a steady ocean current.

With increasing water depth the length of the suspended

pipeline between the pipe laying barge or the lift-off point

from the stinger and the ocean floor becomes greater. This may

cause the stresses in the pipe to become so high that the pipe

buckles or the stresses in the stinger to reach a level where

the stinger breaks. In calm seas these considerations establish

the limits for water depths in which pipes can be layed.

The problem of predicting the suspended geometry and there-

by the stresses of marine pipelines during laying in the ocean

is one of large deflection beam theory, where the length of the

suspended beam is nota priori known. There has been some pre-

vious work dealing with the numerical solution of this problem.

2

Stiffened catenary solutions have been developed [i), [2)

for laying of pipes with relatively small bending stiffness in

extremely deep water. The stiffened catenary solution is ob-

tainod by applying the method of matched asymptotic expansions

irto is based on the assumption that the pipeline takes a shape

which approximates a natural catenary over most of its length

,nd that the influence of the boundary conditions is confined

to small "boundary layers" near the pipe-laying barge and the

ocean floor. Unfortunately, the method is difficult to apply

when the pipe is laid with the use of a stinger, but in those

cases where all the basic assumptions of the theory are satis-

fied the method is attractive because of relatively small

demands onl calculations.

For the analysis of pipes laid in shallower water or of

pipes with larger bending stiffness much more involved finite

element methods [3), finite difference methods [41 - [51, and

methods involving transformation of the non-linear,two-point

boundary value problem into initial-value problems have been

proposed. Common to these methods is that they are based on

successive iterations and that each main iteration step in-

volves a second set of successive iterations for the calcula-

tion of the suspended length, thereby making these methods

relatively expensive as regards computer size and time.

Furthermore, these methods are usually not flexible enough

to handle all the existing types of pipe-laying procedures

without non-trivial modifications.

In this paper a method is presented which provides a

relatively direct solution to cable and pipe-laying problems.

The governing non-linear, two-point boundary value problem is

derived and transformed into a non-dimensional form such that

the a priori unknown suspended length of the pipeline or cableacts as a scaling parameter. The method of solution is then basedon successive integrations. The requirements to computer storage

and time are thereby limited to the same order of magnitude as forthe stiffened catenary calculations. In order to demonstrate

the flexibility of the method, examples are given on the ana-

lysis of the pipe-laying operations, where the pipe is laid

without the use of a stinger or with an articulated stinger,

and with the use of a rigid stinger. Finally pipe abandon!

recovery operations are treated. In the last section of the

paper we demonstrate how cable-laying problems and one-leg

mooring problems can be analysed by the proposed method.

2. LOADING ON THE SUSPENDED PIPE.

We consider a deformed inextensible pipe as shown in

Fig. 2.1. As independent variable in our formulatiqn we

shall employ the arc length s from the point where the

pipe touches the ocean floor. This point will also serve

as origin for the rectangular coordinate system X-Y shown

in Fig. 2.1. The tangent angle to the pipe is denoted O(s).

and we designate the depth of the ocean by H.

1he load on an element of unit length of the suspended

pipe is composed of: the weight wt(s), the buoyancy w0(s)

and,due to a steady ocean current with velocity V(Y) , also

a normal drag force Fe(s) and a tangential drag force Fe(s)

The mass density of the water is given by'

the

gravity by g, and the cross sectional area of the pipe by A.

Y

Fig. 2.1 Loading on the pipe.

The buoyancy load on the pipe due to the water pressure

is determined as follows. Corlsider the segment of length ds

shown in Fig. 2.2. The total buoyancy of the segment with

7open ends equals JgAds

and acts in the Y-direction.

This load has to be corrected

for the lack of pressure at

the ends of the segment. From

Fig. 2.2 it follows that the

resulting buoyancy load w0ds

acts in the direction normal

to the centreline of the pipe

segment and with a magnitude

given byFig. 2.2 Buoyancy on pipe element.

dOW0 = WiCOSO + (H-YY--j

The loading due to the current takes the form

F = p CV2A2v c

where C is the drag coefficient, V the flow velocity, and

Ac a characteristic area. The axial and tangential load per

Unit length can be obtained from (2.2) as

F = pCVVIDsin2O

Ft = 1TJ C VIVIDcOS2OVt

(2.1)

= FcosO - (wü_Fn)sinO(2.4)

p(s) = FsinO + (w0_Fn)coSO -

respectively.

3. GOVERNING EQUATIONS FOR PIPES

In this section we shall set up the governing equations

for the plane, one-dimensional, finite-strain beam theory

which will be used to model the pipe.

As constitutive law for the pipe we will assume a linear

relation between the bending moment M and the curvature

dO/ds. Thus

where EI is the bending stiffness of the pipe.

The moment equilibrium condition for segments of the pipe

(2.2) gives the shear force T(s) as

dOdM-T(s) = ã ds

The shear force T at any section of the pipe can be found

from Fig. 2.1 by equilibrium considerations. We find

d dO(2.3) T(s) - (EI) = _HbsinO(s) + vbcosO(s)

(3.2)

45

where Wb = PgA for O < Y < H and Wb = O for Y > H.M(s) = EIP (3.1)

where D is the diameter of the pipe, and C, C are drag

- Cos8(S)J(5l)dsi + sine(s)J(si)dsi (3.3)

coefficients.where

11.0,V, are the horizontal and vertical force components,

Thus, the resulting horizontal and vertical load inten- respectively, at the support point at the ocean floor. Simi-

sities for the submerged part of the pipe are larly, we find the axial force N(s) at any section of the

6

form

+ coso(s)J P(S1)dS1 - sinO(s)1 p (s )dslxiO JO

and

N(s) = Hbcose(s) + vbsinO(s)

whe re

= Ftcoso(s) + F sinO(s)

7

- siñO(s)J(si)dsi COSO(5)jx(Si)dS1

p(s) = Fsin0(s) - F cosO(s) - (WL_wb)

11b + HwbcosOb Vb + HwbsinOb

and N = N + {H_Y(s)}wb

Equation (3.5) shows that the effect of the buoyancy on the

equilibrium curve of the pipe can be accounted for by intro-

ducing the submerged weight of the pipe. However, it will beseen from Eq. (3.6) that taking care of the buoyancy simply by

introducing the submerged weight results in an apparent axialfor N which equals the real axial force N plus the hydro-static force wb(H_y) . Here, we may note that for the eva-

luation of the buckling strength of a pipe it is the real axialforce N that is of importance, whereas for the determination

of a reference stress for a solid cable or mooring line we will

(3.6)

-2 d deA -(y-) = hbsinø - VbCOSO +

+A{cosOJPdi - sinOJPdi}

and Eg. (3.6) takes the form

(3.7)

n() = hbcosO + vbsinO - A{sinOfPdi + coser d1} (3.8)

0 '0

Neglecting the axial extension of the pipe, the relation between

the dimensionless natural coordinates (O,E) and the dimension-

less rectangular coordinates (x,y) are

dy = sinOd and dx = cosOd (3.9)

The boundary conditions at the ocean floor are

y(0) = x(0) = O (3.10)

O(0) = (3.11)

(de\(3. 12)

The boundary conditions at the upper end of the suspended pipe

depend on the method of operation (for example the type of

stinger used) . But we note, for future use, that the dimen-

= s/L {x, y) = {X, Y)/L A = L/B

The equations (3.3) and (3.4) can also be written in theEIY = ---- x' Pyf =

d dO - wtH= HbsinO(s) - VbcosO(s) and

(s Ishb Vb, n,t} = (Hb, Vb, j:, T}/(wH)

(3.5)

0.where w is a characteristic value of the weight per unit

length w of the pipe. Then Eg. (3.5) takes the form

be concerned with the adjusted axial force which here issuspended pipe as

denoted N

N(s) = HbcosO(s) + VbsinO(s) In order to isolate the unknown suspended length L of

the pipe let us then introduce the following dimensionless

_sinOsJys1dsl - cOsOsJxsidsi (3.4)quantities:

o

and that the vertical component of the tension is given by

i

vi v - A

The non-linear boundary value problem will be solved nu-

meri.cally by successive iterations. The physical background of

of the numerical method is as follows: first the loading on

the pipe (p,p) associated with an arbitrary deflection curve

and is determined; based on this loading a new

deflection curve is found (8. ,x. ,y , and À. } which is3+1 j+1 j+l j+1

then used In the next iteration step for the calculation of

the loading on the pipe. This continues until a certain re-

quired degree of equilibrium is obtained. Mathematically,

the method Is based on a series of formal integrations of

the thfferential equations, together with a determination

of the dimensionless suspended length A , which acts as ascaling parameter.

In all the examples presented in the following the

iteration procedure is stopped when the following inequa-

lity is fulfilled

(1

{(x -x)2

+ (Y31-Y)21d12

jj+i i

O

(3.14) where t is a small number (lOs).

(4.1)

o

Consequently,tiere the non-dimensional forces h.,v. are related to theto the equilibriumapp)ied forces H.,V. bygration we can

h1 = (H1 + (H-Yj)wbcosOj}/(wH) whereand

f1(.1) =

y. = (V. + (H-Yjw sinO.}/(wH) ini i i b i

The differential equations (3.7) and (3.9) and the bound- f3R)

let us assume that

curve (8..x..Y}.

determine the functions

Ir,

; f.(F,) = I

I y 1 I

- pd1 i fU) =

we have an approximation

Then by numerical inte-

f.(); i = i .....6,

X

Y 1

8

9

sionless applied horizontal tension can be found from the

following equilibrium equation

h. = hb - pxl (3.13)

ary conditions of the problem constitute the non-linear bound- O O

(4.2)ary value problem to be solved.

f5() = (f3sinø. + f4cosO.}d1

4. METHOD OF SOLUTION FOR PIPE-LAYING PROBLEMS, and

f6;) =

f5

O

Following the outlined method of solution we get from

Eq. (3.7) ,wlth the notation (4.2)

dO-2 dA.1 -(ï

d) - hbslnO. - vbcosO.

(4.3)+ A

. (f sinO f cosO3 j 4 j

and from Eq. (3.8) we get

n1(E) = hbcosO. + vbsin8.

(4.4)+ À 1{f3cosß. - f4sin81)

10

Fin1ly,the equations (3.13) and (3.14) result in

h1 = hb + À.1f3(l)

y. = y - A. f (1)i b j+l 4

respectively.

A first step in the determination of the new approximation

to the equilibrium curve is a formal integration of Eq. (4.3),

which yields

do,A21(y lf1) - h y - y x + A f + C

b j b j+1 S

The static boundary condition (3.12) expressing the fact

that the bending moment is zero for = O imposes the value

zero for the integration constant C. Now the tangent angle

can be obtained from (4.7) by a second formal integra-

tien resulting in

+ A2 0bA2 0. (1) hbfl - vbf2 + A.1 f6 j+l1+1 3+1

when the kinematic boundary condition (3.11) is introduced.

In order to proceed we have to introduce the boun-

dary conditions at the upper end of the suspended pipe. We

shall here consider three important cases.

a. Pipe-laying without stinger or with an articulated stinger.

Fig. 4.1 Pipe-laying without stinger.

First we shall consider a case where the pipe is layed

without the use of a stinger. We shall assume that at the

upper end of the suspended pipe we have the kinematic bound-(4.5) ary conditions

(4.6)

(4.7)

where(4.8)

f2)g () = O + (0.-0i b i b f2(l)

f(ì)= hi(f1() - f2(1) f2()}

jo

The as yet unk'own suspended length A1 of the pipe

can now be determine. by solving the transcendental equation

obtained from the bourìary condition (4.10) and Eq. (4.15)

0(L) = 0.1

Y(L) = H + A

where A is the distance between the pipe support ori the

barge and the ocean surface. We shall also assume that the

applied horizontal tension H. at the support is known.

Introducing the kinematic boundary condition (4.9) in

(4.8) gives

= 1(0 -0.)AT2 + hbfl(l) + f6(1)A1)/f2(1)b i. j+1

Introducing (4.11) and (4.5) In (4.8) yields

2 3= g1() + A.1 g2() + A1 g3()

g3(E) =f6(F) +(f3(l)f1(1) f6(1)}f(1) f3(i)f1U)

li

(4.11)

(4.12)

(4. 13)

From (3.9) and (3.10) we get

X.j+1 = jo cosO d1i+1

(4.14)

= sinO (4.15)

(4.9)

(4.10)

12

1

sin(g + A2 g + A3 g3dE1 = 1 + ai j+l 2 j+i,o

where a = A/H.

Thus, starting with an arbitrary integrable approximation

to the equilibrium curve the functions g1, g2, and g3 can

be determined from (4.13) and a new approximation to the sus-pended length of the pipe can be found from (4.16). The

improved approximation to the equilibrium functions are here-

after found from (4.12), (4.14), and (4.15).

The sequence of successive iterations may be started with

an arbitrary regular function satisfying the kinematic boundaryconditions. We can, for example, use the deflection curve

corresponding to a natural catenary or a solution of the

linearized Bernoulli-Euler beam equation that satisfies all

the boundary conditions at the ocean floor and the kinematic

boundary conditions at the upper end of the suspended pipe.

These methods of obtaining the first approximation also supplyus with a first estimate of the suspended length.

The effect of having part of the equilibrium curve abovethe surface of the ocean (A > O), or support buoys along thepipe, or, an articulated stinger, is easily taken care ofin the present formulation by introducing a variation in the

distributed buoyancy and/or weight of the pipe.

As an application of the foregoing, Fig. 4.2 shows the

results of the numerical analysis of a pipe-laying procedurewhere the pipe is laid without the use of a stinger. The waterdepth H is 50 m, the pipe leaves the pipe-laying barge 2 m

above the water surface at an angle equal to 200. The hori-zontal tension H. applied at the barge is 2.200l05N.The uniform bending stiffness EI of the pipe is 2.256'109Nm2,

the buoyancy per unit length in water Wb is l.6l4l04N/m and

the weight per unit length w is l.843l04N/m.

Starting with a deflection curve corresponding to the

solution of the linearized beam equation, where the effectof the applied horizontal tension h1 is neglected, the so-

lution presented in Fig. 4.2 is obtained in 3 iteratIon steps.

(4.16)

A.2so

Lt 2 fl$ t0 'b.'-, 'to' 41.

a t &t4 to' t41.

4, L200 tO'No

Fig. 4.2 Results of numerical analysis of pipe-laying procedurewithout the use of a stinger.

b. Pipe-laying using a rigid stinger.

We will now consider pipelaying with the use of a rigid

stinger with a fixed curvature i/R as shown in Fig. 4.3. Let us

Fig. 4.3 pipe-laying using a rigid stinger

13

assume that the applied horizontal tension is H. at the upper

end of the suspended pipe (the lift-off point from the stinger)

The tangent angle of the stinger at the point where the stinger

is hinged to the barge is denoted O. The angle e will nor-

mally be a non-linear func-

tion of the position of

the lift-off point given

by Y and and the

magnitude of the concen-

trated force T1 per-

pendicular to the stinger

axis at the lift-off point.

Due to the constant curva-

ture of the stinger the

force T1 equals the shear

force in the pipe Just be-

low the lift-off point.The function = 0(Y.,T.) can be determined when the geometry

and the weight distribution of the stinger are known. See Fig.4.4

Besides the static boundary condition expressing the fact

that the horizontal force is H at the lift-off point the bending

moment M1 is also given

= -EI/R (4.18)

Finally, kinematic considerations give us the relationship

I-1-Y1 +A

cosO. = cose -i u R

Let us now construct the iteration algorithm using these

boundary conditions. Applying the boundary condition (4.18) to

the equation (4.7) results in

Fig. 4.4 The stinger supported partof the pipe.

X- (1)

_____ - Ihb Vb y(l) r (l)A141

(l)

f5(l)

(4.19)

(4.20)

where r R/H. The equations (4.20) and (4.5) yield

y.(l) f5(l)f (1)) + )Vb x1(l) 1h1 + A11{(1)3 1+1 r y(l) (4.21)

substituting (4.21) and (4.5) into (4.8) gives

8. () = O + A. g () + 1g2() +b j+l iJ+i

whereY

=r x.(1)

-g () = h.(f () f2 i 1 x.(1)

and1f5(1) y.(l)

g3() =f6() -f3(l)f1() 2jx.(l) _f3(l)3(1)}J i

The unknown dimensionsless suspended length A141 is determined

from the following transcendental equation obtained from the

boundary condition (4.19) together with (4.15)

cose. (1) = cosO - À(l + a - À1y1(l)}3+1 u r

Thus, starting with an arbitrary integrable approximation

to the equilibrium curve one step in the iteration procedure

is to determine the functions g1, g2, and g3 from (4.23) and

from (4.24). The improved approximation to the equi-

librium functions are hereafter given by (4.22), (4.14) and

(4.15). The iteration scheme is stopped when the convergence

criterion (4.1) is fulfilled.

The method outlined is,of course,only valid when the

stinger is so long that the calculated lift-off point is on the

stinger. If this is not the case,a slightly different itera-

tion scheme is called for.

An example of the numerical analysis of a pipe-laying

procedure using a stinger with fixed curvature Is shown In

Fig. 4.5. The stinger radius is assumed to be 300 m and,

in this example, the stinger is assumed to be rigidly connected

to the pipe-laying barge. The water depth and the pipe data

are assumed to be the sanie as in the previous example.

(4.24)

[415

(4.22)

(4.23)

16

El 226 i0 N,..2

i soi io'

8 101o 10' 1W...

ii, 2.200 lO N

-I

Fig. 4.5 Results of numerical analysis of pipe-laying procedurewith the use of a rigid stinger.

c. Pipe abandon/recovery operations.

Pipe abandon and recovery operations may be modelled as

Fig- 4.6 pipe abandon/recoverY operation

shown in Fig. 4.6. These operations can be performed in a

number of different ways. As an example we will assume

that the wire passes over the stinger rollers and that the

horizontal anchor force transmitted to the pipe Tb andthe V-coordinate of the pipe end Vi are known, whereas

the wire tension T is considered as a dependent vari-able.

Taking into account tohe water pressure on the

lid which is normally welded onto the pipe end during

these operations,the boundary conditions for the upper end

of the pipe take the form

Y(L) = Y.i

M(L) = O

FI. = T - w (H-Y.)cosh(L)i b b i

Let us again construct one step in the iteration algo-

rithm using these boundary conditions. From F.gs. (4.5) and

(4.7) and the boundary conditions (4.25) - (4.26) we get

y,(l) f5(l)

i j+13)J +Vb {hk. f (1)x.(l) 3+1 X (1)

J j

Substituting into Eg. (4.8) results in

e. () ej+l = b 1g2() + À1g3()

where g2() and g3(r,) are given by (4.23). The dimen-

sionless suspended length is determined from theboundary conditions (4.25).

Fig. 4.7 shows the results of a numerical analysis of

a pipe abandon or recovery operation. The same pipe data as

in the two previous numerical examples have beell assumed.

The position of the upper end of the suspended length is

specified as 25 m above the ocean floor and the horizontal

anchor force as 105N. The necessary wire tension is found

(4.25)

(4.26)

(4.27)

17

100 200 i,. 300 X

(4 .28)

(4.29)

18

to be 2.148105N.

dN= wsinO - Ft

Fig. 4.7 ResultS of numerical analysis of abandon/recovery operation

Cables are characterized by a negligible bending stiff-

ness (EI O) . The iteration method presented in section 4

for determination of equilibrium curves for pipes fails if

we let the bending stiffness approach zero. Therefore, in

this section we shall present a similar iteration method for

the analysis of equilibrium curves and elastic restoring forces

of cables as that used, for example, for mooring systems, taking

into account drag forces on the cable and variations in cable

weight per unit length.

and that equilibrium in the direction normal to the neutral

axis of the cable is obtained when

F + (w_wb)cosO - wb(H_Y)ds n

Introducing the adjusted normal force

= N + wb(H'SY)

the governing equations (5.j), (5.2) take the simpler form

AdN

(wt_wb) sinO * Ft

=j"ds

ç

(wt_wb)cosø + F)

The boundary conditions for these equations express the

fact that, at the upper end of the Suspended cable,

Y(L) H+A5. STATIC EQUILIBRIUM OF CABLES.

and = H./cosO(L)

where H. is the applied tension, while at the ocean floorwe have

X(0) O and Y(0) = O (5.6a-b)

If we consider a problem where part of the cable is lying

on the ocean floor we have the additional boundary condition

e(o) - (5.7)

In non-dimensional forni,the governing equations (5.3) -

(5.4) take the form

(5.1) pt() = {(wt_wb)sino -AO

and n(F;) = N/wtH

(5.2)

(5.3)

(5.4)

(5. Sa-b)

19

Governing equations.dn Ap() dO A

(5 .8a-b)

It follows from Fig. 2.1 that force equilibrium in the wheretangential direction for an element of the cable is expressed

= f (w-w)cos8 +by

V

El 223S *0PI.I I3 Io' *4 Witt I.V.n

05I SII. IO' *V4.. o u. I0'N

*4 . I0N05..

.o-

- 20

Numerical solution.

An iteration procedure similar to that outlined in

section 4 will now be applied for the numerical solution of

the non-linear boundary value problem given by (5.5) - (5.8).

Assume that an initial approximation to equilibrium curve

is given so that the iteration algorithm can be

started by a formal integration of Eq. (5.8e)

nU) = 1j+l ptdl + C1Jo

It follows from the boundary condition (5.5b) that the inte-

gration constant is determined by

ih.

J- -ÀC1= cosO j+l' ti

i JO

Formal integration of Eg. (5.8b) results in

= À11[ d+n 1

o

Long Cables. If we are dealing with a mooring cable problem

where the length of the cable is so large that part of the

cable lies on the ocean floor,then the integration constant

in (5.10) is given by the inclination of the ocean floor. Fig.5.l.

Fig. 5.1 Mooring-cable geometry, long cable.

(5.9)

(5. 10)

Using the boundary conditions (5.5a) and (5.6b) it is seen that

in this case the unknown,unsupported length of the cable can be

determined from the transcendental equation

IlA sinO d1 = i + aj+l j+l

Thus, the static equilibrium of the cable can be determined

by successive iterations, where each iteration step involves the

calculation of n and A41 using Eqs. (5.9) - (5.11).

The sequence of iterations can be stopped when the convergencecriterion (4.1) is fulfilled.

Shorter Cables. When dealing with a mooring system where thecable is of shorter length so that the point of initial contactis the anchor point, the inclination of the cable at the anchorpoint

0b is usually unknown. However, in this case we will as-sume that the length L of the cable is known. This leaves us

with sufficient information to construct one step in theiteration procedure. See Fig. 5.2.

(5.11)

21

X7,

FIg. 5.2 Mooring-cable geometry, Short cable.

22

Dropping the subscripts on the known dimensionless length A

Eq. (5.10) can be written as

e. () = n. () + Oj+1 +i b

where

n () = A'j+1J0

n

From (5.13) the unknown angle 8b can be determined as

8 = Arcsin( l+a

)Arctan

A/C+C ()h

Thus, in order to determine the static equilibrium positionof a short mooring cable,each iteration step must consist of acalculation of n (5.9) and 8.4k (5.10), where the angle

is determined from (5.14).

(5.12)

(5.14)

6. CONCLUSION

The method of successive integrations presented for the

determination of equilibrium forms and stresses for submarine

pipelines during laying possesses several advantages over

other available methods. The principal advantage is the ex-

tremely modest requirements to computer storage and computer

time. Since, in principle, the method only involves integration

of known functions,the me'thod is well suited for programming

on shipboard computers for control of the actual pipe-laying

procedure. The numerical calculations for the three pipe-

laying examples presented in this paper were done as one job

on an IHM 370/165 computer. The total computer time is 0.88

sec. and the storage requirement,including object code and

array area for this general program,is 32 K8. Another ad-

vantage of the method is its flexibility. For example the

effect of variations of pipeline bending stiffness due to

variations in the coating, current variations with depth,

and auxiliary support buoys can easily be accounted for.

The primary limitation of the present method for the

analysis of equilibrium forms of pipes is failure to converge

for some extreme input values such as very deep water or small

bending stiffness. However, since the present method and the

stiffened catenary method has a common convergence region it

is possible to shift to stiffened catenary solutions if the

pipe data are such that the method of successive integrations

fails to converge. If it is necessary to take ocean current

and variations in pipe data into account the stiffened cate-

nary solution can be based on the numerical method presented

for the analysis of cables.

23

Introducing

in

where

C1 =

(5.12) into the boundary condition

CicosOb + C2slflOb =

Ilsinn. dF1 and C2

I 3+1

(5.4a)

Ilcosri

i j+1û

results

(5. 13)

24

REF ERENC ES

il) PLUNKETT, R.: Static bending stresses in catenaries

and drill strings, Trans. ASME, Journal of Engineering

for Industry, Vol. 89, No. 1, pp 31-36, 1967

(2) DIXON, D.A. and RUTLEDGE, D.R.: Stiffened catenary cal-

culations In pipeline laying problems, Trans. ASME,

Journal of Engineering for Industry, Vol. 90, No. 1,

pp 153-160, 1968

[3] POWERS, J.T. and FINN. L.D.: Stress analysis of off-

shore pipelines during installation, Offshore Technology

Conference, Houston, Texas, Paper No. OTC 1071, May, 1969

[4) PALMER, A.C., HUTCHINSON, G. and ELL.S, J.W.: Configuration

of submarine pipelines during laying operations, ASME,

Paper No. 73-WA/OCT-4, 1973

[5] DAREING, D.W., NEATHERY, R.F.: Marine pipeline analysis

based on Newtons's method with an arctic application,

Trans. ASME, Journal of Engineering for Industry, Vol. 92,

No. 4, pp 827-833, 1970

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